︠2bfa5fb1-984a-4236-8c08-12bd2c3d2d21︠ ︠f45b73b4-6300-4531-aa3d-1de9ccc253f5︠ x = QQ['x'].0 R = genus2reduction(x^3 - 2*x^2 - 2*x + 1, -5*x^5) R.conductor ︡bad76692-a0c6-440f-a357-b554d1f2bd5d︡{"stdout":"1416875\n"}︡ ︠8875b426-00ae-490a-8a69-b722558031b0︠ genus2reduction? ︡02da8a77-8157-40e5-907f-5f192ab8b429︡{"code":{"source":"File: /projects/sage/sage-6.7/src/sage/misc/lazy_import.pyx\nSignature : genus2reduction()\nDocstring :\nConductor and Reduction Types for Genus 2 Curves.\n\nUse \"R = genus2reduction(Q, P)\" to obtain reduction information\nabout the Jacobian of the projective smooth curve defined by y^2 +\nQ(x)y = P(x). Type \"R?\" for further documentation and a description\nof how to interpret the local reduction data.\n\nEXAMPLES:\n\n sage: x = QQ['x'].0\n sage: R = genus2reduction(x^3 - 2*x^2 - 2*x + 1, -5*x^5)\n sage: R.conductor\n 1416875\n sage: factor(R.conductor)\n 5^4 * 2267\n\nThis means that only the odd part of the conductor is known.\n\n sage: R.prime_to_2_conductor_only\n True\n\nThe discriminant is always minimal away from 2, but possibly not at\n2.\n\n sage: factor(R.minimal_disc)\n 2^3 * 5^5 * 2267\n\nPrinting R summarizes all the information computed about the curve\n\n sage: R\n Reduction data about this proper smooth genus 2 curve:\n y^2 + (x^3 - 2*x^2 - 2*x + 1)*y = -5*x^5\n A Minimal Equation (away from 2):\n y^2 = x^6 - 240*x^4 - 2550*x^3 - 11400*x^2 - 24100*x - 19855\n Minimal Discriminant (away from 2): 56675000\n Conductor (away from 2): 1416875\n Local Data:\n p=2\n (potential) stable reduction: (II), j=1\n p=5\n (potential) stable reduction: (I)\n reduction at p: [V] page 156, (3), f=4\n p=2267\n (potential) stable reduction: (II), j=432\n reduction at p: [I{1-0-0}] page 170, (1), f=1\n\nHere are some examples of curves with modular Jacobians:\n\n sage: R = genus2reduction(x^3 + x + 1, -2*x^5 - 3*x^2 + 2*x - 2)\n sage: factor(R.conductor)\n 23^2\n sage: factor(genus2reduction(x^3 + 1, -x^5 - 3*x^4 + 2*x^2 + 2*x - 2).conductor)\n 29^2\n sage: factor(genus2reduction(x^3 + x + 1, x^5 + 2*x^4 + 2*x^3 + x^2 - x - 1).conductor)\n 5^6\n\nEXAMPLE:\n\n sage: genus2reduction(0, x^6 + 3*x^3 + 63)\n Reduction data about this proper smooth genus 2 curve:\n y^2 = x^6 + 3*x^3 + 63\n A Minimal Equation (away from 2):\n y^2 = x^6 + 3*x^3 + 63\n Minimal Discriminant (away from 2): 10628388316852992\n Conductor (away from 2): 2893401\n Local Data:\n p=2\n (potential) stable reduction: (V), j1+j2=0, j1*j2=0\n p=3\n (potential) stable reduction: (I)\n reduction at p: [III{9}] page 184, (3)^2, f=10\n p=7\n (potential) stable reduction: (V), j1+j2=0, j1*j2=0\n reduction at p: [I{0}-II-0] page 159, (1), f=2\n\nIn the above example, Liu remarks that in fact at p=2, the\nreduction is [II-II-0] page 163, (1), f=8. So the conductor of J(C)\nis actually 2 * 2893401=5786802.\n\nA MODULAR CURVE:\n\nConsider the modular curve X_1(13) defined by an equation\n\n y^2 + (x^3-x^2-1)y = x^2 - x.\n\nWe have:\n\n sage: genus2reduction(x^3-x^2-1, x^2 - x)\n Reduction data about this proper smooth genus 2 curve:\n y^2 + (x^3 - x^2 - 1)*y = x^2 - x\n A Minimal Equation (away from 2):\n y^2 = x^6 + 58*x^5 + 1401*x^4 + 18038*x^3 + 130546*x^2 + 503516*x + 808561\n Minimal Discriminant (away from 2): 169\n Conductor: 169\n Local Data:\n p=13\n (potential) stable reduction: (V), j1+j2=0, j1*j2=0\n reduction at p: [I{0}-II-0] page 159, (1), f=2\n\nSo the curve has good reduction at 2. At p=13, the stable reduction\nis union of two elliptic curves, and both of them have 0 as modular\ninvariant. The reduction at 13 is of type [I_0-II-0] (see Namikawa-\nUeno, page 159). It is an elliptic curve with a cusp. The group of\nconnected components of the Neron model of J(C) is trivial, and the\nexponent of the conductor of J(C) at 13 is f=2. The conductor of\nJ(C) is 13^2. (Note: It is a theorem of Conrad-Edixhoven-Stein that\nthe component group of J(X_1(p)) is trivial for all primes p.)","mode":"text/x-rst","lineno":-1,"filename":null}}︡ ︠8f4b9906-5d3c-4a82-8336-9cbd0d7e4fc5︠