Properties

Label 1369.a.50653.1
Conductor 1369
Discriminant 50653
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

Learn more about

Show commands for: Magma / SageMath

This is a model for the modular curve $X_0(37)$.

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 3, -6, 7, -5, 2], R![0, 0, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 3, -6, 7, -5, 2]), R([0, 0, 0, 1]))

$y^2 + x^3y = 2x^5 - 5x^4 + 7x^3 - 6x^2 + 3x - 1$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 1369 \)  =  \( 37^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(50653\)  =  \( 37^{3} \)

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);

\( I_2 \)  =  \(1824\)  =  \( 2^{5} \cdot 3 \cdot 19 \)
\( I_4 \)  =  \(179520\)  =  \( 2^{6} \cdot 3 \cdot 5 \cdot 11 \cdot 17 \)
\( I_6 \)  =  \(140795904\)  =  \( 2^{13} \cdot 3 \cdot 17 \cdot 337 \)
\( I_{10} \)  =  \(207474688\)  =  \( 2^{12} \cdot 37^{3} \)
\( J_2 \)  =  \(228\)  =  \( 2^{2} \cdot 3 \cdot 19 \)
\( J_4 \)  =  \(296\)  =  \( 2^{3} \cdot 37 \)
\( J_6 \)  =  \(-98568\)  =  \( -1 \cdot 2^{3} \cdot 3^{2} \cdot 37^{2} \)
\( J_8 \)  =  \(-5640280\)  =  \( -1 \cdot 2^{3} \cdot 5 \cdot 37^{2} \cdot 103 \)
\( J_{10} \)  =  \(50653\)  =  \( 37^{3} \)
\( g_1 \)  =  \(616132666368/50653\)
\( g_2 \)  =  \(94818816/1369\)
\( g_3 \)  =  \(-3742848/37\)
Alternative geometric invariants: Igusa-Clebsch, Igusa, G2

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
\(\mathrm{Aut}(X)\)\(\simeq\) \(V_4 \) (GAP id : [4,2])
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) \(V_4 \) (GAP id : [4,2])

Rational points

magma: f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsLocallyEverywhere(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);

This curve is locally solvable everywhere.

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

All rational points:

(1 : -1 : 0), (1 : -1 : 1), (1 : 0 : 0), (1 : 0 : 1)

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:

\(1\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*:

square

Tamagawa numbers:

3 (p = 37)

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

Torsion:

\(\Z/{3}\Z\)

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 37.b2
  Elliptic curve 37.a1

Endomorphisms

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

Endomorphism ring over \(\Q\):
\(\End (J_{})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

All endomorphisms of the Jacobian are defined over \(\Q\)