Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x^2 + x + 1)y = -12x^6 - 15x^5 + 9x^4 + 31x^3 + 9x^2 - 15x - 12$ | (homogenize, simplify) |
$y^2 + (x^3 + x^2z + xz^2 + z^3)y = -12x^6 - 15x^5z + 9x^4z^2 + 31x^3z^3 + 9x^2z^4 - 15xz^5 - 12z^6$ | (dehomogenize, simplify) |
$y^2 = -47x^6 - 58x^5 + 39x^4 + 128x^3 + 39x^2 - 58x - 47$ | (homogenize, minimize) |
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-12, -15, 9, 31, 9, -15, -12]), R([1, 1, 1, 1]));
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-12, -15, 9, 31, 9, -15, -12], R![1, 1, 1, 1]);
sage: X = HyperellipticCurve(R([-47, -58, 39, 128, 39, -58, -47]))
magma: X,pi:= SimplifiedModel(C);
Invariants
Conductor: | \( N \) | \(=\) | \(847\) | \(=\) | \( 7 \cdot 11^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-847\) | \(=\) | \( - 7 \cdot 11^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(80408\) | \(=\) | \( 2^{3} \cdot 19 \cdot 23^{2} \) |
\( I_4 \) | \(=\) | \(402403732\) | \(=\) | \( 2^{2} \cdot 2441 \cdot 41213 \) |
\( I_6 \) | \(=\) | \(8094753026048\) | \(=\) | \( 2^{12} \cdot 13 \cdot 67 \cdot 499 \cdot 4547 \) |
\( I_{10} \) | \(=\) | \(3388\) | \(=\) | \( 2^{2} \cdot 7 \cdot 11^{2} \) |
\( J_2 \) | \(=\) | \(40204\) | \(=\) | \( 2^{2} \cdot 19 \cdot 23^{2} \) |
\( J_4 \) | \(=\) | \(281112\) | \(=\) | \( 2^{3} \cdot 3 \cdot 13 \cdot 17 \cdot 53 \) |
\( J_6 \) | \(=\) | \(1967560\) | \(=\) | \( 2^{3} \cdot 5 \cdot 7 \cdot 7027 \) |
\( J_8 \) | \(=\) | \(19956424\) | \(=\) | \( 2^{3} \cdot 2494553 \) |
\( J_{10} \) | \(=\) | \(847\) | \(=\) | \( 7 \cdot 11^{2} \) |
\( g_1 \) | \(=\) | \(105037970421355597057024/847\) | ||
\( g_2 \) | \(=\) | \(18267839107785466368/847\) | ||
\( g_3 \) | \(=\) | \(454326923025280/121\) |
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
This curve has no rational points.
magma: []; // minimal model
magma: []; // simplified model
Number of rational Weierstrass points: \(0\)
magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
This curve is locally solvable except over $\R$.
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: \(\Z/{3}\Z\)
magma: MordellWeilGroupGenus2(Jacobian(C));
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(18x^2 + 29xz + 18z^2\) | \(=\) | \(0,\) | \(162y\) | \(=\) | \(-77xz^2 - 126z^3\) | \(0\) | \(3\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(18x^2 + 29xz + 18z^2\) | \(=\) | \(0,\) | \(162y\) | \(=\) | \(-77xz^2 - 126z^3\) | \(0\) | \(3\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(18x^2 + 29xz + 18z^2\) | \(=\) | \(0,\) | \(162y\) | \(=\) | \(x^3 + x^2z - 153xz^2 - 251z^3\) | \(0\) | \(3\) |
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 1 \) |
Real period: | \( 1.179534 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 3 \) |
Leading coefficient: | \( 0.262118 \) |
Analytic order of Ш: | \( 2 \) (rounded) |
Order of Ш: | twice a square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 2 T + 7 T^{2} )\) | |
\(11\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )( 1 + T )\) |
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? |
---|---|---|
\(2\) | 2.15.2 | no |
\(3\) | 3.720.4 | yes |
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 77.b
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);