Minimal equation
Minimal equation
Simplified equation
$y^2 + xy = x^6 - 5x^5 + 7x^4 - 5x^2 + x + 1$ | (homogenize, simplify) |
$y^2 + xz^2y = x^6 - 5x^5z + 7x^4z^2 - 5x^2z^4 + xz^5 + z^6$ | (dehomogenize, simplify) |
$y^2 = 4x^6 - 20x^5 + 28x^4 - 19x^2 + 4x + 4$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 1, -5, 0, 7, -5, 1]), R([0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 1, -5, 0, 7, -5, 1], R![0, 1]);
sage: X = HyperellipticCurve(R([4, 4, -19, 0, 28, -20, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(13078\) | \(=\) | \( 2 \cdot 13 \cdot 503 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-26156\) | \(=\) | \( - 2^{2} \cdot 13 \cdot 503 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(368\) | \(=\) | \( 2^{4} \cdot 23 \) |
\( I_4 \) | \(=\) | \(31360\) | \(=\) | \( 2^{7} \cdot 5 \cdot 7^{2} \) |
\( I_6 \) | \(=\) | \(1939463\) | \(=\) | \( 19 \cdot 102077 \) |
\( I_{10} \) | \(=\) | \(-104624\) | \(=\) | \( - 2^{4} \cdot 13 \cdot 503 \) |
\( J_2 \) | \(=\) | \(184\) | \(=\) | \( 2^{3} \cdot 23 \) |
\( J_4 \) | \(=\) | \(-3816\) | \(=\) | \( - 2^{3} \cdot 3^{2} \cdot 53 \) |
\( J_6 \) | \(=\) | \(66065\) | \(=\) | \( 5 \cdot 73 \cdot 181 \) |
\( J_8 \) | \(=\) | \(-601474\) | \(=\) | \( - 2 \cdot 311 \cdot 967 \) |
\( J_{10} \) | \(=\) | \(-26156\) | \(=\) | \( - 2^{2} \cdot 13 \cdot 503 \) |
\( g_1 \) | \(=\) | \(-52726521856/6539\) | ||
\( g_2 \) | \(=\) | \(5942946816/6539\) | ||
\( g_3 \) | \(=\) | \(-559174160/6539\) |
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 | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((1 : 0 : 1)\) | \((1 : -1 : 1)\) |
\((2 : -1 : 1)\) | \((3 : 5 : 1)\) | \((-2 : -7 : 3)\) | \((3 : -8 : 1)\) | \((-2 : 25 : 3)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((1 : 0 : 1)\) | \((1 : -1 : 1)\) |
\((2 : -1 : 1)\) | \((3 : 5 : 1)\) | \((-2 : -7 : 3)\) | \((3 : -8 : 1)\) | \((-2 : 25 : 3)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -2 : 0)\) | \((1 : 2 : 0)\) | \((1 : -1 : 1)\) | \((1 : 1 : 1)\) | \((0 : -2 : 1)\) | \((0 : 2 : 1)\) |
\((2 : 0 : 1)\) | \((3 : -13 : 1)\) | \((3 : 13 : 1)\) | \((-2 : -32 : 3)\) | \((-2 : 32 : 3)\) |
magma: [C![-2,-7,3],C![-2,25,3],C![0,-1,1],C![0,1,1],C![1,-1,0],C![1,-1,1],C![1,0,1],C![1,1,0],C![2,-1,1],C![3,-8,1],C![3,5,1]]; // minimal model
magma: [C![-2,-32,3],C![-2,32,3],C![0,-2,1],C![0,2,1],C![1,-2,0],C![1,-1,1],C![1,1,1],C![1,2,0],C![2,0,1],C![3,-13,1],C![3,13,1]]; // 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: \(\Z \oplus \Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1 : 1) + (2 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - 2z) (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.263232\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.046644\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1 : 1) + (2 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - 2z) (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.263232\) | \(\infty\) |
\((0 : -1 : 1) - (1 : 1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.046644\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1 : 1) + (2 : 0 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) | \((x - 2z) (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 - 2z^3\) | \(0.263232\) | \(\infty\) |
\((0 : -2 : 1) - (1 : 2 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2x^3 + xz^2 - 2z^3\) | \(0.046644\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.011869 \) |
Real period: | \( 24.42663 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.579855 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + 2 T^{2} )\) | |
\(13\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 13 T^{2} )\) | |
\(503\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 24 T + 503 T^{2} )\) |
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);