Minimal equation
Minimal equation
Simplified equation
$y^2 + y = x^6 - x^4$ | (homogenize, simplify) |
$y^2 + z^3y = x^6 - x^4z^2$ | (dehomogenize, simplify) |
$y^2 = 4x^6 - 4x^4 + 1$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 0, 0, -1, 0, 1]), R([1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 0, 0, -1, 0, 1], R![1]);
sage: X = HyperellipticCurve(R([1, 0, 0, 0, -4, 0, 4]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(968\) | \(=\) | \( 2^{3} \cdot 11^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-1936\) | \(=\) | \( - 2^{4} \cdot 11^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(120\) | \(=\) | \( 2^{3} \cdot 3 \cdot 5 \) |
\( I_4 \) | \(=\) | \(357\) | \(=\) | \( 3 \cdot 7 \cdot 17 \) |
\( I_6 \) | \(=\) | \(14937\) | \(=\) | \( 3 \cdot 13 \cdot 383 \) |
\( I_{10} \) | \(=\) | \(242\) | \(=\) | \( 2 \cdot 11^{2} \) |
\( J_2 \) | \(=\) | \(120\) | \(=\) | \( 2^{3} \cdot 3 \cdot 5 \) |
\( J_4 \) | \(=\) | \(362\) | \(=\) | \( 2 \cdot 181 \) |
\( J_6 \) | \(=\) | \(-1344\) | \(=\) | \( - 2^{6} \cdot 3 \cdot 7 \) |
\( J_8 \) | \(=\) | \(-73081\) | \(=\) | \( - 107 \cdot 683 \) |
\( J_{10} \) | \(=\) | \(1936\) | \(=\) | \( 2^{4} \cdot 11^{2} \) |
\( g_1 \) | \(=\) | \(1555200000/121\) | ||
\( g_2 \) | \(=\) | \(39096000/121\) | ||
\( g_3 \) | \(=\) | \(-1209600/121\) |
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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Rational points
All points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) |
\((-1 : -1 : 1)\) | \((1 : -1 : 1)\) |
All points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) |
\((-1 : -1 : 1)\) | \((1 : -1 : 1)\) |
All points | |||||
---|---|---|---|---|---|
\((1 : -2 : 0)\) | \((1 : 2 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : -1 : 1)\) | \((1 : 1 : 1)\) |
magma: [C![-1,-1,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,-1,1],C![1,0,1],C![1,1,0]]; // minimal model
magma: [C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-2,0],C![1,-1,1],C![1,1,1],C![1,2,0]]; // 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: \(\Z \oplus \Z/{5}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1 : 1) - (1 : 1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3\) | \(0.080528\) | \(\infty\) |
\(2 \cdot(0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(5\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1 : 1) - (1 : 1 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3\) | \(0.080528\) | \(\infty\) |
\(2 \cdot(0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(5\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -1 : 1) - (1 : 2 : 0)\) | \(z (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2x^3 + z^3\) | \(0.080528\) | \(\infty\) |
\(2 \cdot(0 : 1 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(z^3\) | \(0\) | \(5\) |
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 0.080528 \) |
Real period: | \( 26.98675 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 5 \) |
Leading coefficient: | \( 0.173856 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(3\) | \(4\) | \(2\) | \(1 + 2 T + 2 T^{2}\) | |
\(11\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )( 1 + T )\) |
Galois representations
For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.
For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.
Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
---|---|---|
\(2\) | 2.60.2 | no |
\(3\) | 3.90.1 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $\mathrm{SU}(2)\times\mathrm{SU}(2)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over \(\Q\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 88.a
Elliptic curve isogeny class 11.a
magma: HeuristicDecompositionFactors(C);
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
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);