Minimal equation
Minimal equation
Simplified equation
$y^2 + x^3y = x^5 - x^4 - 5x^3 + 5x + 2$ | (homogenize, simplify) |
$y^2 + x^3y = x^5z - x^4z^2 - 5x^3z^3 + 5xz^5 + 2z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 + 4x^5 - 4x^4 - 20x^3 + 20x + 8$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 5, 0, -5, -1, 1]), R([0, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, 5, 0, -5, -1, 1], R![0, 0, 0, 1]);
sage: X = HyperellipticCurve(R([8, 20, 0, -20, -4, 4, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(23631\) | \(=\) | \( 3 \cdot 7877 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(638037\) | \(=\) | \( 3^{4} \cdot 7877 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(920\) | \(=\) | \( 2^{3} \cdot 5 \cdot 23 \) |
\( I_4 \) | \(=\) | \(34144\) | \(=\) | \( 2^{5} \cdot 11 \cdot 97 \) |
\( I_6 \) | \(=\) | \(8577464\) | \(=\) | \( 2^{3} \cdot 7 \cdot 89 \cdot 1721 \) |
\( I_{10} \) | \(=\) | \(2552148\) | \(=\) | \( 2^{2} \cdot 3^{4} \cdot 7877 \) |
\( J_2 \) | \(=\) | \(460\) | \(=\) | \( 2^{2} \cdot 5 \cdot 23 \) |
\( J_4 \) | \(=\) | \(3126\) | \(=\) | \( 2 \cdot 3 \cdot 521 \) |
\( J_6 \) | \(=\) | \(-596\) | \(=\) | \( - 2^{2} \cdot 149 \) |
\( J_8 \) | \(=\) | \(-2511509\) | \(=\) | \( - 7 \cdot 11 \cdot 13^{2} \cdot 193 \) |
\( J_{10} \) | \(=\) | \(638037\) | \(=\) | \( 3^{4} \cdot 7877 \) |
\( g_1 \) | \(=\) | \(20596297600000/638037\) | ||
\( g_2 \) | \(=\) | \(101424112000/212679\) | ||
\( g_3 \) | \(=\) | \(-126113600/638037\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((-1 : 0 : 1)\) | \((-1 : 1 : 1)\) | \((1 : 1 : 1)\) | \((-1 : -1 : 2)\) |
\((1 : -2 : 1)\) | \((-1 : 2 : 2)\) | \((2 : -2 : 1)\) | \((-2 : 4 : 1)\) | \((2 : -6 : 1)\) | \((-17 : 1741 : 4)\) |
\((-17 : 3172 : 4)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((-1 : 0 : 1)\) | \((-1 : 1 : 1)\) | \((1 : 1 : 1)\) | \((-1 : -1 : 2)\) |
\((1 : -2 : 1)\) | \((-1 : 2 : 2)\) | \((2 : -2 : 1)\) | \((-2 : 4 : 1)\) | \((2 : -6 : 1)\) | \((-17 : 1741 : 4)\) |
\((-17 : 3172 : 4)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) | \((-2 : 0 : 1)\) | \((1 : -3 : 1)\) |
\((1 : 3 : 1)\) | \((-1 : -3 : 2)\) | \((-1 : 3 : 2)\) | \((2 : -4 : 1)\) | \((2 : 4 : 1)\) | \((-17 : -1431 : 4)\) |
\((-17 : 1431 : 4)\) |
magma: [C![-17,1741,4],C![-17,3172,4],C![-2,4,1],C![-1,-1,2],C![-1,0,1],C![-1,1,1],C![-1,2,2],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,1,1],C![2,-6,1],C![2,-2,1]]; // minimal model
magma: [C![-17,-1431,4],C![-17,1431,4],C![-2,0,1],C![-1,-3,2],C![-1,-1,1],C![-1,1,1],C![-1,3,2],C![1,-3,1],C![1,-1,0],C![1,1,0],C![1,3,1],C![2,-4,1],C![2,4,1]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\((1 : 1 : 1) - (1 : 0 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + 2z^3\) | \(0.103728\) | \(\infty\) |
\((-1 : 0 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 - z^3\) | \(0.135089\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : 1 : 1) - (1 : 0 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + 2z^3\) | \(0.103728\) | \(\infty\) |
\((-1 : 0 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 - z^3\) | \(0.135089\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : 3 : 1) - (1 : 1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + 4z^3\) | \(0.103728\) | \(\infty\) |
\((-1 : -1 : 1) + (1 : -3 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 2xz^2 - 2z^3\) | \(0.135089\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.010209 \) |
Real period: | \( 17.67786 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.721960 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(1\) | \(4\) | \(4\) | \(( 1 - T )( 1 + 2 T + 3 T^{2} )\) | |
\(7877\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 7 T + 7877 T^{2} )\) |
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);