Properties

 Label 1369.a.50653.1 Conductor $1369$ Discriminant $50653$ Mordell-Weil group $$\Z \oplus \Z/{3}\Z$$ Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

Related objects

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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) $y^2 = x^6 + 8x^5 - 20x^4 + 28x^3 - 24x^2 + 12x - 4$ (homogenize, minimize)

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

 $$I_2$$ $$=$$ $$456$$ $$=$$ $$2^{3} \cdot 3 \cdot 19$$ $$I_4$$ $$=$$ $$11220$$ $$=$$ $$2^{2} \cdot 3 \cdot 5 \cdot 11 \cdot 17$$ $$I_6$$ $$=$$ $$2199936$$ $$=$$ $$2^{7} \cdot 3 \cdot 17 \cdot 337$$ $$I_{10}$$ $$=$$ $$202612$$ $$=$$ $$2^{2} \cdot 37^{3}$$ $$J_2$$ $$=$$ $$228$$ $$=$$ $$2^{2} \cdot 3 \cdot 19$$ $$J_4$$ $$=$$ $$296$$ $$=$$ $$2^{3} \cdot 37$$ $$J_6$$ $$=$$ $$-98568$$ $$=$$ $$- 2^{3} \cdot 3^{2} \cdot 37^{2}$$ $$J_8$$ $$=$$ $$-5640280$$ $$=$$ $$- 2^{3} \cdot 5 \cdot 37^{2} \cdot 103$$ $$J_{10}$$ $$=$$ $$50653$$ $$=$$ $$37^{3}$$ $$g_1$$ $$=$$ $$616132666368/50653$$ $$g_2$$ $$=$$ $$94818816/1369$$ $$g_3$$ $$=$$ $$-3742848/37$$

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

magma: [C![1,-1,0],C![1,-1,1],C![1,0,0],C![1,0,1]]; // minimal model

magma: [C![1,-1,0],C![1,-1,1],C![1,1,0],C![1,1,1]]; // simplified model

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: $$\Z \oplus \Z/{3}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

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

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 Cluster picture
$$37$$ $$2$$ $$3$$ $$3$$ $$( 1 - T )( 1 + T )$$

Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.

Prime $$\ell$$ mod-$$\ell$$ image Is torsion prime?
$$2$$ 2.30.4 no
$$3$$ 3.2160.20 yes

Sato-Tate group

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

Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 37.b
Elliptic curve isogeny class 37.a

magma: HeuristicDecompositionFactors(C);

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 $$\Q$$.

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);