Minimal equation
Minimal equation
Simplified equation
$y^2 + y = x^6 - x^5 + 2x^3 - x^2 - x$ | (homogenize, simplify) |
$y^2 + z^3y = x^6 - x^5z + 2x^3z^3 - x^2z^4 - xz^5$ | (dehomogenize, simplify) |
$y^2 = 4x^6 - 4x^5 + 8x^3 - 4x^2 - 4x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -1, 2, 0, -1, 1]), R([1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -1, 2, 0, -1, 1], R![1]);
sage: X = HyperellipticCurve(R([1, -4, -4, 8, 0, -4, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(85403\) | \(=\) | \( 41 \cdot 2083 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-85403\) | \(=\) | \( - 41 \cdot 2083 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(16\) | \(=\) | \( 2^{4} \) |
\( I_4 \) | \(=\) | \(-1676\) | \(=\) | \( - 2^{2} \cdot 419 \) |
\( I_6 \) | \(=\) | \(-416552\) | \(=\) | \( - 2^{3} \cdot 52069 \) |
\( I_{10} \) | \(=\) | \(-341612\) | \(=\) | \( - 2^{2} \cdot 41 \cdot 2083 \) |
\( J_2 \) | \(=\) | \(8\) | \(=\) | \( 2^{3} \) |
\( J_4 \) | \(=\) | \(282\) | \(=\) | \( 2 \cdot 3 \cdot 47 \) |
\( J_6 \) | \(=\) | \(45664\) | \(=\) | \( 2^{5} \cdot 1427 \) |
\( J_8 \) | \(=\) | \(71447\) | \(=\) | \( 37 \cdot 1931 \) |
\( J_{10} \) | \(=\) | \(-85403\) | \(=\) | \( - 41 \cdot 2083 \) |
\( g_1 \) | \(=\) | \(-32768/85403\) | ||
\( g_2 \) | \(=\) | \(-144384/85403\) | ||
\( g_3 \) | \(=\) | \(-2922496/85403\) |
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 | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) |
\((-1 : -1 : 1)\) | \((1 : -1 : 1)\) | \((-2 : -5 : 3)\) | \((2 : 6 : 1)\) | \((2 : -7 : 1)\) | \((-1 : 16 : 5)\) |
\((-2 : -22 : 3)\) | \((-5 : -57 : 8)\) | \((-1 : -141 : 5)\) | \((2 : -377 : 11)\) | \((-5 : -455 : 8)\) | \((2 : -954 : 11)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) |
\((-1 : -1 : 1)\) | \((1 : -1 : 1)\) | \((-2 : -5 : 3)\) | \((2 : 6 : 1)\) | \((2 : -7 : 1)\) | \((-1 : 16 : 5)\) |
\((-2 : -22 : 3)\) | \((-5 : -57 : 8)\) | \((-1 : -141 : 5)\) | \((2 : -377 : 11)\) | \((-5 : -455 : 8)\) | \((2 : -954 : 11)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -2 : 0)\) | \((1 : 2 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : -1 : 1)\) | \((1 : 1 : 1)\) | \((2 : -13 : 1)\) | \((2 : 13 : 1)\) | \((-2 : -17 : 3)\) | \((-2 : 17 : 3)\) |
\((-1 : -157 : 5)\) | \((-1 : 157 : 5)\) | \((-5 : -398 : 8)\) | \((-5 : 398 : 8)\) | \((2 : -577 : 11)\) | \((2 : 577 : 11)\) |
magma: [C![-5,-455,8],C![-5,-57,8],C![-2,-22,3],C![-2,-5,3],C![-1,-141,5],C![-1,-1,1],C![-1,0,1],C![-1,16,5],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,-1,1],C![1,0,1],C![1,1,0],C![2,-954,11],C![2,-377,11],C![2,-7,1],C![2,6,1]]; // minimal model
magma: [C![-5,-398,8],C![-5,398,8],C![-2,-17,3],C![-2,17,3],C![-1,-157,5],C![-1,-1,1],C![-1,1,1],C![-1,157,5],C![0,-1,1],C![0,1,1],C![1,-2,0],C![1,-1,1],C![1,1,1],C![1,2,0],C![2,-577,11],C![2,577,11],C![2,-13,1],C![2,13,1]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -1 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3\) | \(0.516673\) | \(\infty\) |
\((-1 : 0 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.328783\) | \(\infty\) |
\((-1 : 0 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 - z^3\) | \(0.329904\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -1 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3\) | \(0.516673\) | \(\infty\) |
\((-1 : 0 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.328783\) | \(\infty\) |
\((-1 : 0 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2 - z^3\) | \(0.329904\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -1 : 1) - (1 : -2 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2x^3 + z^3\) | \(0.516673\) | \(\infty\) |
\((-1 : 1 : 1) + (1 : 1 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(z^3\) | \(0.328783\) | \(\infty\) |
\((-1 : 1 : 1) + (0 : -1 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2 - z^3\) | \(0.329904\) | \(\infty\) |
2-torsion field: 6.4.5465792.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.049862 \) |
Real period: | \( 16.94122 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.844730 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(41\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 6 T + 41 T^{2} )\) | |
\(2083\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 68 T + 2083 T^{2} )\) |
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);