Properties

Label 363.a.43923.1
Conductor 363
Discriminant -43923
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-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![-2, 1, 10, -7, -13, 11], R![0, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-2, 1, 10, -7, -13, 11]), R([0, 0, 1]))
 

$y^2 + x^2y = 11x^5 - 13x^4 - 7x^3 + 10x^2 + x - 2$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 363 \)  =  \( 3 \cdot 11^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(-43923\)  =  \( -1 \cdot 3 \cdot 11^{4} \)

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);
 

\( I_2 \)  =  \(44384\)  =  \( 2^{5} \cdot 19 \cdot 73 \)
\( I_4 \)  =  \(409792\)  =  \( 2^{6} \cdot 19 \cdot 337 \)
\( I_6 \)  =  \(5649542080\)  =  \( 2^{6} \cdot 5 \cdot 7 \cdot 13 \cdot 19 \cdot 10211 \)
\( I_{10} \)  =  \(-179908608\)  =  \( -1 \cdot 2^{12} \cdot 3 \cdot 11^{4} \)
\( J_2 \)  =  \(5548\)  =  \( 2^{2} \cdot 19 \cdot 73 \)
\( J_4 \)  =  \(1278244\)  =  \( 2^{2} \cdot 11^{2} \cdot 19 \cdot 139 \)
\( J_6 \)  =  \(392069161\)  =  \( 11^{2} \cdot 19 \cdot 170539 \)
\( J_8 \)  =  \(135322995423\)  =  \( 3 \cdot 7 \cdot 11^{2} \cdot 19^{2} \cdot 29 \cdot 5087 \)
\( J_{10} \)  =  \(-43923\)  =  \( -1 \cdot 3 \cdot 11^{4} \)
\( g_1 \)  =  \(-5256325630316243968/43923\)
\( g_2 \)  =  \(-1804005053317888/363\)
\( g_3 \)  =  \(-99735603013264/363\)
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,-1,1],C![1,0,0],C![1,0,1]];
 

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

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

Regulator: 1.0

Real period: 3.7941191990697680648528743005

Tamagawa numbers: 1 (p = 3), 5 (p = 11)

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

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

2-torsion field: 6.0.52272.1

Sato-Tate group

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

Decomposition

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 33.a3
  Elliptic curve 11.a2

Endomorphisms

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

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

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).