# Properties

 Label 36481.a.36481.1 Conductor $36481$ Discriminant $36481$ Mordell-Weil group $$\Z \oplus \Z$$ Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathsf{RM}$$ $$\End(J) \otimes \Q$$ $$\mathsf{RM}$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands: Magma / SageMath

This is a model for the quotient of the modular curve $X_0(191)$ by its Fricke involution involution $w_{191}$; this quotient is also denoted $X_0^+(191)$. This is the largest level $N \in \mathbb{N}$ such that $X_0(N)/ \langle w_N \rangle$ is of genus $2$.

## Simplified equation

 $y^2 + (x^3 + x + 1)y = x^2 - 2x$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = x^2z^4 - 2xz^5$ (dehomogenize, simplify) $y^2 = x^6 + 2x^4 + 2x^3 + 5x^2 - 6x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -2, 1]), R([1, 1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -2, 1], R![1, 1, 0, 1]);

sage: X = HyperellipticCurve(R([1, -6, 5, 2, 2, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$36481$$ $$=$$ $$191^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$36481$$ $$=$$ $$191^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$188$$ $$=$$ $$2^{2} \cdot 47$$ $$I_4$$ $$=$$ $$9409$$ $$=$$ $$97^{2}$$ $$I_6$$ $$=$$ $$508799$$ $$=$$ $$508799$$ $$I_{10}$$ $$=$$ $$-4669568$$ $$=$$ $$- 2^{7} \cdot 191^{2}$$ $$J_2$$ $$=$$ $$47$$ $$=$$ $$47$$ $$J_4$$ $$=$$ $$-300$$ $$=$$ $$- 2^{2} \cdot 3 \cdot 5^{2}$$ $$J_6$$ $$=$$ $$-1708$$ $$=$$ $$- 2^{2} \cdot 7 \cdot 61$$ $$J_8$$ $$=$$ $$-42569$$ $$=$$ $$-42569$$ $$J_{10}$$ $$=$$ $$-36481$$ $$=$$ $$- 191^{2}$$ $$g_1$$ $$=$$ $$-229345007/36481$$ $$g_2$$ $$=$$ $$31146900/36481$$ $$g_3$$ $$=$$ $$3772972/36481$$

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

## Automorphism group

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

## Rational points

All points: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (2 : 0 : 1),\, (2 : -11 : 1)$$
All points: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (2 : 0 : 1),\, (2 : -11 : 1)$$
All points: $$(1 : -1 : 0),\, (1 : 1 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (2 : -11 : 1),\, (2 : 11 : 1)$$

magma: [C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,0,0],C![2,-11,1],C![2,0,1]]; // minimal model

magma: [C![0,-1,1],C![0,1,1],C![1,-1,0],C![1,1,0],C![2,-11,1],C![2,11,1]]; // 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: $$\Z \oplus \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : 0 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.309839$$ $$\infty$$
$$(0 : -1 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.210369$$ $$\infty$$
Generator $D_0$ Height Order
$$(0 : 0 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.309839$$ $$\infty$$
$$(0 : -1 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.210369$$ $$\infty$$
Generator $D_0$ Height Order
$$(0 : 1 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 + xz^2 + z^3$$ $$0.309839$$ $$\infty$$
$$(0 : -1 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 + xz^2 - z^3$$ $$0.210369$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.055286$$ Real period: $$17.35657$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.959581$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$191$$ $$2$$ $$2$$ $$1$$ $$( 1 + T )^{2}$$

## 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?
$$2$$ 2.12.2 no
$$3$$ 3.432.4 no

## 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

Simple over $$\overline{\Q}$$

magma: HeuristicDecompositionFactors(C);

## Endomorphisms of the Jacobian

Of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ $$\Z [\frac{1 + \sqrt{5}}{2}]$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{5})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

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);