Properties

Label 1369.a.50653.1
Conductor 1369
Discriminant 50653
Mordell-Weil group \(\Z \times \Z/{3}\Z\)
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

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

Minimal equation

Minimal equation

Simplified equation

$y^2 + x^3y = 2x^5 - 5x^4 + 7x^3 - 6x^2 + 3x - 1$ (homogenize, simplify)
$y^2 + x^3y = 2x^5z - 5x^4z^2 + 7x^3z^3 - 6x^2z^4 + 3xz^5 - z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 8x^5 - 20x^4 + 28x^3 - 24x^2 + 12x - 4$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 3, -6, 7, -5, 2]), R([0, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 3, -6, 7, -5, 2], R![0, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([-4, 12, -24, 28, -20, 8, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

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

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

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

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

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

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

Number of rational Weierstrass points: \(0\)

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

This curve is locally solvable everywhere.

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);
 

Mordell-Weil group of the Jacobian

Group structure: \(\Z \times \Z/{3}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(z^3\) \(0.102222\) \(\infty\)
\((1 : -1 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(3\)

2-torsion field: 4.0.592.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.102222 \)
Real period: \( 6.516888 \)
Tamagawa product: \( 3 \)
Torsion order:\( 3 \)
Leading coefficient: \( 0.222058 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(37\) \(2\) \(3\) \(3\) \(( 1 - T )( 1 + T )\)

Sato-Tate group

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

Decomposition of the Jacobian

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 the Jacobian

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