Properties

Label 1077.b.1077.2
Conductor 1077
Discriminant 1077
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

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Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 15, -79, 38, 14, 1], R![1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 15, -79, 38, 14, 1]), R([1]))
 

$y^2 + y = x^5 + 14x^4 + 38x^3 - 79x^2 + 15x - 1$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 1077 \)  =  \( 3 \cdot 359 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(1077\)  =  \( 3 \cdot 359 \)

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);
 

\( I_2 \)  =  \(431360\)  =  \( 2^{8} \cdot 5 \cdot 337 \)
\( I_4 \)  =  \(356510464\)  =  \( 2^{8} \cdot 1392619 \)
\( I_6 \)  =  \(49016221772800\)  =  \( 2^{10} \cdot 5^{2} \cdot 659 \cdot 839 \cdot 3463 \)
\( I_{10} \)  =  \(4411392\)  =  \( 2^{12} \cdot 3 \cdot 359 \)
\( J_2 \)  =  \(53920\)  =  \( 2^{5} \cdot 5 \cdot 337 \)
\( J_4 \)  =  \(117426616\)  =  \( 2^{3} \cdot 17 \cdot 71 \cdot 12161 \)
\( J_6 \)  =  \(333407026000\)  =  \( 2^{4} \cdot 5^{3} \cdot 17 \cdot 29 \cdot 338141 \)
\( J_8 \)  =  \(1047074174177136\)  =  \( 2^{4} \cdot 3 \cdot 7 \cdot 17 \cdot 23833 \cdot 7691491 \)
\( J_{10} \)  =  \(1077\)  =  \( 3 \cdot 359 \)
\( g_1 \)  =  \(455773864377135923200000/1077\)
\( g_2 \)  =  \(18408406506675601408000/1077\)
\( g_3 \)  =  \(969336384916326400000/1077\)
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![1,0,0]];
 

All rational points: (1 : 0 : 0)

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

Number of rational Weierstrass points: \(1\)

Invariants of the Jacobian:

Analytic rank: \(0\)

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

2-Selmer rank: \(0\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: square

Regulator: 1.0

Real period: 0.40629143137139960141171637930

Tamagawa numbers: 1 (p = 3), 1 (p = 359)

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

Torsion: \(\mathrm{trivial}\)

2-torsion field: 5.1.17232.1

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 \(\Q\).