Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = 5x^5 + 34x^4 + 80x^3 - x^2 - 90x + 32$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = 5x^5z + 34x^4z^2 + 80x^3z^3 - x^2z^4 - 90xz^5 + 32z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 + 20x^5 + 136x^4 + 322x^3 - 4x^2 - 360x + 129$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([32, -90, -1, 80, 34, 5]), R([1, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![32, -90, -1, 80, 34, 5], R![1, 0, 0, 1]);
sage: X = HyperellipticCurve(R([129, -360, -4, 322, 136, 20, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(1077\) | \(=\) | \( 3 \cdot 359 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(1077\) | \(=\) | \( 3 \cdot 359 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(155924\) | \(=\) | \( 2^{2} \cdot 17 \cdot 2293 \) |
\( I_4 \) | \(=\) | \(161593\) | \(=\) | \( 283 \cdot 571 \) |
\( I_6 \) | \(=\) | \(8379938029\) | \(=\) | \( 8379938029 \) |
\( I_{10} \) | \(=\) | \(137856\) | \(=\) | \( 2^{7} \cdot 3 \cdot 359 \) |
\( J_2 \) | \(=\) | \(38981\) | \(=\) | \( 17 \cdot 2293 \) |
\( J_4 \) | \(=\) | \(63306532\) | \(=\) | \( 2^{2} \cdot 61 \cdot 259453 \) |
\( J_6 \) | \(=\) | \(137068427976\) | \(=\) | \( 2^{3} \cdot 3 \cdot 5711184499 \) |
\( J_8 \) | \(=\) | \(333836849266358\) | \(=\) | \( 2 \cdot 20369 \cdot 8194728491 \) |
\( J_{10} \) | \(=\) | \(1077\) | \(=\) | \( 3 \cdot 359 \) |
\( g_1 \) | \(=\) | \(90004636142290020118901/1077\) | ||
\( g_2 \) | \(=\) | \(3749794358746968581012/1077\) | ||
\( g_3 \) | \(=\) | \(69425997674312689112/359\) |
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);
|
Rational points
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-13 : 1094 : 2),\, (-13 : 1095 : 2)\)
magma: [C![-13,1094,2],C![-13,1095,2],C![1,-1,0],C![1,0,0]]; // minimal model
magma: [C![-13,-1,2],C![-13,1,2],C![1,-1,0],C![1,1,0]]; // 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: \(\Z \oplus \Z/{2}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + 6xz - 4z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-20xz^2 + 11z^3\) | \(0.035632\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + 6xz - 3z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-39xz^2 + 17z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + 6xz - 4z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-20xz^2 + 11z^3\) | \(0.035632\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + 6xz - 3z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-39xz^2 + 17z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 + 6xz - 4z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 40xz^2 + 23z^3\) | \(0.035632\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 + 6xz - 3z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(x^3 - 78xz^2 + 35z^3\) | \(0\) | \(2\) |
2-torsion field: 6.6.890825472.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.035632 \) |
Real period: | \( 21.23503 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 2 \) |
Leading coefficient: | \( 0.189165 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 2 T + 3 T^{2} )\) | |
\(359\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 8 T + 359 T^{2} )\) |
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 |
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);