Minimal equation
Minimal equation
Simplified equation
$y^2 + (x + 1)y = x^6 - x^5 + 2x^4 + x^3 - 3x^2$ | (homogenize, simplify) |
$y^2 + (xz^2 + z^3)y = x^6 - x^5z + 2x^4z^2 + x^3z^3 - 3x^2z^4$ | (dehomogenize, simplify) |
$y^2 = 4x^6 - 4x^5 + 8x^4 + 4x^3 - 11x^2 + 2x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -3, 1, 2, -1, 1]), R([1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -3, 1, 2, -1, 1], R![1, 1]);
sage: X = HyperellipticCurve(R([1, 2, -11, 4, 8, -4, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(14050\) | \(=\) | \( 2 \cdot 5^{2} \cdot 281 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(140500\) | \(=\) | \( 2^{2} \cdot 5^{3} \cdot 281 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(56\) | \(=\) | \( 2^{3} \cdot 7 \) |
\( I_4 \) | \(=\) | \(7720\) | \(=\) | \( 2^{3} \cdot 5 \cdot 193 \) |
\( I_6 \) | \(=\) | \(-739285\) | \(=\) | \( - 5 \cdot 199 \cdot 743 \) |
\( I_{10} \) | \(=\) | \(562000\) | \(=\) | \( 2^{4} \cdot 5^{3} \cdot 281 \) |
\( J_2 \) | \(=\) | \(28\) | \(=\) | \( 2^{2} \cdot 7 \) |
\( J_4 \) | \(=\) | \(-1254\) | \(=\) | \( - 2 \cdot 3 \cdot 11 \cdot 19 \) |
\( J_6 \) | \(=\) | \(92201\) | \(=\) | \( 137 \cdot 673 \) |
\( J_8 \) | \(=\) | \(252278\) | \(=\) | \( 2 \cdot 13 \cdot 31 \cdot 313 \) |
\( J_{10} \) | \(=\) | \(140500\) | \(=\) | \( 2^{2} \cdot 5^{3} \cdot 281 \) |
\( g_1 \) | \(=\) | \(4302592/35125\) | ||
\( g_2 \) | \(=\) | \(-6881952/35125\) | ||
\( g_3 \) | \(=\) | \(18071396/35125\) |
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);
|
\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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Rational points
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) |
\((1 : -2 : 1)\) | \((2 : -20 : 3)\) | \((1 : -21 : 8)\) | \((2 : -25 : 3)\) | \((3 : -92 : 5)\) | \((3 : -108 : 5)\) |
\((1 : -555 : 8)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) |
\((1 : -2 : 1)\) | \((2 : -20 : 3)\) | \((1 : -21 : 8)\) | \((2 : -25 : 3)\) | \((3 : -92 : 5)\) | \((3 : -108 : 5)\) |
\((1 : -555 : 8)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -2 : 0)\) | \((1 : 2 : 0)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((1 : -2 : 1)\) |
\((1 : 2 : 1)\) | \((2 : -5 : 3)\) | \((2 : 5 : 3)\) | \((3 : -16 : 5)\) | \((3 : 16 : 5)\) | \((1 : -534 : 8)\) |
\((1 : 534 : 8)\) |
magma: [C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-555,8],C![1,-21,8],C![1,-2,1],C![1,-1,0],C![1,0,1],C![1,1,0],C![2,-25,3],C![2,-20,3],C![3,-108,5],C![3,-92,5]]; // minimal model
magma: [C![-1,0,1],C![0,-1,1],C![0,1,1],C![1,-534,8],C![1,534,8],C![1,-2,1],C![1,-2,0],C![1,2,1],C![1,2,0],C![2,-5,3],C![2,5,3],C![3,-16,5],C![3,16,5]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.156979\) | \(\infty\) |
\((0 : 0 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2\) | \(0.053729\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.156979\) | \(\infty\) |
\((0 : 0 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2\) | \(0.053729\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) + (1 : 2 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 + z^3\) | \(0.156979\) | \(\infty\) |
\((0 : 1 : 1) + (1 : -2 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-3xz^2 + z^3\) | \(0.053729\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.008416 \) |
Real period: | \( 17.84526 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.600765 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + 2 T^{2} )\) | |
\(5\) | \(2\) | \(3\) | \(2\) | \(1 + 3 T + 5 T^{2}\) | |
\(281\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 20 T + 281 T^{2} )\) |
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);