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

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

-1 * 2^10 * 3^4 * 5^6

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

(x + 3)^2 * (x^2 + 4*x + 1)
3*x^2 + 4*x + 2

[[ 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 $f_2$.

First we compute the splitting field of $f$. It is a totally ramified extension of $\QQ$ of degree $8$. The configuration of the roots of $f$ is as follows: they all lie in a disk of radius $3/4$, and inside in two smaller disks of radius $1$.

(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 $v_1,v_2$ on $K[x]$ corresponding to the configuration of roots.

[ 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 $f$ it is clear that the stable reduction of $Y$ has three components of genus $1$ (Case (c) in Theorem 4.1). The extension $L/K$ over which $Y$ aquires semistable reduction is a (tamely ramified) extension of $L_0$ of degree $3$.

-24
-32

We compute the jumps in the filtration of higher ramification groups of the extension $L$. It turns out that these are $0,3,9,15$. Here $\Gamma_3$ is a dihedral group of order $8$, $\Gamma_9$ is of order $4$, and $\Gamma_{15}$ is the center (of order $2$).

We also compute the subfields of $L$ corresponding to the jumps.

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

[(15, 2, pi12-adic valuation), (9, 4, pi6-adic valuation), (3, 8, pi3-adic valuation), (0, 24, 2-adic valuation)]

For $m=0,\ldots,15$, we have to compute the genus of the curve $\bar{Y}_m:=\bar{Y}/\Gamma_m$. We do this by extending the valuations $v_1,v_2$ from $K(x)$ to the function field $L_m(Y)=L_m(x,y)$ and computing the genera of the residue fields of these extensions.

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

Now we have to compute $f_3$.

Function field in y defined by y^3 - x^4 + 3*x^3 + 24*x^2 + x

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 $Y$ 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 $\mathbb{Q}_3$ of degree $12$. The two other components form an orbit under the Galois group of $\mathbb{F}_9/\mathbb{F}_3$. A splitting field is a Galois extension of degree $24$ of $\mathbb{Q}_3$ with ramification index $12$.

Now we compute the quotients $W_m=W/\Gamma_m$ of these three components.

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

All in all, the conductor of $Y$ is $N_Y = 2^{19}3^{13} = 835884417024.$

835884417024

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

-1 * 2^8 * 3^5 * 5^6

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

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 ]]

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]

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 $Y$ has potentially good reduction over a tamely ramified extension of $\mathbb{Q}_2$ (of degree $18$). However, $\bar{Y}_0=\bar{Y}/\Gamma_0$ has genus $0$. It follows that $f_2 = 6.$

Now we look at $p=3$.

Function field in y defined by y^3 - x^4 + 24*x^2 + 76*x - 24

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...

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

3^10

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 $\bar{Y}$ of $Y$ consists of $4$ components, three of genus $1$ and one of genus $0$ (*type (c)*).

The first component $W_1$ of genus $1$ has a tamely ramified splitting field and its inertial reduction has genus $0$. So $g(W_0)=0$ and $g(W_m)=1$ for $m\geq 1$. The contribution of $W_1$ to the conductor exponent is therefore $2$.

To compute the contribution of the second component $W_2$ fo genus $1$, we analyse the ramificaton filtration of its splitting field.

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[(2, 3, pi4-adic valuation), (0, 12, 3-adic valuation)]

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 $|\Gamma_0|=12$, $|\Gamma_m|=3$ for $m=1,2$ and $\Gamma_m=1$ for $m\geq 3$. Also, $g(W_2/\Gamma_2)=0$ and $g(W_2/\Gamma_3)=1$. It follows that the contribution is $2(1 + 2\frac{1}{4}) = 3.$

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)]

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 $W_3$ we have $|\Gamma_0|=12$, $|\Gamma_m|=3$ for $m=1,\ldots,6$ and $\Gamma_m=1$ for $m\geq 7$.
Also, $g(W_3/\Gamma_m)=0$ for $m=0,\ldots,6$. Hence the contribution of $W_3$ is
$2(1 + 6\cdot\frac{1}{4}) = 5.$

Altogether, we see that the conductor exponent is
$f_3 = 2 + 3 + 5 = 10.$