Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x + 1)y = x^5 + 3x^4$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2 + z^3)y = x^5z + 3x^4z^2$ | (dehomogenize, simplify) |
$y^2 = x^6 + 4x^5 + 14x^4 + 2x^3 + x^2 + 2x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 0, 0, 3, 1]), R([1, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 0, 0, 3, 1], R![1, 1, 0, 1]);
sage: X = HyperellipticCurve(R([1, 2, 1, 2, 14, 4, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(90963\) | \(=\) | \( 3^{4} \cdot 1123 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-818667\) | \(=\) | \( - 3^{6} \cdot 1123 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(20\) | \(=\) | \( 2^{2} \cdot 5 \) |
\( I_4 \) | \(=\) | \(3105\) | \(=\) | \( 3^{3} \cdot 5 \cdot 23 \) |
\( I_6 \) | \(=\) | \(3969\) | \(=\) | \( 3^{4} \cdot 7^{2} \) |
\( I_{10} \) | \(=\) | \(431232\) | \(=\) | \( 2^{7} \cdot 3 \cdot 1123 \) |
\( J_2 \) | \(=\) | \(15\) | \(=\) | \( 3 \cdot 5 \) |
\( J_4 \) | \(=\) | \(-1155\) | \(=\) | \( - 3 \cdot 5 \cdot 7 \cdot 11 \) |
\( J_6 \) | \(=\) | \(3371\) | \(=\) | \( 3371 \) |
\( J_8 \) | \(=\) | \(-320865\) | \(=\) | \( - 3 \cdot 5 \cdot 21391 \) |
\( J_{10} \) | \(=\) | \(818667\) | \(=\) | \( 3^{6} \cdot 1123 \) |
\( g_1 \) | \(=\) | \(3125/3369\) | ||
\( g_2 \) | \(=\) | \(-48125/10107\) | ||
\( g_3 \) | \(=\) | \(84275/90963\) |
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
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -1 : 1)\) | \((1 : 1 : 1)\) |
\((-1 : 2 : 1)\) | \((1 : 1 : 2)\) | \((-1 : 2 : 2)\) | \((-3 : 0 : 1)\) | \((1 : -4 : 1)\) | \((-1 : -5 : 2)\) |
\((2 : 5 : 1)\) | \((-2 : 13 : 5)\) | \((1 : -14 : 2)\) | \((2 : -16 : 1)\) | \((-3 : 29 : 1)\) | \((-2 : -80 : 5)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -1 : 1)\) | \((1 : 1 : 1)\) |
\((-1 : 2 : 1)\) | \((1 : 1 : 2)\) | \((-1 : 2 : 2)\) | \((-3 : 0 : 1)\) | \((1 : -4 : 1)\) | \((-1 : -5 : 2)\) |
\((2 : 5 : 1)\) | \((-2 : 13 : 5)\) | \((1 : -14 : 2)\) | \((2 : -16 : 1)\) | \((-3 : 29 : 1)\) | \((-2 : -80 : 5)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -3 : 1)\) | \((-1 : 3 : 1)\) |
\((1 : -5 : 1)\) | \((1 : 5 : 1)\) | \((-1 : -7 : 2)\) | \((-1 : 7 : 2)\) | \((1 : -15 : 2)\) | \((1 : 15 : 2)\) |
\((2 : -21 : 1)\) | \((2 : 21 : 1)\) | \((-3 : -29 : 1)\) | \((-3 : 29 : 1)\) | \((-2 : -93 : 5)\) | \((-2 : 93 : 5)\) |
magma: [C![-3,0,1],C![-3,29,1],C![-2,-80,5],C![-2,13,5],C![-1,-5,2],C![-1,-1,1],C![-1,2,1],C![-1,2,2],C![0,-1,1],C![0,0,1],C![1,-14,2],C![1,-4,1],C![1,-1,0],C![1,0,0],C![1,1,1],C![1,1,2],C![2,-16,1],C![2,5,1]]; // minimal model
magma: [C![-3,-29,1],C![-3,29,1],C![-2,-93,5],C![-2,93,5],C![-1,-7,2],C![-1,-3,1],C![-1,3,1],C![-1,7,2],C![0,-1,1],C![0,1,1],C![1,-15,2],C![1,-5,1],C![1,-1,0],C![1,1,0],C![1,5,1],C![1,15,2],C![2,-21,1],C![2,21,1]]; // 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: \(\Z \oplus \Z \oplus \Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 2 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-3xz^2 - z^3\) | \(0.248488\) | \(\infty\) |
\((-1 : -1 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.341079\) | \(\infty\) |
\((-1 : -1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.258830\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 2 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-3xz^2 - z^3\) | \(0.248488\) | \(\infty\) |
\((-1 : -1 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.341079\) | \(\infty\) |
\((-1 : -1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.258830\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 3 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 5xz^2 - z^3\) | \(0.248488\) | \(\infty\) |
\((-1 : -3 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 - z^3\) | \(0.341079\) | \(\infty\) |
\((-1 : -3 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 - z^3\) | \(0.258830\) | \(\infty\) |
2-torsion field: 6.0.5821632.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.017320 \) |
Real period: | \( 14.02121 \) |
Tamagawa product: | \( 3 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.728562 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(4\) | \(6\) | \(3\) | \(1 + 3 T + 3 T^{2}\) | |
\(1123\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 28 T + 1123 T^{2} )\) |
Galois representations
The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) .
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);