Properties

Label 100162.b.400648.1
Conductor 100162
Discriminant -400648
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![2, 5, 4, 0, -4, 1], R![0, 1, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 5, 4, 0, -4, 1]), R([0, 1, 1]))

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

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 100162 \)  =  \( 2 \cdot 61 \cdot 821 \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(-400648\)  =  \( -1 \cdot 2^{3} \cdot 61 \cdot 821 \)

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);

\( I_2 \)  =  \(7304\)  =  \( 2^{3} \cdot 11 \cdot 83 \)
\( I_4 \)  =  \(-928124\)  =  \( -1 \cdot 2^{2} \cdot 331 \cdot 701 \)
\( I_6 \)  =  \(-2399465848\)  =  \( -1 \cdot 2^{3} \cdot 13 \cdot 23071787 \)
\( I_{10} \)  =  \(-1641054208\)  =  \( -1 \cdot 2^{15} \cdot 61 \cdot 821 \)
\( J_2 \)  =  \(913\)  =  \( 11 \cdot 83 \)
\( J_4 \)  =  \(44400\)  =  \( 2^{4} \cdot 3 \cdot 5^{2} \cdot 37 \)
\( J_6 \)  =  \(3475524\)  =  \( 2^{2} \cdot 3 \cdot 13 \cdot 22279 \)
\( J_8 \)  =  \(300448353\)  =  \( 3 \cdot 43 \cdot 293 \cdot 7949 \)
\( J_{10} \)  =  \(-400648\)  =  \( -1 \cdot 2^{3} \cdot 61 \cdot 821 \)
\( g_1 \)  =  \(-634386434595793/400648\)
\( g_2 \)  =  \(-4223819158350/50081\)
\( g_3 \)  =  \(-724272266289/100162\)
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,-4,1],C![1,0,0],C![1,2,1]];

All rational points: (1 : -4 : 1), (1 : 0 : 0), (1 : 2 : 1)

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

Number of rational Weierstrass points: \(1\)

Invariants of the Jacobian:

Analytic rank: \(1\)

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

2-Selmer rank: \(1\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*: square

Tamagawa numbers: 3 (p = 2), 1 (p = 61), 1 (p = 821)

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

Torsion: \(\Z/{3}\Z\)

2-torsion field: 5.3.1602592.1

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 \(\Q\).