# 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

# Related objects

Show commands for: SageMath / Magma

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

## 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);

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

## 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 $$\Q$$.