Properties

Label 576.a.576.1
Conductor 576
Discriminant -576
Sato-Tate group $E_2$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Show commands for: Magma / SageMath

Minimal equation

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

$y^2 + (x^3 + x^2 + x + 1)y = -x^3 - x$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 576 \)  =  \( 2^{6} \cdot 3^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(-576\)  =  \( -1 \cdot 2^{6} \cdot 3^{2} \)

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);

\( I_2 \)  =  \(-544\)  =  \( -1 \cdot 2^{5} \cdot 17 \)
\( I_4 \)  =  \(7936\)  =  \( 2^{8} \cdot 31 \)
\( I_6 \)  =  \(-1339392\)  =  \( -1 \cdot 2^{12} \cdot 3 \cdot 109 \)
\( I_{10} \)  =  \(-2359296\)  =  \( -1 \cdot 2^{18} \cdot 3^{2} \)
\( J_2 \)  =  \(-68\)  =  \( -1 \cdot 2^{2} \cdot 17 \)
\( J_4 \)  =  \(110\)  =  \( 2 \cdot 5 \cdot 11 \)
\( J_6 \)  =  \(36\)  =  \( 2^{2} \cdot 3^{2} \)
\( J_8 \)  =  \(-3637\)  =  \( -1 \cdot 3637 \)
\( J_{10} \)  =  \(-576\)  =  \( -1 \cdot 2^{6} \cdot 3^{2} \)
\( g_1 \)  =  \(22717712/9\)
\( g_2 \)  =  \(540430/9\)
\( g_3 \)  =  \(-289\)
Alternative geometric invariants: Igusa-Clebsch, Igusa, G2

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
\(\mathrm{Aut}(X)\)\(\simeq\) \(C_4 \) (GAP id : [4,1])
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) \(D_4 \) (GAP id : [8,3])

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![0,-1,1],C![0,0,1],C![1,-1,0],C![1,0,0]];

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

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

Number of rational Weierstrass points: \(0\)

Invariants of the Jacobian:

Analytic rank: \(0\)

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

2-Selmer rank: \(1\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*: square

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

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

Torsion: \(\Z/{10}\Z\)

2-torsion field: \(\Q(\zeta_{12})\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $E_2$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\)

Decomposition

Splits over the number field \(\Q (b) \simeq \) \(\Q(\sqrt{2}) \) with defining polynomial:
  \(x^{2} - 2\)

Decomposes up to isogeny as the square of the elliptic curve:
  Elliptic curve 2.2.8.1-9.1-a3

Endomorphisms

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):
\(\End (J_{})\)\(\simeq\)\(\Z [\sqrt{-1}]\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-1}) \)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\C\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{2}) \) with defining polynomial \(x^{2} - 2\)

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

Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\)\(\simeq\)a non-Eichler order of index \(4\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)