Properties

Label 100162.a.100162.1
Conductor 100162
Discriminant -100162
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

Learn more about

Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-20, 20, 8, -9, -2, 1], R![1, 0, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-20, 20, 8, -9, -2, 1]), R([1, 0, 0, 1]))

$y^2 + (x^3 + 1)y = x^5 - 2x^4 - 9x^3 + 8x^2 + 20x - 20$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 100162 \)  =  \( 2 \cdot 61 \cdot 821 \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(-100162\)  =  \( -1 \cdot 2 \cdot 61 \cdot 821 \)

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);

\( I_2 \)  =  \(42792\)  =  \( 2^{3} \cdot 3 \cdot 1783 \)
\( I_4 \)  =  \(-42012\)  =  \( -1 \cdot 2^{2} \cdot 3^{3} \cdot 389 \)
\( I_6 \)  =  \(-607587480\)  =  \( -1 \cdot 2^{3} \cdot 3^{5} \cdot 5 \cdot 17 \cdot 3677 \)
\( I_{10} \)  =  \(-410263552\)  =  \( -1 \cdot 2^{13} \cdot 61 \cdot 821 \)
\( J_2 \)  =  \(5349\)  =  \( 3 \cdot 1783 \)
\( J_4 \)  =  \(1192596\)  =  \( 2^{2} \cdot 3 \cdot 23 \cdot 29 \cdot 149 \)
\( J_6 \)  =  \(354674332\)  =  \( 2^{2} \cdot 17 \cdot 5215799 \)
\( J_8 \)  =  \(118716945663\)  =  \( 3 \cdot 37 \cdot 53 \cdot 3407 \cdot 5923 \)
\( J_{10} \)  =  \(-100162\)  =  \( -1 \cdot 2 \cdot 61 \cdot 821 \)
\( g_1 \)  =  \(-4378879451923801749/100162\)
\( g_2 \)  =  \(-91260143303221602/50081\)
\( g_3 \)  =  \(-5073935703495966/50081\)
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![-13,1025,4],C![-13,1108,4],C![1,-1,0],C![1,0,0],C![2,-5,1],C![2,-4,1],C![4,-60,3],C![4,-31,3]];

Known rational points: (-13 : 1025 : 4), (-13 : 1108 : 4), (1 : -1 : 0), (1 : 0 : 0), (2 : -5 : 1), (2 : -4 : 1), (4 : -60 : 3), (4 : -31 : 3)

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

Number of rational Weierstrass points: \(0\)

Invariants of the Jacobian:

Analytic rank*: \(2\)

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

2-Selmer rank: \(2\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*: square

Tamagawa numbers: 1 (p = 2), 1 (p = 61), 1 (p = 821)

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

Torsion: \(\mathrm{trivial}\)

2-torsion field: 6.4.400648.1

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 \(\Q\).