Properties

Label 601179.a.601179.1
Conductor 601179
Discriminant -601179
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![33, 4, -34, -1, 10, 0, -1], R![1, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([33, 4, -34, -1, 10, 0, -1]), R([1, 0, 1]))

$y^2 + (x^2 + 1)y = -x^6 + 10x^4 - x^3 - 34x^2 + 4x + 33$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 601179 \)  =  \( 3 \cdot 19 \cdot 53 \cdot 199 \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(-601179\)  =  \( -1 \cdot 3 \cdot 19 \cdot 53 \cdot 199 \)

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);

\( I_2 \)  =  \(215680\)  =  \( 2^{7} \cdot 5 \cdot 337 \)
\( I_4 \)  =  \(216064\)  =  \( 2^{10} \cdot 211 \)
\( I_6 \)  =  \(9979699648\)  =  \( 2^{6} \cdot 79 \cdot 179 \cdot 11027 \)
\( I_{10} \)  =  \(-2462429184\)  =  \( -1 \cdot 2^{12} \cdot 3 \cdot 19 \cdot 53 \cdot 199 \)
\( J_2 \)  =  \(26960\)  =  \( 2^{4} \cdot 5 \cdot 337 \)
\( J_4 \)  =  \(30282816\)  =  \( 2^{6} \cdot 3 \cdot 109 \cdot 1447 \)
\( J_6 \)  =  \(45359828977\)  =  \( 45359828977 \)
\( J_8 \)  =  \(76463011082516\)  =  \( 2^{2} \cdot 5827 \cdot 3280547927 \)
\( J_{10} \)  =  \(-601179\)  =  \( -1 \cdot 3 \cdot 19 \cdot 53 \cdot 199 \)
\( g_1 \)  =  \(-14242933261785497600000/601179\)
\( g_2 \)  =  \(-197803816433057792000/200393\)
\( g_3 \)  =  \(-32969410669369043200/601179\)
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![3,-21,2],C![3,-5,2]];

Known rational points: (3 : -21 : 2), (3 : -5 : 2)

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

Number of rational Weierstrass points: \(0\)

Invariants of the Jacobian:

Analytic rank: \(1\)

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

2-Selmer rank: \(2\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*: square

Real period: 2.6186402950677080556807326360

Tamagawa numbers: 1 (p = 3), 1 (p = 19), 1 (p = 53), 1 (p = 199)

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

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

2-torsion field: splitting field of \(x^{6} + 64 x^{4} - 1462 x^{3} - 13568 x^{2} + 69496 x - 3938600\) with Galois group $S_4\times C_2$

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 \(\Q\).