Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = -x^5 - 2x^4 + x^3 + 3x^2 - 2x$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = -x^5z - 2x^4z^2 + x^3z^3 + 3x^2z^4 - 2xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 - 4x^5 - 8x^4 + 6x^3 + 12x^2 - 8x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -2, 3, 1, -2, -1]), R([1, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -2, 3, 1, -2, -1], R![1, 0, 0, 1]);
sage: X = HyperellipticCurve(R([1, -8, 12, 6, -8, -4, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(226077\) | \(=\) | \( 3 \cdot 179 \cdot 421 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-678231\) | \(=\) | \( - 3^{2} \cdot 179 \cdot 421 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(1396\) | \(=\) | \( 2^{2} \cdot 349 \) |
\( I_4 \) | \(=\) | \(34585\) | \(=\) | \( 5 \cdot 6917 \) |
\( I_6 \) | \(=\) | \(14310749\) | \(=\) | \( 37 \cdot 386777 \) |
\( I_{10} \) | \(=\) | \(-86813568\) | \(=\) | \( - 2^{7} \cdot 3^{2} \cdot 179 \cdot 421 \) |
\( J_2 \) | \(=\) | \(349\) | \(=\) | \( 349 \) |
\( J_4 \) | \(=\) | \(3634\) | \(=\) | \( 2 \cdot 23 \cdot 79 \) |
\( J_6 \) | \(=\) | \(39340\) | \(=\) | \( 2^{2} \cdot 5 \cdot 7 \cdot 281 \) |
\( J_8 \) | \(=\) | \(130926\) | \(=\) | \( 2 \cdot 3 \cdot 21821 \) |
\( J_{10} \) | \(=\) | \(-678231\) | \(=\) | \( - 3^{2} \cdot 179 \cdot 421 \) |
\( g_1 \) | \(=\) | \(-5177583776749/678231\) | ||
\( g_2 \) | \(=\) | \(-154476067066/678231\) | ||
\( g_3 \) | \(=\) | \(-4791651340/678231\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : -1 : 1)\) | \((-2 : -1 : 1)\) |
\((1 : -3 : 2)\) | \((-3 : -3 : 1)\) | \((1 : -6 : 2)\) | \((-1 : 7 : 2)\) | \((-2 : 8 : 1)\) | \((-1 : -14 : 2)\) |
\((-3 : 29 : 1)\) | \((6 : -111 : 7)\) | \((6 : -448 : 7)\) | \((-425 : -8591400 : 104)\) | \((-425 : 84232161 : 104)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : -1 : 1)\) | \((-2 : -1 : 1)\) |
\((1 : -3 : 2)\) | \((-3 : -3 : 1)\) | \((1 : -6 : 2)\) | \((-1 : 7 : 2)\) | \((-2 : 8 : 1)\) | \((-1 : -14 : 2)\) |
\((-3 : 29 : 1)\) | \((6 : -111 : 7)\) | \((6 : -448 : 7)\) | \((-425 : -8591400 : 104)\) | \((-425 : 84232161 : 104)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((1 : 0 : 1)\) | \((1 : -3 : 2)\) |
\((1 : 3 : 2)\) | \((-2 : -9 : 1)\) | \((-2 : 9 : 1)\) | \((-1 : -21 : 2)\) | \((-1 : 21 : 2)\) | \((-3 : -32 : 1)\) |
\((-3 : 32 : 1)\) | \((6 : -337 : 7)\) | \((6 : 337 : 7)\) | \((-425 : -92823561 : 104)\) | \((-425 : 92823561 : 104)\) |
magma: [C![-425,-8591400,104],C![-425,84232161,104],C![-3,-3,1],C![-3,29,1],C![-2,-1,1],C![-2,8,1],C![-1,-14,2],C![-1,7,2],C![0,-1,1],C![0,0,1],C![1,-6,2],C![1,-3,2],C![1,-1,0],C![1,-1,1],C![1,0,0],C![6,-448,7],C![6,-111,7]]; // minimal model
magma: [C![-425,-92823561,104],C![-425,92823561,104],C![-3,-32,1],C![-3,32,1],C![-2,-9,1],C![-2,9,1],C![-1,-21,2],C![-1,21,2],C![0,-1,1],C![0,1,1],C![1,-3,2],C![1,3,2],C![1,-1,0],C![1,0,1],C![1,1,0],C![6,-337,7],C![6,337,7]]; // simplified model
Number of rational Weierstrass points: \(1\)
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 | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : -1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.831918\) | \(\infty\) |
\((1 : -1 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.288615\) | \(\infty\) |
\((0 : -1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.238480\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : -1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.831918\) | \(\infty\) |
\((1 : -1 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.288615\) | \(\infty\) |
\((0 : -1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.238480\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-2 : -9 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x + 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - z^3\) | \(0.831918\) | \(\infty\) |
\((1 : 0 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - z^3\) | \(0.288615\) | \(\infty\) |
\((0 : -1 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - z^3\) | \(0.238480\) | \(\infty\) |
2-torsion field: 5.3.1205744.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.047804 \) |
Real period: | \( 15.08193 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 1.441968 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + T + 3 T^{2} )\) | |
\(179\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 2 T + 179 T^{2} )\) | |
\(421\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 22 T + 421 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.6.1 | no |
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);