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) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([56, 0, -75, 0, -16, 0, -1]), R([0, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![56, 0, -75, 0, -16, 0, -1], R![0, 1, 0, 1]);
sage: X = HyperellipticCurve(R([224, 0, -299, 0, -62, 0, -3]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(672\) | \(=\) | \( 2^{5} \cdot 3 \cdot 7 \) | magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(672,2),R![1, 1]>*])); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(172032\) | \(=\) | \( 2^{13} \cdot 3 \cdot 7 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(16916\) | \(=\) | \( 2^{2} \cdot 4229 \) |
\( I_4 \) | \(=\) | \(151117825\) | \(=\) | \( 5^{2} \cdot 6044713 \) |
\( I_6 \) | \(=\) | \(232872423961\) | \(=\) | \( 397 \cdot 586580413 \) |
\( I_{10} \) | \(=\) | \(-21504\) | \(=\) | \( - 2^{10} \cdot 3 \cdot 7 \) |
\( J_2 \) | \(=\) | \(16916\) | \(=\) | \( 2^{2} \cdot 4229 \) |
\( J_4 \) | \(=\) | \(-88822256\) | \(=\) | \( - 2^{4} \cdot 103 \cdot 53897 \) |
\( J_6 \) | \(=\) | \(277597802496\) | \(=\) | \( 2^{12} \cdot 3 \cdot 7 \cdot 3227281 \) |
\( J_8 \) | \(=\) | \(-798387183476800\) | \(=\) | \( - 2^{6} \cdot 5^{2} \cdot 107 \cdot 991 \cdot 4705829 \) |
\( J_{10} \) | \(=\) | \(-172032\) | \(=\) | \( - 2^{13} \cdot 3 \cdot 7 \) |
\( g_1 \) | \(=\) | \(-1352659309173012149/168\) | ||
\( g_2 \) | \(=\) | \(419870026410625699/168\) | ||
\( g_3 \) | \(=\) | \(-461744933079368\) |
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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Rational points
This curve has no rational points. |
This curve has no rational points. |
This curve has no rational points. |
magma: []; // minimal model
magma: []; // simplified model
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));
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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 | Cluster picture |
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\(2\) | \(5\) | \(13\) | \(2\) | \(1 + T\) | |
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 2 T + 3 T^{2} )\) | |
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 7 T^{2} )\) |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $\mathrm{SU}(2)\times\mathrm{SU}(2)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) |
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 \(\Q\).