Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x)y = -3x^6 - 13x^5 + 4x^4 + 51x^3 + 4x^2 - 13x - 3$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2)y = -3x^6 - 13x^5z + 4x^4z^2 + 51x^3z^3 + 4x^2z^4 - 13xz^5 - 3z^6$ | (dehomogenize, simplify) |
$y^2 = -12x^6 - 52x^5 + 17x^4 + 206x^3 + 17x^2 - 52x - 12$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-3, -13, 4, 51, 4, -13, -3]), R([0, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-3, -13, 4, 51, 4, -13, -3], R![0, 1, 1]);
sage: X = HyperellipticCurve(R([-12, -52, 17, 206, 17, -52, -12]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(644\) | \(=\) | \( 2^{2} \cdot 7 \cdot 23 \) | magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(644,2),R![1, 2, 1]>*])); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(659456\) | \(=\) | \( 2^{12} \cdot 7 \cdot 23 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(161796\) | \(=\) | \( 2^{2} \cdot 3 \cdot 97 \cdot 139 \) |
\( I_4 \) | \(=\) | \(1070662305\) | \(=\) | \( 3 \cdot 5 \cdot 23 \cdot 1487 \cdot 2087 \) |
\( I_6 \) | \(=\) | \(46065265919409\) | \(=\) | \( 3 \cdot 11 \cdot 1395917149073 \) |
\( I_{10} \) | \(=\) | \(84410368\) | \(=\) | \( 2^{19} \cdot 7 \cdot 23 \) |
\( J_2 \) | \(=\) | \(40449\) | \(=\) | \( 3 \cdot 97 \cdot 139 \) |
\( J_4 \) | \(=\) | \(23560804\) | \(=\) | \( 2^{2} \cdot 199 \cdot 29599 \) |
\( J_6 \) | \(=\) | \(14638854160\) | \(=\) | \( 2^{4} \cdot 5 \cdot 7 \cdot 23 \cdot 1136557 \) |
\( J_8 \) | \(=\) | \(9253881697856\) | \(=\) | \( 2^{6} \cdot 54101 \cdot 2672629 \) |
\( J_{10} \) | \(=\) | \(659456\) | \(=\) | \( 2^{12} \cdot 7 \cdot 23 \) |
\( g_1 \) | \(=\) | \(108277681088425330677249/659456\) | ||
\( g_2 \) | \(=\) | \(389810454818831018649/164864\) | ||
\( g_3 \) | \(=\) | \(9297727292338785/256\) |
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
All points: \((-3 : -3 : 1),\, (-1 : 3 : 3)\)
magma: [C![-3,-3,1],C![-1,3,3]]; // minimal model
magma: [C![-3,0,1],C![-1,0,3]]; // simplified model
Number of rational Weierstrass points: \(2\)
magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
This curve is locally solvable everywhere.
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: \(\Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-3 : -3 : 1) + (-1 : 3 : 3) - D_\infty\) | \((x + 3z) (3x + z)\) | \(=\) | \(0,\) | \(6y\) | \(=\) | \(7xz^2 + 3z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-3 : -3 : 1) + (-1 : 3 : 3) - D_\infty\) | \((x + 3z) (3x + z)\) | \(=\) | \(0,\) | \(6y\) | \(=\) | \(7xz^2 + 3z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \((x + 3z) (3x + z)\) | \(=\) | \(0,\) | \(6y\) | \(=\) | \(x^2z + 15xz^2 + 6z^3\) | \(0\) | \(2\) |
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 1 \) |
Real period: | \( 0.872984 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 0.218246 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(2\) | \(12\) | \(1\) | \(( 1 + T )^{2}\) | |
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 4 T + 7 T^{2} )\) | |
\(23\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 23 T^{2} )\) |
Galois representations
For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.
For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.
Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
---|---|---|
\(2\) | 2.90.3 | yes |
\(3\) | 3.720.5 | no |
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 curve isogeny classes:
Elliptic curve isogeny class 46.a
Elliptic curve isogeny class 14.a
magma: HeuristicDecompositionFactors(C);
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 \(\Q\).
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);