# Properties

 Label 35344.a.565504.1 Conductor $35344$ Discriminant $565504$ 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

## Simplified equation

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

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

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

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

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$35344$$ $$=$$ $$2^{4} \cdot 47^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$565504$$ $$=$$ $$2^{8} \cdot 47^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$58$$ $$=$$ $$2 \cdot 29$$ $$I_4$$ $$=$$ $$64$$ $$=$$ $$2^{6}$$ $$I_6$$ $$=$$ $$562$$ $$=$$ $$2 \cdot 281$$ $$I_{10}$$ $$=$$ $$-2209$$ $$=$$ $$- 47^{2}$$ $$J_2$$ $$=$$ $$116$$ $$=$$ $$2^{2} \cdot 29$$ $$J_4$$ $$=$$ $$390$$ $$=$$ $$2 \cdot 3 \cdot 5 \cdot 13$$ $$J_6$$ $$=$$ $$5116$$ $$=$$ $$2^{2} \cdot 1279$$ $$J_8$$ $$=$$ $$110339$$ $$=$$ $$110339$$ $$J_{10}$$ $$=$$ $$-565504$$ $$=$$ $$- 2^{8} \cdot 47^{2}$$ $$g_1$$ $$=$$ $$-82044596/2209$$ $$g_2$$ $$=$$ $$-4755855/4418$$ $$g_3$$ $$=$$ $$-1075639/8836$$

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
$$(1 : 0 : 0)$$ $$(0 : -1 : 1)$$ $$(0 : 1 : 1)$$ $$(-1 : -1 : 1)$$ $$(-1 : 1 : 1)$$ $$(1 : -1 : 1)$$
$$(1 : 1 : 1)$$ $$(2 : -5 : 1)$$ $$(2 : 5 : 1)$$ $$(4 : -29 : 1)$$ $$(4 : 29 : 1)$$
All points
$$(1 : 0 : 0)$$ $$(0 : -1 : 1)$$ $$(0 : 1 : 1)$$ $$(-1 : -1 : 1)$$ $$(-1 : 1 : 1)$$ $$(1 : -1 : 1)$$
$$(1 : 1 : 1)$$ $$(2 : -5 : 1)$$ $$(2 : 5 : 1)$$ $$(4 : -29 : 1)$$ $$(4 : 29 : 1)$$
All points
$$(1 : 0 : 0)$$ $$(0 : -1/2 : 1)$$ $$(0 : 1/2 : 1)$$ $$(-1 : -1/2 : 1)$$ $$(-1 : 1/2 : 1)$$ $$(1 : -1/2 : 1)$$
$$(1 : 1/2 : 1)$$ $$(2 : -5/2 : 1)$$ $$(2 : 5/2 : 1)$$ $$(4 : -29/2 : 1)$$ $$(4 : 29/2 : 1)$$

magma: [C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-1,1],C![1,0,0],C![1,1,1],C![2,-5,1],C![2,5,1],C![4,-29,1],C![4,29,1]]; // minimal model

magma: [C![-1,-1/2,1],C![-1,1/2,1],C![0,-1/2,1],C![0,1/2,1],C![1,-1/2,1],C![1,0,0],C![1,1/2,1],C![2,-5/2,1],C![2,5/2,1],C![4,-29/2,1],C![4,29/2,1]]; // simplified model

Number of rational Weierstrass points: $$1$$

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
$$(0 : 1 : 1) + (1 : 1 : 1) - 2 \cdot(1 : 0 : 0)$$ $$x (x - z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$z^3$$ $$0.127053$$ $$\infty$$
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.096317$$ $$\infty$$
Generator $D_0$ Height Order
$$(0 : 1 : 1) + (1 : 1 : 1) - 2 \cdot(1 : 0 : 0)$$ $$x (x - z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$z^3$$ $$0.127053$$ $$\infty$$
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.096317$$ $$\infty$$
Generator $D_0$ Height Order
$$(0 : 1/2 : 1) + (1 : 1/2 : 1) - 2 \cdot(1 : 0 : 0)$$ $$x (x - z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$1/2z^3$$ $$0.127053$$ $$\infty$$
$$(0 : -1/2 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-1/2z^3$$ $$0.096317$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.011292$$ Real period: $$11.51929$$ Tamagawa product: $$9$$ Torsion order: $$1$$ Leading coefficient: $$1.170768$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$4$$ $$8$$ $$9$$ $$1$$
$$47$$ $$2$$ $$2$$ $$1$$ $$( 1 - T )^{2}$$

## 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?
$$2$$ 2.72.1 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);

The Jacobian of this curve is an optimal quotient of $J_0(188)$. The set of rational points on this genus 2 curve with rank 2 Jacobian was computed by Balakrishnan--Dogra--MÃ¼ller--Tuitman--Vonk using quadratic Chabauty and the Mordell--Weil sieve.