Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = -x^5 + 7x^2 - 8x + 2$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = -x^5z + 7x^2z^4 - 8xz^5 + 2z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 - 4x^5 + 2x^3 + 28x^2 - 32x + 9$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, -8, 7, 0, 0, -1]), R([1, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, -8, 7, 0, 0, -1], R![1, 0, 0, 1]);
sage: X = HyperellipticCurve(R([9, -32, 28, 2, 0, -4, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(110499\) | \(=\) | \( 3 \cdot 36833 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(331497\) | \(=\) | \( 3^{2} \cdot 36833 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(1492\) | \(=\) | \( 2^{2} \cdot 373 \) |
\( I_4 \) | \(=\) | \(222409\) | \(=\) | \( 11 \cdot 20219 \) |
\( I_6 \) | \(=\) | \(72914981\) | \(=\) | \( 72914981 \) |
\( I_{10} \) | \(=\) | \(42431616\) | \(=\) | \( 2^{7} \cdot 3^{2} \cdot 36833 \) |
\( J_2 \) | \(=\) | \(373\) | \(=\) | \( 373 \) |
\( J_4 \) | \(=\) | \(-3470\) | \(=\) | \( - 2 \cdot 5 \cdot 347 \) |
\( J_6 \) | \(=\) | \(67588\) | \(=\) | \( 2^{2} \cdot 61 \cdot 277 \) |
\( J_8 \) | \(=\) | \(3292356\) | \(=\) | \( 2^{2} \cdot 3 \cdot 17 \cdot 16139 \) |
\( J_{10} \) | \(=\) | \(331497\) | \(=\) | \( 3^{2} \cdot 36833 \) |
\( g_1 \) | \(=\) | \(7220115733093/331497\) | ||
\( g_2 \) | \(=\) | \(-180076055990/331497\) | ||
\( g_3 \) | \(=\) | \(9403450852/331497\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 1 : 1)\) | \((1 : 0 : 1)\) | \((0 : -2 : 1)\) | \((1 : -2 : 1)\) |
\((1 : -3 : 2)\) | \((2 : -3 : 1)\) | \((-2 : -6 : 1)\) | \((1 : -6 : 2)\) | \((2 : -6 : 1)\) | \((-2 : 13 : 1)\) |
\((-4 : -15 : 1)\) | \((-4 : 78 : 1)\) | \((23 : -344 : 24)\) | \((11 : -516 : 3)\) | \((11 : -842 : 3)\) | \((9 : -1183 : 14)\) |
\((9 : -2290 : 14)\) | \((23 : -25647 : 24)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 1 : 1)\) | \((1 : 0 : 1)\) | \((0 : -2 : 1)\) | \((1 : -2 : 1)\) |
\((1 : -3 : 2)\) | \((2 : -3 : 1)\) | \((-2 : -6 : 1)\) | \((1 : -6 : 2)\) | \((2 : -6 : 1)\) | \((-2 : 13 : 1)\) |
\((-4 : -15 : 1)\) | \((-4 : 78 : 1)\) | \((23 : -344 : 24)\) | \((11 : -516 : 3)\) | \((11 : -842 : 3)\) | \((9 : -1183 : 14)\) |
\((9 : -2290 : 14)\) | \((23 : -25647 : 24)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((1 : -2 : 1)\) | \((1 : 2 : 1)\) | \((0 : -3 : 1)\) | \((0 : 3 : 1)\) |
\((1 : -3 : 2)\) | \((1 : 3 : 2)\) | \((2 : -3 : 1)\) | \((2 : 3 : 1)\) | \((-2 : -19 : 1)\) | \((-2 : 19 : 1)\) |
\((-4 : -93 : 1)\) | \((-4 : 93 : 1)\) | \((11 : -326 : 3)\) | \((11 : 326 : 3)\) | \((9 : -1107 : 14)\) | \((9 : 1107 : 14)\) |
\((23 : -25303 : 24)\) | \((23 : 25303 : 24)\) |
magma: [C![-4,-15,1],C![-4,78,1],C![-2,-6,1],C![-2,13,1],C![0,-2,1],C![0,1,1],C![1,-6,2],C![1,-3,2],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![2,-6,1],C![2,-3,1],C![9,-2290,14],C![9,-1183,14],C![11,-842,3],C![11,-516,3],C![23,-25647,24],C![23,-344,24]]; // minimal model
magma: [C![-4,-93,1],C![-4,93,1],C![-2,-19,1],C![-2,19,1],C![0,-3,1],C![0,3,1],C![1,-3,2],C![1,3,2],C![1,-2,1],C![1,-1,0],C![1,1,0],C![1,2,1],C![2,-3,1],C![2,3,1],C![9,-1107,14],C![9,1107,14],C![11,-326,3],C![11,326,3],C![23,-25303,24],C![23,25303,24]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -2 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2z^3\) | \(0.391929\) | \(\infty\) |
\((1 : -2 : 1) + (2 : -6 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - 2z) (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-4xz^2 + 2z^3\) | \(0.454238\) | \(\infty\) |
\((1 : -3 : 2) - (1 : 0 : 0)\) | \(z (2x - z)\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(-4x^3 - z^3\) | \(0.169029\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -2 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2z^3\) | \(0.391929\) | \(\infty\) |
\((1 : -2 : 1) + (2 : -6 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - 2z) (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-4xz^2 + 2z^3\) | \(0.454238\) | \(\infty\) |
\((1 : -3 : 2) - (1 : 0 : 0)\) | \(z (2x - z)\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(-4x^3 - z^3\) | \(0.169029\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -2 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 3z^3\) | \(0.391929\) | \(\infty\) |
\((1 : -2 : 1) + (2 : -3 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - 2z) (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 8xz^2 + 5z^3\) | \(0.454238\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(z (2x - z)\) | \(=\) | \(0,\) | \(4y\) | \(=\) | \(-7x^3 - z^3\) | \(0.169029\) | \(\infty\) |
2-torsion field: 6.2.2357312.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.027992 \) |
Real period: | \( 18.92576 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 1.059548 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + T + 3 T^{2} )\) | |
\(36833\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 278 T + 36833 T^{2} )\) |
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);