Properties

Label 20736.l.373248.1
Conductor 20736
Discriminant 373248
Sato-Tate group $J(E_4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{QM}\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -3, -1, 9, 6], R![1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -3, -1, 9, 6]), R([1]))

$y^2 + y = 6x^5 + 9x^4 - x^3 - 3x^2$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 20736 \)  =  \( 2^{8} \cdot 3^{4} \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(373248\)  =  \( 2^{9} \cdot 3^{6} \)

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);

\( I_2 \)  =  \(7008\)  =  \( 2^{5} \cdot 3 \cdot 73 \)
\( I_4 \)  =  \(1700352\)  =  \( 2^{9} \cdot 3^{4} \cdot 41 \)
\( I_6 \)  =  \(3259367424\)  =  \( 2^{17} \cdot 3^{4} \cdot 307 \)
\( I_{10} \)  =  \(1528823808\)  =  \( 2^{21} \cdot 3^{6} \)
\( J_2 \)  =  \(876\)  =  \( 2^{2} \cdot 3 \cdot 73 \)
\( J_4 \)  =  \(14262\)  =  \( 2 \cdot 3 \cdot 2377 \)
\( J_6 \)  =  \(207364\)  =  \( 2^{2} \cdot 47 \cdot 1103 \)
\( J_8 \)  =  \(-5438445\)  =  \( -1 \cdot 3 \cdot 5 \cdot 37 \cdot 41 \cdot 239 \)
\( J_{10} \)  =  \(373248\)  =  \( 2^{9} \cdot 3^{6} \)
\( g_1 \)  =  \(4146143186/3\)
\( g_2 \)  =  \(924693409/36\)
\( g_3 \)  =  \(276260689/648\)
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,-4,2],C![0,-1,1],C![0,0,1],C![1,0,0]];

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

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

Number of rational Weierstrass points: \(2\)

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

Real period: 18.600159265208624917879119180

Tamagawa numbers: 2 (p = 2), 4 (p = 3)

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

Torsion: \(\Z/{2}\Z \times \Z/{6}\Z\)

2-torsion field: \(\Q(\sqrt{2}, \sqrt{3})\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $J(E_4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\)

Decomposition

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

Endomorphisms

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

Endomorphism ring over \(\Q\):
\(\End (J_{})\)\(\simeq\)\(\Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 8.0.339738624.10 with defining polynomial \(x^{8} + 4 x^{6} + 10 x^{4} + 24 x^{2} + 36\)

Not of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\)\(\simeq\)a non-Eichler order of index \(6\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)the quaternion algebra over \(\Q\) of discriminant 6
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)

Remainder of the endomorphism lattice by field

Over subfield \(F \simeq \) \(\Q(\sqrt{-3}) \) with generator \(-\frac{1}{15} a^{6} - \frac{1}{6} a^{4} - \frac{2}{3} a^{2} - \frac{3}{5}\) with minimal polynomial \(x^{2} - x + 1\):
\(\End (J_{F})\)\(\simeq\)\(\Z [3\sqrt{-1}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-1}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C\)
Sato Tate group: $E_4$
of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{-2}) \) with generator \(-\frac{1}{10} a^{6} - \frac{2}{5}\) with minimal polynomial \(x^{2} + 2\):
\(\End (J_{F})\)\(\simeq\)\(\Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R\)
Sato Tate group: $J(E_2)$
not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{6}) \) with generator \(-\frac{1}{6} a^{6} - \frac{2}{3} a^{4} - \frac{2}{3} a^{2} - 2\) with minimal polynomial \(x^{2} - 6\):
\(\End (J_{F})\)\(\simeq\)\(\Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R\)
Sato Tate group: $J(E_2)$
not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{-2}, \sqrt{-3})\) with generator \(-\frac{2}{15} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{6}{5}\) with minimal polynomial \(x^{4} - 2 x^{2} + 4\):
\(\End (J_{F})\)\(\simeq\)\(\Z [3\sqrt{-1}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-1}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C\)
Sato Tate group: $E_2$
of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 4.0.6144.1 with generator \(\frac{2}{45} a^{7} - \frac{1}{18} a^{5} + \frac{1}{9} a^{3} + \frac{11}{15} a\) with minimal polynomial \(x^{4} - 4 x^{2} + 6\):
\(\End (J_{F})\)\(\simeq\)\(\Z [\sqrt{6}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{6}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
Sato Tate group: $J(E_1)$
of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 4.2.18432.2 with generator \(\frac{7}{90} a^{7} + \frac{5}{18} a^{5} + \frac{4}{9} a^{3} + \frac{23}{15} a\) with minimal polynomial \(x^{4} + 4 x^{2} - 2\):
\(\End (J_{F})\)\(\simeq\)\(\Z [\sqrt{3}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
Sato Tate group: $J(E_1)$
of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 4.2.18432.2 with generator \(\frac{1}{18} a^{7} + \frac{1}{18} a^{5} - \frac{1}{9} a^{3} - \frac{1}{3} a\) with minimal polynomial \(x^{4} + 4 x^{2} - 2\):
\(\End (J_{F})\)\(\simeq\)\(\Z [\sqrt{3}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
Sato Tate group: $J(E_1)$
of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 4.0.6144.1 with generator \(-\frac{1}{15} a^{7} - \frac{1}{6} a^{5} - \frac{2}{3} a^{3} - \frac{3}{5} a\) with minimal polynomial \(x^{4} - 4 x^{2} + 6\):
\(\End (J_{F})\)\(\simeq\)\(\Z [\sqrt{6}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{6}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
Sato Tate group: $J(E_1)$
of \(\GL_2\)-type, simple