Minimal equation
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]))
$y^2 + (x^3 + x)y = -x^6 - 16x^4 - 75x^2 + 56$
Invariants
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magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(672,2),R![1, 1]>*])); Factorization($1);
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| \( N \) | = | \( 672 \) | = | \( 2^{5} \cdot 3 \cdot 7 \) | |
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magma: Discriminant(C); Factorization(Integers()!$1);
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| \( \Delta \) | = | \(172032\) | = | \( 2^{13} \cdot 3 \cdot 7 \) | |
Igusa-Clebsch invariants
magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
Igusa invariants
magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
G2 invariants
magma: G2Invariants(C);
| \( I_2 \) | = | \(-135328\) | = | \( -1 \cdot 2^{5} \cdot 4229 \) |
| \( I_4 \) | = | \(9671540800\) | = | \( 2^{6} \cdot 5^{2} \cdot 6044713 \) |
| \( I_6 \) | = | \(-119230681068032\) | = | \( -1 \cdot 2^{9} \cdot 397 \cdot 586580413 \) |
| \( I_{10} \) | = | \(704643072\) | = | \( 2^{25} \cdot 3 \cdot 7 \) |
| \( J_2 \) | = | \(-16916\) | = | \( -1 \cdot 2^{2} \cdot 4229 \) |
| \( J_4 \) | = | \(-88822256\) | = | \( -1 \cdot 2^{4} \cdot 103 \cdot 53897 \) |
| \( J_6 \) | = | \(-277597802496\) | = | \( -1 \cdot 2^{12} \cdot 3 \cdot 7 \cdot 3227281 \) |
| \( J_8 \) | = | \(-798387183476800\) | = | \( -1 \cdot 2^{6} \cdot 5^{2} \cdot 107 \cdot 991 \cdot 4705829 \) |
| \( J_{10} \) | = | \(172032\) | = | \( 2^{13} \cdot 3 \cdot 7 \) |
| \( g_1 \) | = | \(-1352659309173012149/168\) | ||
| \( g_2 \) | = | \(419870026410625699/168\) | ||
| \( g_3 \) | = | \(-461744933079368\) |
Automorphism group
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magma: AutomorphismGroup(C); IdentifyGroup($1);
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| \(\mathrm{Aut}(X)\) | \(\simeq\) | \(V_4 \) | (GAP id : [4,2]) | ||
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magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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| \(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | \(V_4 \) | (GAP id : [4,2]) | ||
Rational points
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);
This curve is locally solvable except over $\Q_{2}$.
magma: [];
There are no rational points.
magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
Number of rational Weierstrass points: \(0\)
Invariants of the Jacobian:
Analytic rank: \(0\)
magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);
2-Selmer rank: \(2\)
magma: HasSquareSha(Jacobian(C));
Order of Ш*: twice a square
Regulator: 1.0
Real period: 1.1133493018149438309175533043
Tamagawa numbers: 2 (p = 2), 1 (p = 3), 1 (p = 7)
magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);
Torsion: \(\Z/{4}\Z\)
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $G_{3,3}$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) |
Decomposition
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 \(\GL_2\)-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 \(\Q\).