# Properties

 Label 4225.a.274625.1 Conductor $4225$ Discriminant $-274625$ Mordell-Weil group $$\Z/{6}\Z$$ 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

 $y^2 + (x^3 + 1)y = x^5 + 4x^4 + 6x^3 + 10x^2 + 3x + 1$ (homogenize, simplify) $y^2 + (x^3 + z^3)y = x^5z + 4x^4z^2 + 6x^3z^3 + 10x^2z^4 + 3xz^5 + z^6$ (dehomogenize, simplify) $y^2 = x^6 + 4x^5 + 16x^4 + 26x^3 + 40x^2 + 12x + 5$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 3, 10, 6, 4, 1]), R([1, 0, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 3, 10, 6, 4, 1], R![1, 0, 0, 1]);

sage: X = HyperellipticCurve(R([5, 12, 40, 26, 16, 4, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$4225$$ $$=$$ $$5^{2} \cdot 13^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-274625$$ $$=$$ $$- 5^{3} \cdot 13^{3}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$2732$$ $$=$$ $$2^{2} \cdot 683$$ $$I_4$$ $$=$$ $$286321$$ $$=$$ $$7 \cdot 40903$$ $$I_6$$ $$=$$ $$209098567$$ $$=$$ $$209098567$$ $$I_{10}$$ $$=$$ $$35152000$$ $$=$$ $$2^{7} \cdot 5^{3} \cdot 13^{3}$$ $$J_2$$ $$=$$ $$683$$ $$=$$ $$683$$ $$J_4$$ $$=$$ $$7507$$ $$=$$ $$7507$$ $$J_6$$ $$=$$ $$96775$$ $$=$$ $$5^{2} \cdot 7^{2} \cdot 79$$ $$J_8$$ $$=$$ $$2435569$$ $$=$$ $$2435569$$ $$J_{10}$$ $$=$$ $$274625$$ $$=$$ $$5^{3} \cdot 13^{3}$$ $$g_1$$ $$=$$ $$148628987203643/274625$$ $$g_2$$ $$=$$ $$2391820186409/274625$$ $$g_3$$ $$=$$ $$1805778919/10985$$

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

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

magma: [C![1,-1,0],C![1,1,0]]; // 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: $$\Z/{6}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$2x^2 + 2xz + z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-4xz^2 - z^3$$ $$0$$ $$6$$
Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$2x^2 + 2xz + z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-4xz^2 - z^3$$ $$0$$ $$6$$
Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$2x^2 + 2xz + z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$x^3 - 8xz^2 - z^3$$ $$0$$ $$6$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$8.210257$$ Tamagawa product: $$4$$ Torsion order: $$6$$ Leading coefficient: $$0.912250$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$5$$ $$2$$ $$3$$ $$2$$ $$( 1 + T )^{2}$$
$$13$$ $$2$$ $$3$$ $$2$$ $$( 1 - T )^{2}$$

## 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?
$$2$$ 2.90.1 yes
$$3$$ 3.1920.1 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

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

magma: HeuristicDecompositionFactors(C);

## Endomorphisms of the Jacobian

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

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ $$\Z [\sqrt{3}]$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{3})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

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);