Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x^2)y = x^4 + x^3 - 3$ | (homogenize, simplify) |
$y^2 + (x^3 + x^2z)y = x^4z^2 + x^3z^3 - 3z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 + 2x^5 + 5x^4 + 4x^3 - 12$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-3, 0, 0, 1, 1]), R([0, 0, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-3, 0, 0, 1, 1], R![0, 0, 1, 1]);
sage: X = HyperellipticCurve(R([-12, 0, 0, 4, 5, 2, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(20736\) | \(=\) | \( 2^{8} \cdot 3^{4} \) | magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(20736,2),R![1, 1]>*])); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(995328\) | \(=\) | \( 2^{12} \cdot 3^{5} \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(372\) | \(=\) | \( 2^{2} \cdot 3 \cdot 31 \) |
\( I_4 \) | \(=\) | \(2898\) | \(=\) | \( 2 \cdot 3^{2} \cdot 7 \cdot 23 \) |
\( I_6 \) | \(=\) | \(384570\) | \(=\) | \( 2 \cdot 3^{2} \cdot 5 \cdot 4273 \) |
\( I_{10} \) | \(=\) | \(124416\) | \(=\) | \( 2^{9} \cdot 3^{5} \) |
\( J_2 \) | \(=\) | \(372\) | \(=\) | \( 2^{2} \cdot 3 \cdot 31 \) |
\( J_4 \) | \(=\) | \(3834\) | \(=\) | \( 2 \cdot 3^{3} \cdot 71 \) |
\( J_6 \) | \(=\) | \(-23036\) | \(=\) | \( - 2^{2} \cdot 13 \cdot 443 \) |
\( J_8 \) | \(=\) | \(-5817237\) | \(=\) | \( - 3 \cdot 47 \cdot 41257 \) |
\( J_{10} \) | \(=\) | \(995328\) | \(=\) | \( 2^{12} \cdot 3^{5} \) |
\( g_1 \) | \(=\) | \(28629151/4\) | ||
\( g_2 \) | \(=\) | \(6345483/32\) | ||
\( g_3 \) | \(=\) | \(-5534399/1728\) |
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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Rational points
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (1 : -1 : 1),\, (-2 : -1 : 1),\, (-2 : 5 : 1)\)
magma: [C![-2,-1,1],C![-2,5,1],C![1,-1,0],C![1,-1,1],C![1,0,0]]; // minimal model
magma: [C![-2,-6,1],C![-2,6,1],C![1,-1,0],C![1,0,1],C![1,1,0]]; // simplified model
Number of rational Weierstrass points: \(1\)
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: \(\Z \oplus \Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1 : 1) - (1 : 0 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3\) | \(0.074592\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 + z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1 : 1) - (1 : 0 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3\) | \(0.074592\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 + z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : 0 : 1) - (1 : 1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + x^2z\) | \(0.074592\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z + 2xz^2 + 2z^3\) | \(0\) | \(2\) |
2-torsion field: 6.0.1492992.6
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.074592 \) |
Real period: | \( 6.704716 \) |
Tamagawa product: | \( 8 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 1.000241 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(8\) | \(12\) | \(4\) | \(1 + T\) | |
\(3\) | \(4\) | \(5\) | \(2\) | \(1 + T\) |
Galois representations
The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.
Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
---|---|---|
\(2\) | 2.60.1 | yes |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $\mathrm{USp}(4)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{USp}(4)\) |
Decomposition of the Jacobian
Simple over \(\overline{\Q}\)
magma: HeuristicDecompositionFactors(C);
Endomorphisms of the Jacobian
Not of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | \(\Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R\) |
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);