Properties

Label 65520.b.131040.1
Conductor 65520
Discriminant -131040
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

Learn more about

Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-210, 0, -227, 0, -82, 0, -10], R![0, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-210, 0, -227, 0, -82, 0, -10]), R([0, 1, 0, 1]))
 

$y^2 + (x^3 + x)y = -10x^6 - 82x^4 - 227x^2 - 210$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 65520 \)  =  \( 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(-131040\)  =  \( -1 \cdot 2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)

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);
 

\( I_2 \)  =  \(-12593312\)  =  \( -1 \cdot 2^{5} \cdot 393541 \)
\( I_4 \)  =  \(11343232\)  =  \( 2^{7} \cdot 23 \cdot 3853 \)
\( I_6 \)  =  \(-47604155826432\)  =  \( -1 \cdot 2^{8} \cdot 3 \cdot 97 \cdot 9431 \cdot 67757 \)
\( I_{10} \)  =  \(-536739840\)  =  \( -1 \cdot 2^{17} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)
\( J_2 \)  =  \(-1574164\)  =  \( -1 \cdot 2^{2} \cdot 393541 \)
\( J_4 \)  =  \(103249560962\)  =  \( 2 \cdot 17 \cdot 83 \cdot 5927 \cdot 6173 \)
\( J_6 \)  =  \(-9029520569946240\)  =  \( -1 \cdot 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \cdot 31 \cdot 555698369 \)
\( J_8 \)  =  \(888368594905774644479\)  =  \( 888368594905774644479 \)
\( J_{10} \)  =  \(-131040\)  =  \( -1 \cdot 2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)
\( g_1 \)  =  \(302064649214662608101539958432/4095\)
\( g_2 \)  =  \(12586012647194024913614166004/4095\)
\( g_3 \)  =  \(170750018582492394877376\)
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 except over $\R$.

magma: [];
 

There are no rational points.

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

Number of rational Weierstrass points: \(0\)

Invariants of the Jacobian:

Analytic rank: \(1\)

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

2-Selmer rank: \(6\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: twice a square

Regulator: 0.532646580956

Real period: 3.1861498316533743187900576499

Tamagawa numbers: 2 (p = 2), 1 (p = 3), 1 (p = 5), 1 (p = 7), 1 (p = 13)

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

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

2-torsion field: splitting field of \(x^{8} - 4 x^{7} + 466 x^{6} - 1384 x^{5} + 79827 x^{4} - 157352 x^{3} + 5948566 x^{2} - 5870120 x + 163148164\) with Galois group $C_2^3$

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 312.f2
  Elliptic curve 210.a2

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 \(\Q\).