# Properties

 Label 5547.b.16641.1 Conductor $5547$ Discriminant $16641$ Mordell-Weil group $$\Z \oplus \Z$$ Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands: Magma / SageMath

This is a model for the quotient of the modular curve $X_0(129)$ by the action of the group of order 4 generated by the Atkin-Lehner involutions $w_3$ and $w_{43}$. This quotient is also denoted $X_0^*(129)$.

## Simplified equation

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -1, 3, 1, -3, 1]), R([1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -1, 3, 1, -3, 1], R![1]);

sage: X = HyperellipticCurve(R([1, -4, -4, 12, 4, -12, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$5547$$ $$=$$ $$3 \cdot 43^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$16641$$ $$=$$ $$3^{2} \cdot 43^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$520$$ $$=$$ $$2^{3} \cdot 5 \cdot 13$$ $$I_4$$ $$=$$ $$6292$$ $$=$$ $$2^{2} \cdot 11^{2} \cdot 13$$ $$I_6$$ $$=$$ $$896816$$ $$=$$ $$2^{4} \cdot 23 \cdot 2437$$ $$I_{10}$$ $$=$$ $$66564$$ $$=$$ $$2^{2} \cdot 3^{2} \cdot 43^{2}$$ $$J_2$$ $$=$$ $$260$$ $$=$$ $$2^{2} \cdot 5 \cdot 13$$ $$J_4$$ $$=$$ $$1768$$ $$=$$ $$2^{3} \cdot 13 \cdot 17$$ $$J_6$$ $$=$$ $$16776$$ $$=$$ $$2^{3} \cdot 3^{2} \cdot 233$$ $$J_8$$ $$=$$ $$308984$$ $$=$$ $$2^{3} \cdot 13 \cdot 2971$$ $$J_{10}$$ $$=$$ $$16641$$ $$=$$ $$3^{2} \cdot 43^{2}$$ $$g_1$$ $$=$$ $$1188137600000/16641$$ $$g_2$$ $$=$$ $$31074368000/16641$$ $$g_3$$ $$=$$ $$126006400/1849$$

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
$$(1 : -1 : 0)$$ $$(1 : 1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : 0 : 1)$$ $$(-1 : 1 : 1)$$
$$(1 : -1 : 1)$$ $$(-1 : -2 : 1)$$ $$(2 : 1 : 1)$$ $$(2 : -2 : 1)$$ $$(-5 : 20 : 7)$$ $$(12 : 20 : 7)$$
$$(-5 : -363 : 7)$$ $$(12 : -363 : 7)$$
All points
$$(1 : -1 : 0)$$ $$(1 : 1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : 0 : 1)$$ $$(-1 : 1 : 1)$$
$$(1 : -1 : 1)$$ $$(-1 : -2 : 1)$$ $$(2 : 1 : 1)$$ $$(2 : -2 : 1)$$ $$(-5 : 20 : 7)$$ $$(12 : 20 : 7)$$
$$(-5 : -363 : 7)$$ $$(12 : -363 : 7)$$
All points
$$(1 : -2 : 0)$$ $$(1 : 2 : 0)$$ $$(0 : -1 : 1)$$ $$(0 : 1 : 1)$$ $$(1 : -1 : 1)$$ $$(1 : 1 : 1)$$
$$(-1 : -3 : 1)$$ $$(-1 : 3 : 1)$$ $$(2 : -3 : 1)$$ $$(2 : 3 : 1)$$ $$(-5 : -383 : 7)$$ $$(-5 : 383 : 7)$$
$$(12 : -383 : 7)$$ $$(12 : 383 : 7)$$

magma: [C![-5,-363,7],C![-5,20,7],C![-1,-2,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,-1,1],C![1,0,1],C![1,1,0],C![2,-2,1],C![2,1,1],C![12,-363,7],C![12,20,7]]; // minimal model

magma: [C![-5,-383,7],C![-5,383,7],C![-1,-3,1],C![-1,3,1],C![0,-1,1],C![0,1,1],C![1,-2,0],C![1,-1,1],C![1,1,1],C![1,2,0],C![2,-3,1],C![2,3,1],C![12,-383,7],C![12,383,7]]; // 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
$$(0 : 0 : 1) + (2 : -2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x (x - 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2$$ $$0.081387$$ $$\infty$$
$$(-1 : -2 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$(x - z) (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2 - z^3$$ $$0.081387$$ $$\infty$$
Generator $D_0$ Height Order
$$(0 : 0 : 1) + (2 : -2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x (x - 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2$$ $$0.081387$$ $$\infty$$
$$(-1 : -2 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$(x - z) (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2 - z^3$$ $$0.081387$$ $$\infty$$
Generator $D_0$ Height Order
$$(0 : 1 : 1) + (2 : -3 : 1) - (1 : -2 : 0) - (1 : 2 : 0)$$ $$x (x - 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2xz^2 + z^3$$ $$0.081387$$ $$\infty$$
$$(-1 : -3 : 1) + (1 : 1 : 1) - (1 : -2 : 0) - (1 : 2 : 0)$$ $$(x - z) (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$2xz^2 - z^3$$ $$0.081387$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.006279$$ Real period: $$24.46037$$ Tamagawa product: $$2$$ Torsion order: $$1$$ Leading coefficient: $$0.307177$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$3$$ $$1$$ $$2$$ $$2$$ $$( 1 + T )( 1 + 2 T + 3 T^{2} )$$
$$43$$ $$2$$ $$2$$ $$1$$ $$( 1 + T )^{2}$$

## 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?
$$2$$ 2.30.2 no
$$3$$ 3.90.1 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

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 129.a
Elliptic curve isogeny class 43.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);