Properties

Label 688.a.704512.2
Conductor 688
Discriminant -704512
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

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Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 4, 2, -8, -7, 2], R![]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 4, 2, -8, -7, 2]), R([]))
 

$y^2 = 2x^5 - 7x^4 - 8x^3 + 2x^2 + 4x + 1$

Invariants

magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(688,2),R![1]>*])); Factorization($1);
 
\( N \)  =  \( 688 \)  =  \( 2^{4} \cdot 43 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(-704512\)  =  \( -1 \cdot 2^{14} \cdot 43 \)

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);
 

\( I_2 \)  =  \(14848\)  =  \( 2^{9} \cdot 29 \)
\( I_4 \)  =  \(-253952\)  =  \( -1 \cdot 2^{13} \cdot 31 \)
\( I_6 \)  =  \(-1297678336\)  =  \( -1 \cdot 2^{16} \cdot 19801 \)
\( I_{10} \)  =  \(-2885681152\)  =  \( -1 \cdot 2^{26} \cdot 43 \)
\( J_2 \)  =  \(1856\)  =  \( 2^{6} \cdot 29 \)
\( J_4 \)  =  \(146176\)  =  \( 2^{8} \cdot 571 \)
\( J_6 \)  =  \(15688704\)  =  \( 2^{10} \cdot 3 \cdot 5107 \)
\( J_8 \)  =  \(1937702912\)  =  \( 2^{16} \cdot 29567 \)
\( J_{10} \)  =  \(-704512\)  =  \( -1 \cdot 2^{14} \cdot 43 \)
\( g_1 \)  =  \(-1344218660864/43\)
\( g_2 \)  =  \(-57041383424/43\)
\( g_3 \)  =  \(-3298550016/43\)
Alternative geometric invariants: Igusa-Clebsch, Igusa, G2

Automorphism group

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

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: [C![-1,0,2],C![0,-1,1],C![0,1,1],C![1,0,0]];
 

All rational points: (-1 : 0 : 2), (0 : -1 : 1), (0 : 1 : 1), (1 : 0 : 0)

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

Number of rational Weierstrass points: \(2\)

Invariants of the Jacobian:

Analytic rank: \(0\)

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

2-Selmer rank: \(1\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: square

Regulator: 1.0

Real period: 6.4268245374767942528083517346

Tamagawa numbers: 5 (p = 2), 1 (p = 43)

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

Torsion: \(\Z/{10}\Z\)

2-torsion field: 4.2.688.1

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition

Simple over \(\overline{\Q}\)

Endomorphisms

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

Endomorphism ring over \(\Q\):
\(\End (J_{})\)\(\simeq\)\(\Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).