︠f45b73b4-6300-4531-aa3d-1de9ccc253f5︠ x = QQ['x'].0 R = genus2reduction(x^3 - 2*x^2 - 2*x + 1, -5*x^5) R.conductor

1416875

genus2reduction?

File: /projects/sage/sage-6.7/src/sage/misc/lazy_import.pyx Signature : genus2reduction() Docstring : Conductor and Reduction Types for Genus 2 Curves. Use "R = genus2reduction(Q, P)" to obtain reduction information about the Jacobian of the projective smooth curve defined by y^2 + Q(x)y = P(x). Type "R?" for further documentation and a description of how to interpret the local reduction data. EXAMPLES: sage: x = QQ['x'].0 sage: R = genus2reduction(x^3 - 2*x^2 - 2*x + 1, -5*x^5) sage: R.conductor 1416875 sage: factor(R.conductor) 5^4 * 2267 This means that only the odd part of the conductor is known. sage: R.prime_to_2_conductor_only True The discriminant is always minimal away from 2, but possibly not at 2. sage: factor(R.minimal_disc) 2^3 * 5^5 * 2267 Printing R summarizes all the information computed about the curve sage: R Reduction data about this proper smooth genus 2 curve: y^2 + (x^3 - 2*x^2 - 2*x + 1)*y = -5*x^5 A Minimal Equation (away from 2): y^2 = x^6 - 240*x^4 - 2550*x^3 - 11400*x^2 - 24100*x - 19855 Minimal Discriminant (away from 2): 56675000 Conductor (away from 2): 1416875 Local Data: p=2 (potential) stable reduction: (II), j=1 p=5 (potential) stable reduction: (I) reduction at p: [V] page 156, (3), f=4 p=2267 (potential) stable reduction: (II), j=432 reduction at p: [I{1-0-0}] page 170, (1), f=1 Here are some examples of curves with modular Jacobians: sage: R = genus2reduction(x^3 + x + 1, -2*x^5 - 3*x^2 + 2*x - 2) sage: factor(R.conductor) 23^2 sage: factor(genus2reduction(x^3 + 1, -x^5 - 3*x^4 + 2*x^2 + 2*x - 2).conductor) 29^2 sage: factor(genus2reduction(x^3 + x + 1, x^5 + 2*x^4 + 2*x^3 + x^2 - x - 1).conductor) 5^6 EXAMPLE: sage: genus2reduction(0, x^6 + 3*x^3 + 63) Reduction data about this proper smooth genus 2 curve: y^2 = x^6 + 3*x^3 + 63 A Minimal Equation (away from 2): y^2 = x^6 + 3*x^3 + 63 Minimal Discriminant (away from 2): 10628388316852992 Conductor (away from 2): 2893401 Local Data: p=2 (potential) stable reduction: (V), j1+j2=0, j1*j2=0 p=3 (potential) stable reduction: (I) reduction at p: [III{9}] page 184, (3)^2, f=10 p=7 (potential) stable reduction: (V), j1+j2=0, j1*j2=0 reduction at p: [I{0}-II-0] page 159, (1), f=2 In the above example, Liu remarks that in fact at p=2, the reduction is [II-II-0] page 163, (1), f=8. So the conductor of J(C) is actually 2 * 2893401=5786802. A MODULAR CURVE: Consider the modular curve X_1(13) defined by an equation y^2 + (x^3-x^2-1)y = x^2 - x. We have: sage: genus2reduction(x^3-x^2-1, x^2 - x) Reduction data about this proper smooth genus 2 curve: y^2 + (x^3 - x^2 - 1)*y = x^2 - x A Minimal Equation (away from 2): y^2 = x^6 + 58*x^5 + 1401*x^4 + 18038*x^3 + 130546*x^2 + 503516*x + 808561 Minimal Discriminant (away from 2): 169 Conductor: 169 Local Data: p=13 (potential) stable reduction: (V), j1+j2=0, j1*j2=0 reduction at p: [I{0}-II-0] page 159, (1), f=2 So the curve has good reduction at 2. At p=13, the stable reduction is union of two elliptic curves, and both of them have 0 as modular invariant. The reduction at 13 is of type [I_0-II-0] (see Namikawa- Ueno, page 159). It is an elliptic curve with a cusp. The group of connected components of the Neron model of J(C) is trivial, and the exponent of the conductor of J(C) at 13 is f=2. The conductor of J(C) is 13^2. (Note: It is a theorem of Conrad-Edixhoven-Stein that the component group of J(X_1(p)) is trivial for all primes p.)