Minimal equation
Minimal equation
Simplified equation
$y^2 + y = x^6 - x^5 - 3x^4 + x^3 + 3x^2 + x$ | (homogenize, simplify) |
$y^2 + z^3y = x^6 - x^5z - 3x^4z^2 + x^3z^3 + 3x^2z^4 + xz^5$ | (dehomogenize, simplify) |
$y^2 = 4x^6 - 4x^5 - 12x^4 + 4x^3 + 12x^2 + 4x + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 3, 1, -3, -1, 1]), R([1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 3, 1, -3, -1, 1], R![1]);
sage: X = HyperellipticCurve(R([1, 4, 12, 4, -12, -4, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(54983\) | \(=\) | \( 54983 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-54983\) | \(=\) | \( -54983 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(200\) | \(=\) | \( 2^{3} \cdot 5^{2} \) |
\( I_4 \) | \(=\) | \(21076\) | \(=\) | \( 2^{2} \cdot 11 \cdot 479 \) |
\( I_6 \) | \(=\) | \(648368\) | \(=\) | \( 2^{4} \cdot 7^{2} \cdot 827 \) |
\( I_{10} \) | \(=\) | \(-219932\) | \(=\) | \( - 2^{2} \cdot 54983 \) |
\( J_2 \) | \(=\) | \(100\) | \(=\) | \( 2^{2} \cdot 5^{2} \) |
\( J_4 \) | \(=\) | \(-3096\) | \(=\) | \( - 2^{3} \cdot 3^{2} \cdot 43 \) |
\( J_6 \) | \(=\) | \(27848\) | \(=\) | \( 2^{3} \cdot 59^{2} \) |
\( J_8 \) | \(=\) | \(-1700104\) | \(=\) | \( - 2^{3} \cdot 7^{2} \cdot 4337 \) |
\( J_{10} \) | \(=\) | \(-54983\) | \(=\) | \( -54983 \) |
\( g_1 \) | \(=\) | \(-10000000000/54983\) | ||
\( g_2 \) | \(=\) | \(3096000000/54983\) | ||
\( g_3 \) | \(=\) | \(-278480000/54983\) |
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 | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -1 : 1)\) |
\((1 : 1 : 1)\) | \((1 : -2 : 1)\) | \((2 : 2 : 1)\) | \((-1 : -2 : 3)\) | \((2 : -3 : 1)\) | \((-4 : -4 : 5)\) |
\((-1 : -25 : 3)\) | \((-4 : -121 : 5)\) | \((5 : 3721 : 24)\) | \((5 : -17545 : 24)\) | \((54 : 129735 : 11)\) | \((54 : -131066 : 11)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -1 : 1)\) |
\((1 : 1 : 1)\) | \((1 : -2 : 1)\) | \((2 : 2 : 1)\) | \((-1 : -2 : 3)\) | \((2 : -3 : 1)\) | \((-4 : -4 : 5)\) |
\((-1 : -25 : 3)\) | \((-4 : -121 : 5)\) | \((5 : 3721 : 24)\) | \((5 : -17545 : 24)\) | \((54 : 129735 : 11)\) | \((54 : -131066 : 11)\) |
Known points | |||||
---|---|---|---|---|---|
\((1 : -2 : 0)\) | \((1 : 2 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : -3 : 1)\) | \((1 : 3 : 1)\) | \((2 : -5 : 1)\) | \((2 : 5 : 1)\) | \((-1 : -23 : 3)\) | \((-1 : 23 : 3)\) |
\((-4 : -117 : 5)\) | \((-4 : 117 : 5)\) | \((5 : -21266 : 24)\) | \((5 : 21266 : 24)\) | \((54 : -260801 : 11)\) | \((54 : 260801 : 11)\) |
magma: [C![-4,-121,5],C![-4,-4,5],C![-1,-25,3],C![-1,-2,3],C![-1,-1,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,1,0],C![1,1,1],C![2,-3,1],C![2,2,1],C![5,-17545,24],C![5,3721,24],C![54,-131066,11],C![54,129735,11]]; // minimal model
magma: [C![-4,-117,5],C![-4,117,5],C![-1,-23,3],C![-1,23,3],C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-3,1],C![1,-2,0],C![1,2,0],C![1,3,1],C![2,-5,1],C![2,5,1],C![5,-21266,24],C![5,21266,24],C![54,-260801,11],C![54,260801,11]]; // 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 | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -2 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 3z^3\) | \(0.628570\) | \(\infty\) |
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - z^3\) | \(0.432242\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 - xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.145844\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -2 : 1) - (1 : -1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 3z^3\) | \(0.628570\) | \(\infty\) |
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - z^3\) | \(0.432242\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 - xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.145844\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -3 : 1) - (1 : -2 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2x^3 - 5z^3\) | \(0.628570\) | \(\infty\) |
\((0 : -1 : 1) - (1 : -2 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(2x^3 - z^3\) | \(0.432242\) | \(\infty\) |
\(D_0 - (1 : -2 : 0) - (1 : 2 : 0)\) | \(x^2 - xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.145844\) | \(\infty\) |
2-torsion field: 6.0.3518912.1
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(3\) (upper bound) |
Mordell-Weil rank: | \(3\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.032929 \) |
Real period: | \( 21.51843 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.708582 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(54983\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 176 T + 54983 T^{2} )\) |
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);