Properties

Label 1077.b.1077.2
Conductor 1077
Discriminant 1077
Mordell-Weil group trivial
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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Minimal equation

Minimal equation

Simplified 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]))
 

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 15, -79, 38, 14, 1], R![1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-3, 60, -316, 152, 56, 4]))
 

$y^2 + y = x^5 + 14x^4 + 38x^3 - 79x^2 + 15x - 1$ (homogenize, simplify)
$y^2 + z^3y = x^5z + 14x^4z^2 + 38x^3z^3 - 79x^2z^4 + 15xz^5 - z^6$ (dehomogenize, simplify)
$y^2 = 4x^5 + 56x^4 + 152x^3 - 316x^2 + 60x - 3$ (minimize, homogenize)

Invariants

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

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);
 

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

Automorphism group

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

Rational points

magma: [C![1,0,0]];
 

Points: \((1 : 0 : 0)\)

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

Number of rational Weierstrass points: \(1\)

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.

Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Group structure: trivial

2-torsion field: 5.1.17232.1

BSD invariants

Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 0.406291 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.406291 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(3\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + T + 3 T^{2} )\)
\(359\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 30 T + 359 T^{2} )\)

Sato-Tate group

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

Decomposition of the Jacobian

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

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

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

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