Properties

Label 9792.a.9792.1
Conductor $9792$
Discriminant $-9792$
Mordell-Weil group \(\Z/{2}\Z\)
Sato-Tate group $N(\mathrm{SU}(2)\times\mathrm{SU}(2))$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type no

Related objects

Downloads

Learn more

Show commands: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

$y^2 + y = -6x^6 - 4x^5 - 4x^4 - 5x^3 - 2x^2 - x - 1$ (homogenize, simplify)
$y^2 + z^3y = -6x^6 - 4x^5z - 4x^4z^2 - 5x^3z^3 - 2x^2z^4 - xz^5 - z^6$ (dehomogenize, simplify)
$y^2 = -24x^6 - 16x^5 - 16x^4 - 20x^3 - 8x^2 - 4x - 3$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, -1, -2, -5, -4, -4, -6]), R([1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, -1, -2, -5, -4, -4, -6], R![1]);
 
sage: X = HyperellipticCurve(R([-3, -4, -8, -20, -16, -16, -24]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(9792\) \(=\) \( 2^{6} \cdot 3^{2} \cdot 17 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-9792\) \(=\) \( - 2^{6} \cdot 3^{2} \cdot 17 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1796\) \(=\)  \( 2^{2} \cdot 449 \)
\( I_4 \)  \(=\) \(158332\) \(=\)  \( 2^{2} \cdot 23 \cdot 1721 \)
\( I_6 \)  \(=\) \(73791160\) \(=\)  \( 2^{3} \cdot 5 \cdot 31 \cdot 59509 \)
\( I_{10} \)  \(=\) \(1224\) \(=\)  \( 2^{3} \cdot 3^{2} \cdot 17 \)
\( J_2 \)  \(=\) \(1796\) \(=\)  \( 2^{2} \cdot 449 \)
\( J_4 \)  \(=\) \(28846\) \(=\)  \( 2 \cdot 14423 \)
\( J_6 \)  \(=\) \(478108\) \(=\)  \( 2^{2} \cdot 17 \cdot 79 \cdot 89 \)
\( J_8 \)  \(=\) \(6647563\) \(=\)  \( 13 \cdot 511351 \)
\( J_{10} \)  \(=\) \(9792\) \(=\)  \( 2^{6} \cdot 3^{2} \cdot 17 \)
\( g_1 \)  \(=\) \(291979047635984/153\)
\( g_2 \)  \(=\) \(2611106718254/153\)
\( g_3 \)  \(=\) \(1417456631/9\)

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);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

This curve has no rational points.
This curve has no 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 $\R$ and $\Q_{2}$.

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: \(\Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(6x^2 - 2xz + 3z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(6x^2 - 2xz + 3z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(6x^2 - 2xz + 3z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-z^3\) \(0\) \(2\)

2-torsion field: 8.0.1731891456.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 3.636835 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 1.818417 \)
Analytic order of Ш: \( 2 \)   (rounded)
Order of Ш:twice a square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(6\) \(6\) \(1\) \(1 - 2 T + 2 T^{2}\)
\(3\) \(2\) \(2\) \(1\) \(1 + T^{2}\)
\(17\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 6 T + 17 T^{2} )\)

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?
\(2\) 2.90.5 yes
\(3\) 3.45.1 no

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $N(\mathrm{SU}(2)\times\mathrm{SU}(2))$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\sqrt{2}) \) with defining polynomial:
  \(x^{2} - 2\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 2.2.8.1-153.2-a
  Elliptic curve isogeny class 2.2.8.1-153.1-a

magma: HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{2}) \) with defining polynomial \(x^{2} - 2\)

Of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):

\(\End (J_{\overline{\Q}})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\R \times \R\)

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);