Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + x)y = -x^6 - 16x^4 - 75x^2 + 56$ | (homogenize, simplify) |
| $y^2 + (x^3 + xz^2)y = -x^6 - 16x^4z^2 - 75x^2z^4 + 56z^6$ | (dehomogenize, simplify) |
| $y^2 = -3x^6 - 62x^4 - 299x^2 + 224$ | (minimize, homogenize) |
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![56, 0, -75, 0, -16, 0, -1], R![0, 1, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([56, 0, -75, 0, -16, 0, -1]), R([0, 1, 0, 1]));
magma: X,pi:= SimplifiedModel(C);
sage: X = HyperellipticCurve(R([224, 0, -299, 0, -62, 0, -3]))
Invariants
| Conductor: | \( N \) | = | \(672\) | = | \( 2^{5} \cdot 3 \cdot 7 \) | magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(672,2),R![1, 1]>*])); Factorization($1);
|
| Discriminant: | \( \Delta \) | = | \(172032\) | = | \( 2^{13} \cdot 3 \cdot 7 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | = | \(-135328\) | = | \( - 2^{5} \cdot 4229 \) |
| \( I_4 \) | = | \(9671540800\) | = | \( 2^{6} \cdot 5^{2} \cdot 6044713 \) |
| \( I_6 \) | = | \(-119230681068032\) | = | \( - 2^{9} \cdot 397 \cdot 586580413 \) |
| \( I_{10} \) | = | \(704643072\) | = | \( 2^{25} \cdot 3 \cdot 7 \) |
| \( J_2 \) | = | \(-16916\) | = | \( - 2^{2} \cdot 4229 \) |
| \( J_4 \) | = | \(-88822256\) | = | \( - 2^{4} \cdot 103 \cdot 53897 \) |
| \( J_6 \) | = | \(-277597802496\) | = | \( - 2^{12} \cdot 3 \cdot 7 \cdot 3227281 \) |
| \( J_8 \) | = | \(-798387183476800\) | = | \( - 2^{6} \cdot 5^{2} \cdot 107 \cdot 991 \cdot 4705829 \) |
| \( J_{10} \) | = | \(172032\) | = | \( 2^{13} \cdot 3 \cdot 7 \) |
| \( g_1 \) | = | \(-1352659309173012149/168\) | ||
| \( g_2 \) | = | \(419870026410625699/168\) | ||
| \( g_3 \) | = | \(-461744933079368\) |
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
Automorphism group
| \(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
|
| \(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
This curve has no rational points.
magma: [];
Number of rational Weierstrass points: \(0\)
magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
This curve is locally solvable except over $\Q_{2}$.
magma: f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);
Mordell-Weil group of the Jacobian:
Group structure: \(\Z/{4}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - D_\infty\) | \(x^2 + 11z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(5xz^2 + z^3\) | \(0\) | \(4\) |
2-torsion field: 8.0.199148544.2
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 1 \) |
| Real period: | \( 1.113349 \) |
| Tamagawa product: | \( 2 \) |
| Torsion order: | \( 4 \) |
| Leading coefficient: | \( 0.278337 \) |
| Analytic order of Ш: | \( 2 \) (rounded) |
| Order of Ш: | twice a square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor |
|---|---|---|---|---|
| \(2\) | \(13\) | \(5\) | \(2\) | \(1 + T\) |
| \(3\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 2 T + 3 T^{2} )\) |
| \(7\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 7 T^{2} )\) |
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $G_{3,3}$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over \(\Q\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 48.a1
Elliptic curve 14.a1
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
| \(\End (J_{})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
| \(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
| \(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).