Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x)y = -2x^6 - x^5 - 21x^4 - 8x^3 - 80x^2 - 16x - 103$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2)y = -2x^6 - x^5z - 21x^4z^2 - 8x^3z^3 - 80x^2z^4 - 16xz^5 - 103z^6$ | (dehomogenize, simplify) |
$y^2 = -7x^6 - 4x^5 - 82x^4 - 32x^3 - 319x^2 - 64x - 412$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-103, -16, -80, -8, -21, -1, -2]), R([0, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-103, -16, -80, -8, -21, -1, -2], R![0, 1, 0, 1]);
sage: X = HyperellipticCurve(R([-412, -64, -319, -32, -82, -4, -7]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(1269\) | \(=\) | \( 3^{3} \cdot 47 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-102789\) | \(=\) | \( - 3^{7} \cdot 47 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(91192\) | \(=\) | \( 2^{3} \cdot 11399 \) |
\( I_4 \) | \(=\) | \(19900\) | \(=\) | \( 2^{2} \cdot 5^{2} \cdot 199 \) |
\( I_6 \) | \(=\) | \(603982075\) | \(=\) | \( 5^{2} \cdot 24159283 \) |
\( I_{10} \) | \(=\) | \(1692\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 47 \) |
\( J_2 \) | \(=\) | \(136788\) | \(=\) | \( 2^{2} \cdot 3 \cdot 11399 \) |
\( J_4 \) | \(=\) | \(779593356\) | \(=\) | \( 2^{2} \cdot 3^{3} \cdot 7218457 \) |
\( J_6 \) | \(=\) | \(5923938871071\) | \(=\) | \( 3^{4} \cdot 13 \cdot 52567 \cdot 107021 \) |
\( J_8 \) | \(=\) | \(50639487394179303\) | \(=\) | \( 3^{5} \cdot 163 \cdot 149561 \cdot 8548247 \) |
\( J_{10} \) | \(=\) | \(102789\) | \(=\) | \( 3^{7} \cdot 47 \) |
\( g_1 \) | \(=\) | \(197075993647247827966976/423\) | ||
\( g_2 \) | \(=\) | \(2737061778548953841408/141\) | ||
\( g_3 \) | \(=\) | \(152047414479420367856/141\) |
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
This curve has no rational points.
magma: []; // minimal model
magma: []; // simplified model
Number of rational Weierstrass points: \(0\)
magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
This curve is locally solvable except over $\R$.
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: \(\Z/{10}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(4x^2 + 17z^2\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(7xz^2 - z^3\) | \(0\) | \(10\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(4x^2 + 17z^2\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(7xz^2 - z^3\) | \(0\) | \(10\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(4x^2 + 17z^2\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(x^3 + 15xz^2 - 2z^3\) | \(0\) | \(10\) |
2-torsion field: 6.4.45805824.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1 \) |
Real period: | \( 4.110304 \) |
Tamagawa product: | \( 5 \) |
Torsion order: | \( 10 \) |
Leading coefficient: | \( 0.411030 \) |
Analytic order of Ш: | \( 2 \) (rounded) |
Order of Ш: | twice a square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(3\) | \(7\) | \(5\) | \(1 - T\) | |
\(47\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 8 T + 47 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.15.1 | yes |
\(5\) | not computed | 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);