CoCalc Shared Filesmain_project / semistable_reduction / picard / Picard_example1.sagewsOpen in CoCalc with one click!
Authors: Jeroen Sijsling, Stefan Wewers
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Description: Computation of conductor exponent for some Picard curves

Computing the conductor of Picard curves

1. Example

We consider the Picard curve Y:  y3=f(x)=x4+14x2+72x41. Y:\; y^3 = f(x) = x^4 + 14x^2 + 72x -41. The discriminant of ff is Δ(f)=2103456\Delta(f)=-2^{10}3^45^6.

Our goal is to compute the conductor NY=2f23f35f5N_Y = 2^{f_2}3^{f_3}5^{f_5}.

K = QQ R.<x> = K[] f = x^4 +14*x^2 + 72*x -41 f.discriminant().factor()
-1 * 2^10 * 3^4 * 5^6

We check that p=5p=5 is an exceptional prime. It follows that f5=0f_5=0. (Note that v5(Δ(f))=6v_5(\Delta(f))=6.)

f.change_ring(GF(5)).factor() (f(-58+5^3*x)/5^6).change_ring(GF(5))
(x + 3)^2 * (x^2 + 4*x + 1) 3*x^2 + 4*x + 2
vK = pAdicValuation(K, 5) vK.mac_lane_approximants(f)
[[ Gauss valuation induced by 5-adic valuation, v(x + 58) = 3, v(x^2 + 491*x + 87614) = 7 ], [ Gauss valuation induced by 5-adic valuation, v(x^2 + 134*x + 356) = 4 ]]

Now we want to compute f2f_2.

First we compute the splitting field of ff. It is a totally ramified extension of Q\QQ of degree 88. The configuration of the roots of ff is as follows: they all lie in a disk of radius 3/43/4, and inside in two smaller disks of radius 11.

f.change_ring(GF(2)).factor() vK = pAdicValuation(K, 2) vK.montes_factorization(f) # shows that f is irreducible over QQ_2 L0.<alpha0> = K.extension(f) L0.<alpha>= L0.galois_closure() L0 vL0 = vK.extension(L0) vL0(2) fL0 = f.change_ring(L0) a = fL0.roots(multiplicities=None) [vL0(a[0]-a[i]) for i in range(4)]
(x + 1)^4 x^4 + 14*x^2 + 72*x - 41 Number Field in alpha with defining polynomial x^8 + 40*x^6 + 2400*x^4 + 81472*x^2 + 1000000 8 [+Infinity, 6, 6, 8]

We compute the MacLane valuations v1,v2v_1,v_2 on K[x]K[x] corresponding to the configuration of roots.

v0 = GaussValuation(R, vK) v = v0.mac_lane_step(f)[0] v1 = v.extension(f, 3) v1 v2 = v.extension(f, 7/2) v2
[ Gauss valuation induced by 2-adic valuation, v(x + 1) = 1/4, v(x^4 + 14*x^2 + 72*x - 41) = 3 ] [ Gauss valuation induced by 2-adic valuation, v(x + 1) = 1/4, v(x^4 + 14*x^2 + 72*x - 41) = 7/2 ]

From the configuration of roots of ff it is clear that the stable reduction of YY has three components of genus 11 (Case (c) in Theorem 4.1). The extension L/KL/K over which YY aquires semistable reduction is a (tamely ramified) extension of L0L_0 of degree 33.

v0 = GaussValuation(fL0.parent(), vL0) v0(f((x-a[0])/(a[1]-a[0]))) v0(f((x-a[0])/(a[3]-a[0]))) # since this is not divisible by 3, we need L/L_0 to be ramified
-24 -32

We compute the jumps in the filtration of higher ramification groups of the extension LL. It turns out that these are 0,3,9,150,3,9,15. Here Γ3\Gamma_3 is a dihedral group of order 88, Γ9\Gamma_9 is of order 44, and Γ15\Gamma_{15} is the center (of order 22).

We also compute the subfields of LL corresponding to the jumps.

load("../../padic_extensions/padic_extensions.sage") vL0 = padic_splitting_field(vK, f) L0 = vL0.domain() L0 P = L0.polynomial() P = P(P.parent().gen()^3) P
Number Field in pi8 with defining polynomial x^8 + 20*x^7 + 8*x^6 + 72*x^5 + 70*x^4 + 48*x^3 + 76*x^2 + 16*x + 82 x^24 + 20*x^21 + 8*x^18 + 72*x^15 + 70*x^12 + 48*x^9 + 76*x^6 + 16*x^3 + 82
load("../../padic_extensions/padic_extensions.sage") L.<pi> = K.extension(P) vL = vK.extension(L) filt = padic_ramification_filtration(vL, compute_subfields=True) filt
[(15, 2, pi12-adic valuation), (9, 4, pi6-adic valuation), (3, 8, pi3-adic valuation), (0, 24, 2-adic valuation)]

For m=0,,15m=0,\ldots,15, we have to compute the genus of the curve Yˉm:=Yˉ/Γm\bar{Y}_m:=\bar{Y}/\Gamma_m. We do this by extending the valuations v1,v2v_1,v_2 from K(x)K(x) to the function field Lm(Y)=Lm(x,y)L_m(Y)=L_m(x,y) and computing the genera of the residue fields of these extensions.

load("../../MacLane_valuations/maclane.sage") def reduction_component(v, vLm, f): p = vLm.residue_field().characteristic() Lm = vLm.domain() FXL.<z> = FunctionField(Lm) h = FXL._ring(v.phi()) V = maclane_basefield_extensions(v, vLm, h) S.<T> = FXL[] FYL.<y> = FXL.extension(T^3-FXL(f)) for w in V: w = RationalFunctionFieldValuation(FXL, w) w = w.extension(FYL) print " Reduction component:" print " ", w.residue_field() print " ", w print " ground field: ", Lm print " ------------"
V = [v1, v2] print "Computing stable reduction Yb:" for v in V: print "over ", v reduction_component(v, vL, f) for i in range(len(filt)): m = filt[i][0]+1 vLm = filt[i][2] print "Computing Y_m for m=%s:"%m for v in V: print "over ", v reduction_component(v, vLm, f)
Computing stable reduction Yb: over [ Gauss valuation induced by 2-adic valuation, v(x + 1) = 1/4, v(x^4 + 14*x^2 + 72*x - 41) = 3 ] Reduction component: Function field in u2 defined by u2^3 + z^4 + z^2 Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi-adic valuation, v(z + pi^9 + pi^6 + 1) = 18 ], v(y) = 24, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] ground field: Number Field in pi with defining polynomial x^24 + 20*x^21 + 8*x^18 + 72*x^15 + 70*x^12 + 48*x^9 + 76*x^6 + 16*x^3 + 82 ------------ over [ Gauss valuation induced by 2-adic valuation, v(x + 1) = 1/4, v(x^4 + 14*x^2 + 72*x - 41) = 7/2 ] Reduction component: Function field in u2 defined by u2^3 + z^2 + z Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi-adic valuation, v(z + pi^21 + pi^18 + pi^9 + pi^6 + 1) = 24 ], v(y) = 28, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] ground field: Number Field in pi with defining polynomial x^24 + 20*x^21 + 8*x^18 + 72*x^15 + 70*x^12 + 48*x^9 + 76*x^6 + 16*x^3 + 82 ------------ Reduction component: Function field in u2 defined by u2^3 + z^2 + z Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi-adic valuation, v(z + pi^21 + pi^9 + pi^6 + 1) = 24 ], v(y) = 28, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] ground field: Number Field in pi with defining polynomial x^24 + 20*x^21 + 8*x^18 + 72*x^15 + 70*x^12 + 48*x^9 + 76*x^6 + 16*x^3 + 82 ------------ Computing Y_m for m=16: over [ Gauss valuation induced by 2-adic valuation, v(x + 1) = 1/4, v(x^4 + 14*x^2 + 72*x - 41) = 3 ] Reduction component: Function field in u2 defined by u2^3 + z^2 + z Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi12-adic valuation, v(z + pi12^3 + 1) = 9/2, v(z^2 + (-4*pi12^9 - 6*pi12^6 - 2*pi12^3)*z + 7*pi12^9 + 15*pi12^6 + 12*pi12^3 + 7) = 18 ], v(y) = 12, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] ground field: Number Field in pi12 with defining polynomial x^12 + 4*x^9 + 6*x^6 + 4*x^3 + 2 ------------ over [ Gauss valuation induced by 2-adic valuation, v(x + 1) = 1/4, v(x^4 + 14*x^2 + 72*x - 41) = 7/2 ] Reduction component: Function field in u2 defined by u2^3 + z + 1 Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi12-adic valuation, v(z + pi12^3 + 1) = 9/2, v(z^2 + (6*pi12^9 + 14*pi12^6 + 12*pi12^3 + 8)*z - 23*pi12^9 - 57*pi12^6 - 38*pi12^3 - 25) = 24 ], v(y) = 14, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] ground field: Number Field in pi12 with defining polynomial x^12 + 4*x^9 + 6*x^6 + 4*x^3 + 2 ------------ Reduction component: Function field in u2 defined by u2^3 + z Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi12-adic valuation, v(z + pi12^3 + 1) = 9/2, v(z^2 + (6*pi12^9 + 14*pi12^6 + 12*pi12^3 + 8)*z + 31*pi12^9 + 77*pi12^6 + 54*pi12^3 + 35) = 24 ], v(y) = 14, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] ground field: Number Field in pi12 with defining polynomial x^12 + 4*x^9 + 6*x^6 + 4*x^3 + 2 ------------ Computing Y_m for m=10: over [ Gauss valuation induced by 2-adic valuation, v(x + 1) = 1/4, v(x^4 + 14*x^2 + 72*x - 41) = 3 ] Reduction component: Function field in u2 defined by u2^3 + z Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi6-adic valuation, v(z + 1) = 3/2, v(z^2 + (pi6^3 + 2)*z + 2*pi6^3 + 1) = 21/4, v(z^4 + 14*z^2 + 72*z - 41) = 18 ], v(y) = 6, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] ground field: Number Field in pi6 with defining polynomial x^6 + 2*x^3 + 2 ------------ over [ Gauss valuation induced by 2-adic valuation, v(x + 1) = 1/4, v(x^4 + 14*x^2 + 72*x - 41) = 7/2 ] Reduction component: Function field in u2 defined by u2^3 + z Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi6-adic valuation, v(z + 1) = 3/2, v(z^2 + (pi6^3 + 2)*z + 2*pi6^3 + 1) = 21/4, v(z^4 + 14*z^2 + 72*z - 41) = 21 ], v(y) = 7, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] ground field: Number Field in pi6 with defining polynomial x^6 + 2*x^3 + 2 ------------ Computing Y_m for m=4: over [ Gauss valuation induced by 2-adic valuation, v(x + 1) = 1/4, v(x^4 + 14*x^2 + 72*x - 41) = 3 ] Reduction component: Function field in u2 defined by u2^3 + z Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi3-adic valuation, v(z + 1) = 3/4, v(z^4 + 14*z^2 + 72*z - 41) = 9 ], v(y) = 3, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] ground field: Number Field in pi3 with defining polynomial x^3 + 2 ------------ over [ Gauss valuation induced by 2-adic valuation, v(x + 1) = 1/4, v(x^4 + 14*x^2 + 72*x - 41) = 7/2 ] Reduction component: Function field in u2 defined by u2^3 + z Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi3-adic valuation, v(z + 1) = 3/4, v(z^4 + 14*z^2 + 72*z - 41) = 21/2 ], v(y) = 7/2, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] ground field: Number Field in pi3 with defining polynomial x^3 + 2 ------------ Computing Y_m for m=1: over [ Gauss valuation induced by 2-adic valuation, v(x + 1) = 1/4, v(x^4 + 14*x^2 + 72*x - 41) = 3 ] Reduction component: Function field in u2 defined by u2^3 + z Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by 2-adic valuation, v(z + 1) = 1/4, v(z^4 + 14*z^2 + 72*z - 41) = 3 ], v(y) = 1, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] ground field: Rational Field ------------ over [ Gauss valuation induced by 2-adic valuation, v(x + 1) = 1/4, v(x^4 + 14*x^2 + 72*x - 41) = 7/2 ] Reduction component: Rational function field in z over Finite Field of size 2 Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by 2-adic valuation, v(z + 1) = 1/4, v(z^4 + 14*z^2 + 72*z - 41) = 7/2 ], v(y) = 7/6, v(y^3 - z^4 - 14*z^2 - 72*z + 41) = +Infinity ] ground field: Rational Field ------------

We see that g(Yˉm)=0g(\bar{Y}_m)=0 for m=0,,9m=0,\ldots,9, =1=1 for m=10,,15m=10,\ldots,15 and =3=3 for m16m\geq 16. It follows that f2=m=0ΓmΓ0(2gy2gYˉm)=6(1+313+516)+4(6112)=19. f_2 = \sum_{m=0}^\infty \frac{|\Gamma_m|}{|\Gamma_0|}(2g_y-2g_{\bar{Y}_m}) = 6(1+3\cdot\frac{1}{3}+5\cdot\frac{1}{6})+4(6\cdot\frac{1}{12}) = 19.

Now we have to compute f3f_3.

p=3 v_K = pAdicValuation(K,p) load("../../semistable_reduction/superp.sage")
Y = Superp(f, v_K, p) Y.FY
Function field in y defined by y^3 - x^4 + 3*x^3 + 24*x^2 + x
Y.compute_semistable_reduction()
L0 = Rational Field L1 = Number Field in pi2 with defining polynomial x^2 + 3 Inertial reduction component: Function field in u2 defined by x*u2^3 + u2 + 2 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 1, v(x + 8) = 5/4 ] -------------- Reduction component: Function field in u2 defined by u2^3 + 2*u2 + 2*z^2 constant field: Finite Field of size 3 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 1, v(x + 8) = 5/4 ] splitting field: Number Field in pi4 with defining polynomial x^4 + 3 ------------ L0 = Rational Field L1 = Number Field in pi3 with defining polynomial x^3 + 2181*x^2 + 12*x + 2181 Inertial reduction component: Rational function field in x over Finite Field of size 3 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 1 ] -------------- Reduction component: Function field in u2 defined by u2^3 + 2*z^4 + 2*z^3 + 2*z^2 constant field: Finite Field of size 3 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 1 ] splitting field: Number Field in pi3 with defining polynomial x^3 + 2181*x^2 + 12*x + 2181 ------------ L0 = Rational Field L1 = Number Field in pi3 with defining polynomial x^3 + 2181*x^2 + 12*x + 2181 Inertial reduction component: Rational function field in x over Finite Field of size 3 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 5/4 ] -------------- Reduction component: Function field in u2 defined by u2^3 + 2*u2 + 2*z^2 constant field: Finite Field of size 3 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 5/4 ] splitting field: Number Field in pi12 with defining polynomial x^12 + 2181*x^8 + 12*x^4 + 2181 ------------ L0 = Number Field in pi6 with defining polynomial x^6 + 2184 L1 = Number Field in pi6 with defining polynomial x^6 + 2184 Inertial reduction component: Function field in u2 defined by 2*x^2*u2^3 + 2 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 1, v(x + 5) = 5/3, v(x^3 + 15*x^2 + 75*x + 368) = 21/4 ] -------------- Reduction component: Function field in u2 defined by u2^3 + u2 + 2*z^2 constant field: Finite Field of size 3 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 1, v(x + 5) = 5/3, v(x^3 + 15*x^2 + 75*x + 368) = 21/4 ] splitting field: Number Field in pi12 with defining polynomial x^12 + 2184 ------------

We see that the stable reduction of YY is of type (c), i.e. with three tail components of genus one. One of these components has a splitting field which is a totally ramified Galois extension of Q3\mathbb{Q}_3 of degree 1212. The two other components form an orbit under the Galois group of F9/F3\mathbb{F}_9/\mathbb{F}_3. A splitting field is a Galois extension of degree 2424 of Q3\mathbb{Q}_3 with ramification index 1212.

Now we compute the quotients Wm=W/ΓmW_m=W/\Gamma_m of these three components.

for c in Y.components: c.compute_ramification_filtration_reduction()
components over [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 1, v(x + 8) = 5/4 ] the splitting field is Number Field in pi4 with defining polynomial x^4 + 3 [(0, 4, 3-adic valuation)] The ramification breaks are: [1] Computing Y_m for m=0 and |Gm|=4: intermediate field: Rational Field with ramification index 1 Reduction component: Function field in u2 defined by z*u2^3 + u2 + 2 Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by 3-adic valuation, v(z + 8) = 5/4 ], v(y + 2*z + 19) = 3/2, v(y^3 - z^4 + 3*z^3 + 24*z^2 + z) = +Infinity ] components over [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 1 ] the splitting field is Number Field in pi3 with defining polynomial x^3 + 2181*x^2 + 12*x + 2181 [(1/2, 3, 3-adic valuation)] The ramification breaks are: [3/2] Computing Y_m for m=1/2 and |Gm|=3: intermediate field: Rational Field with ramification index 1 Reduction component: Rational function field in z over Finite Field of size 3 Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by 3-adic valuation, v(z + 2) = 1 ], v(y + 2*z + 10) = 4/3, v(y^3 - z^4 + 3*z^3 + 24*z^2 + z) = +Infinity ] components over [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 5/4 ] the splitting field is Number Field in pi12 with defining polynomial x^12 + 2181*x^8 + 12*x^4 + 2181 [(2, 3, pi4-adic valuation), (0, 12, 3-adic valuation)] The ramification breaks are: [3, 1] Computing Y_m for m=2 and |Gm|=3: intermediate field: Number Field in pi4 with defining polynomial x^4 + 3 with ramification index 4 Reduction component: Rational function field in z over Finite Field of size 3 Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi4-adic valuation, v(z + 2) = 5 ], v(y + 2*z + 1) = 16/3, v(y^3 - z^4 + 3*z^3 + 24*z^2 + z) = +Infinity ] Computing Y_m for m=0 and |Gm|=12: intermediate field: Rational Field with ramification index 1 Reduction component: Rational function field in z over Finite Field of size 3 Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by 3-adic valuation, v(z + 2) = 5/4 ], v(y + 2*z + 10) = 4/3, v(y^3 - z^4 + 3*z^3 + 24*z^2 + z) = +Infinity ] components over [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 1, v(x + 5) = 5/3, v(x^3 + 15*x^2 + 75*x + 368) = 21/4 ] the splitting field is Number Field in pi12 with defining polynomial x^12 + 2184 [(6, 3, pi4-adic valuation), (0, 12, 3-adic valuation)] The ramification breaks are: [7, 1] Computing Y_m for m=6 and |Gm|=3: intermediate field: Number Field in pi4 with defining polynomial x^4 + 1419 with ramification index 4 Reduction component: Function field in u2 defined by 2*u2^3 + 2*z Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi4-adic valuation, v(z - 1417) = 20/3, v(z^3 + 15*z^2 + 75*z + 368) = 21 ], v(y + 1/2013561*z^2 - 2834/2013561*z + 2007889/2013561) = 17/3, v(y^3 - z^4 + 3*z^3 + 24*z^2 + z) = +Infinity ] Computing Y_m for m=0 and |Gm|=12: intermediate field: Rational Field with ramification index 1 Reduction component: Function field in u2 defined by 2*z^2*u2^3 + 2 Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by 3-adic valuation, v(z + 5) = 5/3, v(z^3 + 15*z^2 + 75*z + 368) = 21/4 ], v(y + 1/9*z^2 + 10/9*z + 25/9) = 17/12, v(y^3 - z^4 + 3*z^3 + 24*z^2 + z) = +Infinity ]

The splitting field of the first component has ramification breaks 1,31,3, with Γ0=12|\Gamma_0|=12 and Γ1=3|\Gamma_1|=3. We have g(W3)=0g(W_3)=0, hence the contribution to the conductor is 2(1+214)=3. 2(1+2\cdot\frac{1}{4}) = 3. The splitting field of the each of the other two components has ramification breaks 1,71,7, with Γ0=12|\Gamma_0|=12 and Γ6=3|\Gamma_6|=3. So the contribution of one of these components is 2(1+614)=5. 2(1+6\cdot \frac{1}{4}) = 5. So the conductor exponents at p=3p=3 is f3=3+25=13. f_3 = 3 + 2\cdot 5 = 13.

All in all, the conductor of YY is NY=219313=835884417024. N_Y = 2^{19}3^{13} = 835884417024.

2^19*3^13
835884417024

2. Example

A similar example is the curve Y:;y3=x424x276x+24. Y:; y^3 = x^4 - 24x^2 - 76x +24. Here Δ(f)=283556\Delta(f) = 2^83^55^6.

f = x^4-24*x^2-76*x+24 f.discriminant().factor()
-1 * 2^8 * 3^5 * 5^6

Let us first see what happens for p=2p=2.

f.change_ring(GF(2)).factor() p = 2 vK = pAdicValuation(K, 2) V = vK.mac_lane_approximants(f) V[0].mac_lane_step(f) V[1].mac_lane_step(f)
x^4 [[ Gauss valuation induced by 2-adic valuation, v(x + 2) = 3 ]] [[ Gauss valuation induced by 2-adic valuation, v(x) = 2/3, v(x^3 + 4) = 7/3 ]]
L0.<alpha> = f.splitting_field() L0 vL0 = vK.extension(L0) vL0.residue_field() vL0(2) fL0 = f.change_ring(L0) a = fL0.roots(multiplicities=None) [vL0(a[0]-a[i]) for i in range(4)]
Number Field in alpha with defining polynomial x^6 + 15*x^5 + 96*x^4 + 311*x^3 + 492*x^2 + 339*x + 247 Finite Field in u1 of size 2^2 3 [+Infinity, 2, 2, 2]
vL = padic_sufficiently_ramified_extension(vL0, 9) vL v0 = GaussValuation(R, vK) v1 = v0.extension(x, 2/3) reduction_component(v1, vL, f)
pi18-adic valuation Reduction component: Function field in u2 defined by u2^3 + z^4 + z Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi18-adic valuation, v(z) = 6 ], v(y) = 8, v(y^3 - z^4 + 24*z^2 + 76*z - 24) = +Infinity ] ground field: Number Field in pi18 with defining polynomial x^18 + 20*x^15 + 28*x^12 + 76*x^9 + 104*x^6 + 112*x^3 + 68 ------------

We see that YY has potentially good reduction over a tamely ramified extension of Q2\mathbb{Q}_2 (of degree 1818). However, Yˉ0=Yˉ/Γ0\bar{Y}_0=\bar{Y}/\Gamma_0 has genus 00. It follows that f2=6. f_2 = 6.

Now we look at p=3p=3.

p=3 v_K = pAdicValuation(K,p) load("../../semistable_reduction/superp.sage") Y = Superp(f, v_K, p) Y.FY
Function field in y defined by y^3 - x^4 + 24*x^2 + 76*x - 24
Y.compute_semistable_reduction()
Inertial reduction component: Function field in u2 defined by x^2*u2^3 + 1 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 1/3, v(x^3 + 6*x^2 + 12*x + 14) = 5/3, v(x^3 + 9*x^2 + 24*x + 26) = 2, v(x^3 + 9*x^2 + 24*x + 44) = 19/9, v(x^9 + 27*x^8 + 315*x^7 + 2157*x^6 + 9936*x^5 + 32580*x^4 + 76656*x^3 + 128304*x^2 + 140850*x + 88100) = 20/3, v(x^9 + 27*x^8 + 315*x^7 + 2157*x^6 + 9936*x^5 + 32580*x^4 + 76656*x^3 + 129033*x^2 + 143766*x + 91016) = 27/4 ] --------------

Unfortunately, the computation of the splitting field leads to an error...

Example 3:

One more example: Y:  y3=f(x)=x43x324x2x. Y:\; y^3 = f(x) = x^4-3x^3-24x^2-x. We have Δ(f)=310\Delta(f) = 3^{10}, so we only have to look at p=3p=3.

f = x^4-3*x^3-24*x^2-x f.discriminant().factor()
3^10
load("../../semistable_reduction/superp.sage") Y = Superp(f, v_K, p) Y.FY Y.compute_semistable_reduction()
Function field in y defined by y^3 - x^4 + 3*x^3 + 24*x^2 + x Inertial reduction component: Function field in u2 defined by x*u2^3 + u2 + 2 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 1, v(x + 8) = 5/4 ] -------------- Inertial reduction component: Rational function field in x over Finite Field of size 3 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 1 ] -------------- Inertial reduction component: Rational function field in x over Finite Field of size 3 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 5/4 ] -------------- Inertial reduction component: Function field in u2 defined by 2*x^2*u2^3 + 2 [ Gauss valuation induced by 3-adic valuation, v(x + 2) = 1, v(x + 5) = 5/3, v(x^3 + 15*x^2 + 75*x + 368) = 21/4 ] --------------

We see that the stable reduction Yˉ\bar{Y} of YY consists of 44 components, three of genus 11 and one of genus 00 (type (c)).

The first component W1W_1 of genus 11 has a tamely ramified splitting field and its inertial reduction has genus 00. So g(W0)=0g(W_0)=0 and g(Wm)=1g(W_m)=1 for m1m\geq 1. The contribution of W1W_1 to the conductor exponent is therefore 22.

To compute the contribution of the second component W2W_2 fo genus 11, we analyse the ramificaton filtration of its splitting field.

f01f02b5-0b5c-4fa7-b987-9c43995a7f09
c = Y.components[2] v = c.v vL, w = c.reduction[0] vL.domain() w.residue_field()
Error in lines 1-1 Traceback (most recent call last): File "/usr/local/lib/python2.7/dist-packages/smc_sagews/sage_server.py", line 995, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> IndexError: list index out of range
filt = padic_ramification_filtration(vL, compute_subfields=True) filt
[(2, 3, pi4-adic valuation), (0, 12, 3-adic valuation)]
vK2 = filt[0][2] vK2.domain() reduction_component(v, vK2, f)
Number Field in pi4 with defining polynomial x^4 + 3 Reduction component: Rational function field in z over Finite Field of size 3 Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi4-adic valuation, v(z + 2) = 5 ], v(y + 2*z + 1) = 16/3, v(y^3 - z^4 + 3*z^3 + 24*z^2 + z) = +Infinity ] ground field: Number Field in pi4 with defining polynomial x^4 + 3 ------------

We see that Γ0=12|\Gamma_0|=12, Γm=3|\Gamma_m|=3 for m=1,2m=1,2 and Γm=1\Gamma_m=1 for m3m\geq 3. Also, g(W2/Γ2)=0g(W_2/\Gamma_2)=0 and g(W2/Γ3)=1g(W_2/\Gamma_3)=1. It follows that the contribution is 2(1+214)=3. 2(1 + 2\frac{1}{4}) = 3.

c = Y.components[3] v = c.v vL, w = c.reduction[0] vL.domain() w.residue_field() filt = padic_ramification_filtration(vL, compute_subfields=True) filt
Number Field in pi12 with defining polynomial x^12 + 2184 Function field in u2 defined by u2^3 + u2 + 2*z^2 [(6, 3, pi4-adic valuation), (0, 12, 3-adic valuation)]
vK2 = filt[0][2] vK2.domain() reduction_component(v, vK2, f)
Number Field in pi4 with defining polynomial x^4 + 1419 Reduction component: Function field in u2 defined by 2*u2^3 + 2*z Valuation on rational function field induced by [ Gauss valuation induced by Valuation on rational function field induced by [ Gauss valuation induced by pi4-adic valuation, v(z - 1417) = 20/3, v(z^3 + 15*z^2 + 75*z + 368) = 21 ], v(y + 1/2013561*z^2 - 2834/2013561*z + 2007889/2013561) = 17/3, v(y^3 - z^4 + 3*z^3 + 24*z^2 + z) = +Infinity ] ground field: Number Field in pi4 with defining polynomial x^4 + 1419 ------------

For the splitting field of the last component W3W_3 we have Γ0=12|\Gamma_0|=12, Γm=3|\Gamma_m|=3 for m=1,,6m=1,\ldots,6 and Γm=1\Gamma_m=1 for m7m\geq 7. Also, g(W3/Γm)=0g(W_3/\Gamma_m)=0 for m=0,,6m=0,\ldots,6. Hence the contribution of W3W_3 is 2(1+614)=5. 2(1 + 6\cdot\frac{1}{4}) = 5.
Altogether, we see that the conductor exponent is f3=2+3+5=10. f_3 = 2 + 3 + 5 = 10.