Properties

Label 830.a.830000.1
Conductor 830
Discriminant -830000
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![0, 1, 8, 16, -2, 1], R![0, 1, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 8, 16, -2, 1]), R([0, 1, 1]))

$y^2 + (x^2 + x)y = x^5 - 2x^4 + 16x^3 + 8x^2 + x$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 830 \)  =  \( 2 \cdot 5 \cdot 83 \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(-830000\)  =  \( -1 \cdot 2^{4} \cdot 5^{4} \cdot 83 \)

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);

\( I_2 \)  =  \(30472\)  =  \( 2^{3} \cdot 13 \cdot 293 \)
\( I_4 \)  =  \(-917948\)  =  \( -1 \cdot 2^{2} \cdot 229487 \)
\( I_6 \)  =  \(-9181166648\)  =  \( -1 \cdot 2^{3} \cdot 5779 \cdot 198589 \)
\( I_{10} \)  =  \(-3399680000\)  =  \( -1 \cdot 2^{16} \cdot 5^{4} \cdot 83 \)
\( J_2 \)  =  \(3809\)  =  \( 13 \cdot 293 \)
\( J_4 \)  =  \(614082\)  =  \( 2 \cdot 3 \cdot 7 \cdot 14621 \)
\( J_6 \)  =  \(133745600\)  =  \( 2^{6} \cdot 5^{2} \cdot 83591 \)
\( J_8 \)  =  \(33085071919\)  =  \( 33085071919 \)
\( J_{10} \)  =  \(-830000\)  =  \( -1 \cdot 2^{4} \cdot 5^{4} \cdot 83 \)
\( g_1 \)  =  \(-801779343712318049/830000\)
\( g_2 \)  =  \(-16967946642572289/415000\)
\( g_3 \)  =  \(-4851113741084/2075\)
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![-1,6,4],C![0,0,1],C![1,-6,1],C![1,0,0],C![1,4,1]];

All rational points: (-1 : 6 : 4), (0 : 0 : 1), (1 : -6 : 1), (1 : 0 : 0), (1 : 4 : 1)

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

Number of rational Weierstrass points: \(3\)

Invariants of the Jacobian:

Analytic rank: \(0\)

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

2-Selmer rank: \(2\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*: square

Tamagawa numbers: 4 (p = 2), 4 (p = 5), 1 (p = 83)

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

Torsion: \(\Z/{2}\Z \times \Z/{8}\Z\)

2-torsion field: 3.1.83.1

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 \(\Q\).