Properties

Label 810.a.196830.1
Conductor 810
Discriminant -196830
Sato-Tate group $N(G_{1,3})$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\C \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{CM} \times \Q\)
\(\overline{\Q}\)-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![-8, 94, -297, 20, 15, 1], R![1, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-8, 94, -297, 20, 15, 1]), R([1, 1]))

$y^2 + (x + 1)y = x^5 + 15x^4 + 20x^3 - 297x^2 + 94x - 8$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 810 \)  =  \( 2 \cdot 3^{4} \cdot 5 \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(-196830\)  =  \( -1 \cdot 2 \cdot 3^{9} \cdot 5 \)

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);

\( I_2 \)  =  \(1238400\)  =  \( 2^{7} \cdot 3^{2} \cdot 5^{2} \cdot 43 \)
\( I_4 \)  =  \(13269432960\)  =  \( 2^{7} \cdot 3^{6} \cdot 5 \cdot 7 \cdot 17 \cdot 239 \)
\( I_6 \)  =  \(4967784069073344\)  =  \( 2^{6} \cdot 3^{5} \cdot 17 \cdot 18790032941 \)
\( I_{10} \)  =  \(-806215680\)  =  \( -1 \cdot 2^{13} \cdot 3^{9} \cdot 5 \)
\( J_2 \)  =  \(154800\)  =  \( 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 43 \)
\( J_4 \)  =  \(860236740\)  =  \( 2^{2} \cdot 3^{3} \cdot 5 \cdot 11 \cdot 97 \cdot 1493 \)
\( J_6 \)  =  \(5905731060081\)  =  \( 3^{4} \cdot 31 \cdot 2351943871 \)
\( J_8 \)  =  \(43549979813677800\)  =  \( 2^{3} \cdot 3^{10} \cdot 5^{2} \cdot 7753 \cdot 475637 \)
\( J_{10} \)  =  \(-196830\)  =  \( -1 \cdot 2 \cdot 3^{9} \cdot 5 \)
\( g_1 \)  =  \(-451609936896000000000\)
\( g_2 \)  =  \(-16212110811776000000\)
\( g_3 \)  =  \(-2156977131869584000/3\)
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![1,0,0]];

All rational points: (1 : 0 : 0)

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

Number of rational Weierstrass points: \(1\)

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), 4 (p = 3), 1 (p = 5)

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

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

2-torsion field: 6.2.186624000.3

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $N(G_{1,3})$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{U}(1)\times\mathrm{SU}(2)\)

Decomposition

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 30.a1
  Elliptic curve 27.a1

Endomorphisms

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

Endomorphism ring over \(\Q\):
\(\End (J_{})\)\(\simeq\)an order of index \(3\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-3}) \) with defining polynomial \(x^{2} - x + 1\)

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

Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\)\(\simeq\)an order of index \(27\) in \(\Z \times \Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q(\sqrt{-3}) \)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\R \times \C\)