Minimal equation
Minimal equation
Simplified equation
$y^2 + (x + 1)y = x^5 - 13x^3 + 14x^2 + 22x - 30$ | (homogenize, simplify) |
$y^2 + (xz^2 + z^3)y = x^5z - 13x^3z^3 + 14x^2z^4 + 22xz^5 - 30z^6$ | (dehomogenize, simplify) |
$y^2 = 4x^5 - 52x^3 + 57x^2 + 90x - 119$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-30, 22, 14, -13, 0, 1]), R([1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-30, 22, 14, -13, 0, 1], R![1, 1]);
sage: X = HyperellipticCurve(R([-119, 90, 57, -52, 0, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(1702\) | \(=\) | \( 2 \cdot 23 \cdot 37 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(39146\) | \(=\) | \( 2 \cdot 23^{2} \cdot 37 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(7656\) | \(=\) | \( 2^{3} \cdot 3 \cdot 11 \cdot 29 \) |
\( I_4 \) | \(=\) | \(14184\) | \(=\) | \( 2^{3} \cdot 3^{2} \cdot 197 \) |
\( I_6 \) | \(=\) | \(36303939\) | \(=\) | \( 3^{2} \cdot 7 \cdot 59 \cdot 9767 \) |
\( I_{10} \) | \(=\) | \(156584\) | \(=\) | \( 2^{3} \cdot 23^{2} \cdot 37 \) |
\( J_2 \) | \(=\) | \(3828\) | \(=\) | \( 2^{2} \cdot 3 \cdot 11 \cdot 29 \) |
\( J_4 \) | \(=\) | \(608202\) | \(=\) | \( 2 \cdot 3^{3} \cdot 7 \cdot 1609 \) |
\( J_6 \) | \(=\) | \(128326985\) | \(=\) | \( 5 \cdot 1607 \cdot 15971 \) |
\( J_8 \) | \(=\) | \(30331506444\) | \(=\) | \( 2^{2} \cdot 3 \cdot 19 \cdot 283 \cdot 470081 \) |
\( J_{10} \) | \(=\) | \(39146\) | \(=\) | \( 2 \cdot 23^{2} \cdot 37 \) |
\( g_1 \) | \(=\) | \(410988481022237184/19573\) | ||
\( g_2 \) | \(=\) | \(17058217029682752/19573\) | ||
\( g_3 \) | \(=\) | \(940225127082120/19573\) |
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: \((1 : 0 : 0),\, (2 : -1 : 1),\, (2 : -2 : 1),\, (16 : -978 : 9),\, (16 : -1047 : 9)\)
magma: [C![1,0,0],C![2,-2,1],C![2,-1,1],C![16,-1047,9],C![16,-978,9]]; // minimal model
magma: [C![1,0,0],C![2,-1,1],C![2,1,1],C![16,-69,9],C![16,69,9]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 - 7xz + 9z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-15xz^2 + 24z^3\) | \(0.036854\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + 2xz - 7z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2 - z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 - 7xz + 9z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-15xz^2 + 24z^3\) | \(0.036854\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + 2xz - 7z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2 - z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 - 7xz + 9z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-29xz^2 + 49z^3\) | \(0.036854\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + 2xz - 7z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2 - z^3\) | \(0\) | \(2\) |
2-torsion field: 6.6.2803712.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.036854 \) |
Real period: | \( 15.53588 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 0.286283 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 2 T^{2} )\) | |
\(23\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + 4 T + 23 T^{2} )\) | |
\(37\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 4 T + 37 T^{2} )\) |
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);