Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = x^5 - 2x^4 - 9x^3 + 8x^2 + 20x - 20$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = x^5z - 2x^4z^2 - 9x^3z^3 + 8x^2z^4 + 20xz^5 - 20z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 + 4x^5 - 8x^4 - 34x^3 + 32x^2 + 80x - 79$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-20, 20, 8, -9, -2, 1]), R([1, 0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-20, 20, 8, -9, -2, 1], R![1, 0, 0, 1]);
sage: X = HyperellipticCurve(R([-79, 80, 32, -34, -8, 4, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(100162\) | \(=\) | \( 2 \cdot 61 \cdot 821 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-100162\) | \(=\) | \( - 2 \cdot 61 \cdot 821 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(21396\) | \(=\) | \( 2^{2} \cdot 3 \cdot 1783 \) |
\( I_4 \) | \(=\) | \(-10503\) | \(=\) | \( - 3^{3} \cdot 389 \) |
\( I_6 \) | \(=\) | \(-75948435\) | \(=\) | \( - 3^{5} \cdot 5 \cdot 17 \cdot 3677 \) |
\( I_{10} \) | \(=\) | \(-12820736\) | \(=\) | \( - 2^{8} \cdot 61 \cdot 821 \) |
\( J_2 \) | \(=\) | \(5349\) | \(=\) | \( 3 \cdot 1783 \) |
\( J_4 \) | \(=\) | \(1192596\) | \(=\) | \( 2^{2} \cdot 3 \cdot 23 \cdot 29 \cdot 149 \) |
\( J_6 \) | \(=\) | \(354674332\) | \(=\) | \( 2^{2} \cdot 17 \cdot 5215799 \) |
\( J_8 \) | \(=\) | \(118716945663\) | \(=\) | \( 3 \cdot 37 \cdot 53 \cdot 3407 \cdot 5923 \) |
\( J_{10} \) | \(=\) | \(-100162\) | \(=\) | \( - 2 \cdot 61 \cdot 821 \) |
\( g_1 \) | \(=\) | \(-4378879451923801749/100162\) | ||
\( g_2 \) | \(=\) | \(-91260143303221602/50081\) | ||
\( g_3 \) | \(=\) | \(-5073935703495966/50081\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((2 : -4 : 1)\) | \((2 : -5 : 1)\) | \((4 : -31 : 3)\) | \((4 : -60 : 3)\) |
\((-13 : 1025 : 4)\) | \((-13 : 1108 : 4)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((2 : -4 : 1)\) | \((2 : -5 : 1)\) | \((4 : -31 : 3)\) | \((4 : -60 : 3)\) |
\((-13 : 1025 : 4)\) | \((-13 : 1108 : 4)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((2 : -1 : 1)\) | \((2 : 1 : 1)\) | \((4 : -29 : 3)\) | \((4 : 29 : 3)\) |
\((-13 : -83 : 4)\) | \((-13 : 83 : 4)\) |
magma: [C![-13,1025,4],C![-13,1108,4],C![1,-1,0],C![1,0,0],C![2,-5,1],C![2,-4,1],C![4,-60,3],C![4,-31,3]]; // minimal model
magma: [C![-13,-83,4],C![-13,83,4],C![1,-1,0],C![1,1,0],C![2,-1,1],C![2,1,1],C![4,-29,3],C![4,29,3]]; // 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: \(\Z \oplus \Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + 2xz - 5z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-4xz^2 + 4z^3\) | \(0.541687\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + 2xz - 4z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-4xz^2 + 3z^3\) | \(0.385982\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + 2xz - 5z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-4xz^2 + 4z^3\) | \(0.541687\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2 + 2xz - 4z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-4xz^2 + 3z^3\) | \(0.385982\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 + 2xz - 5z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 8xz^2 + 9z^3\) | \(0.541687\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 + 2xz - 4z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 8xz^2 + 7z^3\) | \(0.385982\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.174345 \) |
Real period: | \( 6.880813 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 1.199636 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + T + 2 T^{2} )\) | |
\(61\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 10 T + 61 T^{2} )\) | |
\(821\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 821 T^{2} )\) |
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);