Minimal equation
Minimal equation
Simplified equation
$y^2 + x^2y = 11x^5 - 13x^4 - 7x^3 + 10x^2 + x - 2$ | (homogenize, simplify) |
$y^2 + x^2zy = 11x^5z - 13x^4z^2 - 7x^3z^3 + 10x^2z^4 + xz^5 - 2z^6$ | (dehomogenize, simplify) |
$y^2 = 44x^5 - 51x^4 - 28x^3 + 40x^2 + 4x - 8$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-2, 1, 10, -7, -13, 11]), R([0, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-2, 1, 10, -7, -13, 11], R![0, 0, 1]);
sage: X = HyperellipticCurve(R([-8, 4, 40, -28, -51, 44]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(363\) | \(=\) | \( 3 \cdot 11^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(-43923\) | \(=\) | \( - 3 \cdot 11^{4} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(11096\) | \(=\) | \( 2^{3} \cdot 19 \cdot 73 \) |
\( I_4 \) | \(=\) | \(25612\) | \(=\) | \( 2^{2} \cdot 19 \cdot 337 \) |
\( I_6 \) | \(=\) | \(88274095\) | \(=\) | \( 5 \cdot 7 \cdot 13 \cdot 19 \cdot 10211 \) |
\( I_{10} \) | \(=\) | \(-175692\) | \(=\) | \( - 2^{2} \cdot 3 \cdot 11^{4} \) |
\( J_2 \) | \(=\) | \(5548\) | \(=\) | \( 2^{2} \cdot 19 \cdot 73 \) |
\( J_4 \) | \(=\) | \(1278244\) | \(=\) | \( 2^{2} \cdot 11^{2} \cdot 19 \cdot 139 \) |
\( J_6 \) | \(=\) | \(392069161\) | \(=\) | \( 11^{2} \cdot 19 \cdot 170539 \) |
\( J_8 \) | \(=\) | \(135322995423\) | \(=\) | \( 3 \cdot 7 \cdot 11^{2} \cdot 19^{2} \cdot 29 \cdot 5087 \) |
\( J_{10} \) | \(=\) | \(-43923\) | \(=\) | \( - 3 \cdot 11^{4} \) |
\( g_1 \) | \(=\) | \(-5256325630316243968/43923\) | ||
\( g_2 \) | \(=\) | \(-1804005053317888/363\) | ||
\( g_3 \) | \(=\) | \(-99735603013264/363\) |
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
All points: \((1 : 0 : 0),\, (1 : 0 : 1),\, (1 : -1 : 1)\)
magma: [C![1,-1,1],C![1,0,0],C![1,0,1]]; // minimal model
magma: [C![1,-1,1],C![1,0,0],C![1,1,1]]; // 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: \(\Z/{10}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(11x^2 - 3xz - 6z^2\) | \(=\) | \(0,\) | \(11y\) | \(=\) | \(-2xz^2 - 4z^3\) | \(0\) | \(10\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(11x^2 - 3xz - 6z^2\) | \(=\) | \(0,\) | \(11y\) | \(=\) | \(-2xz^2 - 4z^3\) | \(0\) | \(10\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(11x^2 - 3xz - 6z^2\) | \(=\) | \(0,\) | \(11y\) | \(=\) | \(x^2z - 4xz^2 - 8z^3\) | \(0\) | \(10\) |
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 1 \) |
Real period: | \( 3.794119 \) |
Tamagawa product: | \( 5 \) |
Torsion order: | \( 10 \) |
Leading coefficient: | \( 0.189705 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + T + 3 T^{2} )\) | |
\(11\) | \(2\) | \(4\) | \(5\) | \(( 1 - T )^{2}\) |
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? |
---|---|---|
\(2\) | 2.60.1 | yes |
\(3\) | 3.80.4 | 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 33.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 \(3\) 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);