Properties

Label 114240.d.114240.1
Conductor 114240
Discriminant 114240
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-9, 0, 103, 0, -315, 0, 51], R![1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-9, 0, 103, 0, -315, 0, 51]), R([1, 0, 1]))
 

$y^2 + (x^2 + 1)y = 51x^6 - 315x^4 + 103x^2 - 9$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 114240 \)  =  \( 2^{6} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(114240\)  =  \( 2^{6} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \)

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
 

G2 invariants

magma: G2Invariants(C);
 

\( I_2 \)  =  \(10053216\)  =  \( 2^{5} \cdot 3^{2} \cdot 67 \cdot 521 \)
\( I_4 \)  =  \(3341086802112\)  =  \( 2^{6} \cdot 3 \cdot 29 \cdot 600051509 \)
\( I_6 \)  =  \(9160538337022040064\)  =  \( 2^{11} \cdot 3^{2} \cdot 151 \cdot 3847 \cdot 855557891 \)
\( I_{10} \)  =  \(467927040\)  =  \( 2^{18} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \)
\( J_2 \)  =  \(1256652\)  =  \( 2^{2} \cdot 3^{2} \cdot 67 \cdot 521 \)
\( J_4 \)  =  \(30995939524\)  =  \( 2^{2} \cdot 11 \cdot 704453171 \)
\( J_6 \)  =  \(838652756430720\)  =  \( 2^{7} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 17 \cdot 7789 \cdot 42841 \)
\( J_8 \)  =  \(23286599174677950716\)  =  \( 2^{2} \cdot 11 \cdot 1217233 \cdot 434790126733 \)
\( J_{10} \)  =  \(114240\)  =  \( 2^{6} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \)
\( g_1 \)  =  \(16322019979578992366124730896/595\)
\( g_2 \)  =  \(320367650677941067851565476/595\)
\( g_3 \)  =  \(11592952003636922822112\)
Alternative geometric invariants: Igusa-Clebsch, Igusa, G2

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X)\)\(\simeq\) \(V_4 \) (GAP id : [4,2])
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) \(V_4 \) (GAP id : [4,2])

Rational points

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);
 

This curve is locally solvable everywhere.

magma: [];
 

There are no rational points.

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

Number of rational Weierstrass points: \(0\)

Invariants of the Jacobian:

Analytic rank: \(0\)

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);
 

2-Selmer rank: \(6\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: square

Regulator: 1.0

Real period: 0.16518202509712685856962351073

Tamagawa numbers: 1 (p = 2), 1 (p = 3), 1 (p = 5), 1 (p = 7), 1 (p = 17)

magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);
 

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

2-torsion field: \(\Q(\sqrt{2}, \sqrt{1785})\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $G_{3,3}$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 1120.j1
  Elliptic curve 102.c1

Endomorphisms

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 \(\Q\).