Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x + 1)y = x^3 - x^2 - 2x$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2 + z^3)y = x^3z^3 - x^2z^4 - 2xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 + 2x^4 + 6x^3 - 3x^2 - 6x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -2, -1, 1]), R([1, 1, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -2, -1, 1], R![1, 1, 0, 1]);
sage: X = HyperellipticCurve(R([1, -6, -3, 6, 2, 0, 1]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(25913\) | \(=\) | \( 25913 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(25913\) | \(=\) | \( 25913 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(36\) | \(=\) | \( 2^{2} \cdot 3^{2} \) |
\( I_4 \) | \(=\) | \(4857\) | \(=\) | \( 3 \cdot 1619 \) |
\( I_6 \) | \(=\) | \(-524835\) | \(=\) | \( - 3^{2} \cdot 5 \cdot 107 \cdot 109 \) |
\( I_{10} \) | \(=\) | \(3316864\) | \(=\) | \( 2^{7} \cdot 25913 \) |
\( J_2 \) | \(=\) | \(9\) | \(=\) | \( 3^{2} \) |
\( J_4 \) | \(=\) | \(-199\) | \(=\) | \( -199 \) |
\( J_6 \) | \(=\) | \(7797\) | \(=\) | \( 3 \cdot 23 \cdot 113 \) |
\( J_8 \) | \(=\) | \(7643\) | \(=\) | \( 7643 \) |
\( J_{10} \) | \(=\) | \(25913\) | \(=\) | \( 25913 \) |
\( g_1 \) | \(=\) | \(59049/25913\) | ||
\( g_2 \) | \(=\) | \(-145071/25913\) | ||
\( g_3 \) | \(=\) | \(631557/25913\) |
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 | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : -1 : 1)\) | \((2 : 0 : 1)\) | \((-2 : 1 : 1)\) | \((1 : -2 : 1)\) | \((-1 : 5 : 2)\) | \((-3 : 7 : 2)\) |
\((-2 : 8 : 1)\) | \((-1 : -8 : 2)\) | \((2 : -11 : 1)\) | \((-3 : 24 : 2)\) | \((-7 : 28 : 3)\) | \((-7 : 351 : 3)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : -1 : 1)\) | \((2 : 0 : 1)\) | \((-2 : 1 : 1)\) | \((1 : -2 : 1)\) | \((-1 : 5 : 2)\) | \((-3 : 7 : 2)\) |
\((-2 : 8 : 1)\) | \((-1 : -8 : 2)\) | \((2 : -11 : 1)\) | \((-3 : 24 : 2)\) | \((-7 : 28 : 3)\) | \((-7 : 351 : 3)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : -1 : 1)\) | \((1 : 1 : 1)\) | \((-2 : -7 : 1)\) | \((-2 : 7 : 1)\) | \((2 : -11 : 1)\) | \((2 : 11 : 1)\) |
\((-1 : -13 : 2)\) | \((-1 : 13 : 2)\) | \((-3 : -17 : 2)\) | \((-3 : 17 : 2)\) | \((-7 : -323 : 3)\) | \((-7 : 323 : 3)\) |
magma: [C![-7,28,3],C![-7,351,3],C![-3,7,2],C![-3,24,2],C![-2,1,1],C![-2,8,1],C![-1,-8,2],C![-1,0,1],C![-1,1,1],C![-1,5,2],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![2,-11,1],C![2,0,1]]; // minimal model
magma: [C![-7,-323,3],C![-7,323,3],C![-3,-17,2],C![-3,17,2],C![-2,-7,1],C![-2,7,1],C![-1,-13,2],C![-1,-1,1],C![-1,1,1],C![-1,13,2],C![0,-1,1],C![0,1,1],C![1,-1,1],C![1,-1,0],C![1,1,1],C![1,1,0],C![2,-11,1],C![2,11,1]]; // 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: \(\Z \oplus \Z \oplus \Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.317898\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.375585\) | \(\infty\) |
\((-1 : 0 : 1) - (1 : 0 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.275189\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.317898\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.375585\) | \(\infty\) |
\((-1 : 0 : 1) - (1 : 0 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 - z^3\) | \(0.275189\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : 1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + xz^2 - z^3\) | \(0.317898\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 + z^3\) | \(0.375585\) | \(\infty\) |
\((-1 : -1 : 1) - (1 : 1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + xz^2 - z^3\) | \(0.275189\) | \(\infty\) |
2-torsion field: 6.2.1658432.2
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.019222 \) |
Real period: | \( 20.35187 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.391214 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(25913\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 61 T + 25913 T^{2} )\) |
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);
Additional information
The conductor $25913$ of the Jacobian of this curve is the smallest known for a genus $2$ curve with analytic rank $3$.