Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x)y = -x^6 + 16x^5 - 78x^4 + 88x^3 + 107x^2 + 32x + 3$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2)y = -x^6 + 16x^5z - 78x^4z^2 + 88x^3z^3 + 107x^2z^4 + 32xz^5 + 3z^6$ | (dehomogenize, simplify) |
$y^2 = -4x^6 + 64x^5 - 311x^4 + 354x^3 + 429x^2 + 128x + 12$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([3, 32, 107, 88, -78, 16, -1]), R([0, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![3, 32, 107, 88, -78, 16, -1], R![0, 1, 1]);
sage: X = HyperellipticCurve(R([12, 128, 429, 354, -311, 64, -4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(537225\) | \(=\) | \( 3 \cdot 5^{2} \cdot 13 \cdot 19 \cdot 29 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(537225\) | \(=\) | \( 3 \cdot 5^{2} \cdot 13 \cdot 19 \cdot 29 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(1612900\) | \(=\) | \( 2^{2} \cdot 5^{2} \cdot 127^{2} \) |
\( I_4 \) | \(=\) | \(496441\) | \(=\) | \( 11 \cdot 45131 \) |
\( I_6 \) | \(=\) | \(266939427453\) | \(=\) | \( 3 \cdot 67 \cdot 443 \cdot 2997871 \) |
\( I_{10} \) | \(=\) | \(68764800\) | \(=\) | \( 2^{7} \cdot 3 \cdot 5^{2} \cdot 13 \cdot 19 \cdot 29 \) |
\( J_2 \) | \(=\) | \(403225\) | \(=\) | \( 5^{2} \cdot 127^{2} \) |
\( J_4 \) | \(=\) | \(6774579341\) | \(=\) | \( 89 \cdot 76118869 \) |
\( J_6 \) | \(=\) | \(151759059126301\) | \(=\) | \( 11 \cdot 293 \cdot 67153 \cdot 701179 \) |
\( J_8 \) | \(=\) | \(3824530342174681611\) | \(=\) | \( 3 \cdot 29 \cdot 1049 \cdot 4093 \cdot 10238624729 \) |
\( J_{10} \) | \(=\) | \(537225\) | \(=\) | \( 3 \cdot 5^{2} \cdot 13 \cdot 19 \cdot 29 \) |
\( g_1 \) | \(=\) | \(426380411356794348300390625/21489\) | ||
\( g_2 \) | \(=\) | \(17765796228320971774968125/21489\) | ||
\( g_3 \) | \(=\) | \(986982648872733682573525/21489\) |
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
No rational points are known for this curve.
magma: []; // minimal model
magma: []; // simplified model
Number of rational Weierstrass points: \(0\)
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: \(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(2x^2 - 5xz - z^2\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(21xz^2 + 3z^3\) | \(3.017046\) | \(\infty\) |
\(D_0 - D_\infty\) | \(x^2 - 7xz - 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-4xz^2 - z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(2x^2 - 11xz - 3z^2\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(-13xz^2 - 3z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(2x^2 - 5xz - z^2\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(21xz^2 + 3z^3\) | \(3.017046\) | \(\infty\) |
\(D_0 - D_\infty\) | \(x^2 - 7xz - 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-4xz^2 - z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(2x^2 - 11xz - 3z^2\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(-13xz^2 - 3z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(2x^2 - 5xz - z^2\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(x^2z + 43xz^2 + 6z^3\) | \(3.017046\) | \(\infty\) |
\(D_0 - D_\infty\) | \(x^2 - 7xz - 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z - 7xz^2 - 2z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(2x^2 - 11xz - 3z^2\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(x^2z - 25xz^2 - 6z^3\) | \(0\) | \(2\) |
2-torsion field: 8.8.133273818424405400625.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(5\) |
Regulator: | \( 3.017046 \) |
Real period: | \( 7.280991 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 5.491773 \) |
Analytic order of Ш: | \( 4 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 2 T + 3 T^{2} )\) | |
\(5\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )^{2}\) | |
\(13\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 6 T + 13 T^{2} )\) | |
\(19\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 4 T + 19 T^{2} )\) | |
\(29\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 6 T + 29 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.90.6 | 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);