Minimal equation
Minimal equation
Simplified equation
$y^2 + x^3y = -x^4 + 3x^2 - 4x + 2$ | (homogenize, simplify) |
$y^2 + x^3y = -x^4z^2 + 3x^2z^4 - 4xz^5 + 2z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 - 4x^4 + 12x^2 - 16x + 8$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, -4, 3, 0, -1]), R([0, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, -4, 3, 0, -1], R![0, 0, 0, 1]);
sage: X = HyperellipticCurve(R([8, -16, 12, 0, -4, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(20736\) | \(=\) | \( 2^{8} \cdot 3^{4} \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-41472\) | \(=\) | \( - 2^{9} \cdot 3^{4} \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(72\) | \(=\) | \( 2^{3} \cdot 3^{2} \) |
\( I_4 \) | \(=\) | \(225\) | \(=\) | \( 3^{2} \cdot 5^{2} \) |
\( I_6 \) | \(=\) | \(5031\) | \(=\) | \( 3^{2} \cdot 13 \cdot 43 \) |
\( I_{10} \) | \(=\) | \(162\) | \(=\) | \( 2 \cdot 3^{4} \) |
\( J_2 \) | \(=\) | \(144\) | \(=\) | \( 2^{4} \cdot 3^{2} \) |
\( J_4 \) | \(=\) | \(264\) | \(=\) | \( 2^{3} \cdot 3 \cdot 11 \) |
\( J_6 \) | \(=\) | \(-4864\) | \(=\) | \( - 2^{8} \cdot 19 \) |
\( J_8 \) | \(=\) | \(-192528\) | \(=\) | \( - 2^{4} \cdot 3^{2} \cdot 7 \cdot 191 \) |
\( J_{10} \) | \(=\) | \(41472\) | \(=\) | \( 2^{9} \cdot 3^{4} \) |
\( g_1 \) | \(=\) | \(1492992\) | ||
\( g_2 \) | \(=\) | \(19008\) | ||
\( g_3 \) | \(=\) | \(-2432\) |
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
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (1 : 0 : 1),\, (1 : -1 : 1)\)
magma: [C![1,-1,0],C![1,-1,1],C![1,0,0],C![1,0,1]]; // minimal model
magma: [C![1,-1,0],C![1,-1,1],C![1,1,0],C![1,1,1]]; // 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: \(\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.022946\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.022946\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 2z^3\) | \(0.022946\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 0.022946 \) |
Real period: | \( 18.28609 \) |
Tamagawa product: | \( 3 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 1.258813 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(8\) | \(9\) | \(3\) | \(1\) | |
\(3\) | \(4\) | \(4\) | \(1\) | \(1 + T + 3 T^{2}\) |
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? |
---|---|---|
\(2\) | 2.40.3 | 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);