# Properties

 Label 555.a.8325.1 Conductor $555$ Discriminant $8325$ Mordell-Weil group $$\Z/{2}\Z \oplus \Z/{10}\Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands: Magma / SageMath

## Simplified equation

 $y^2 + (x + 1)y = 3x^5 - 2x^4 - 4x^3 + x^2 + x$ (homogenize, simplify) $y^2 + (xz^2 + z^3)y = 3x^5z - 2x^4z^2 - 4x^3z^3 + x^2z^4 + xz^5$ (dehomogenize, simplify) $y^2 = 12x^5 - 8x^4 - 16x^3 + 5x^2 + 6x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 1, -4, -2, 3]), R([1, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 1, -4, -2, 3], R![1, 1]);

sage: X = HyperellipticCurve(R([1, 6, 5, -16, -8, 12]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$555$$ $$=$$ $$3 \cdot 5 \cdot 37$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$8325$$ $$=$$ $$3^{2} \cdot 5^{2} \cdot 37$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$1264$$ $$=$$ $$2^{4} \cdot 79$$ $$I_4$$ $$=$$ $$18124$$ $$=$$ $$2^{2} \cdot 23 \cdot 197$$ $$I_6$$ $$=$$ $$6869487$$ $$=$$ $$3 \cdot 2289829$$ $$I_{10}$$ $$=$$ $$33300$$ $$=$$ $$2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 37$$ $$J_2$$ $$=$$ $$632$$ $$=$$ $$2^{3} \cdot 79$$ $$J_4$$ $$=$$ $$13622$$ $$=$$ $$2 \cdot 7^{2} \cdot 139$$ $$J_6$$ $$=$$ $$351361$$ $$=$$ $$351361$$ $$J_8$$ $$=$$ $$9125317$$ $$=$$ $$9125317$$ $$J_{10}$$ $$=$$ $$8325$$ $$=$$ $$3^{2} \cdot 5^{2} \cdot 37$$ $$g_1$$ $$=$$ $$100828984082432/8325$$ $$g_2$$ $$=$$ $$3438682756096/8325$$ $$g_3$$ $$=$$ $$140342016064/8325$$

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: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -1 : 1),\, (-1 : -9 : 3)$$
All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -1 : 1),\, (-1 : -9 : 3)$$
All points: $$(1 : 0 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (1 : 0 : 1),\, (-1 : 0 : 3)$$

magma: [C![-1,-9,3],C![0,-1,1],C![0,0,1],C![1,-1,1],C![1,0,0]]; // minimal model

magma: [C![-1,0,3],C![0,-1,1],C![0,1,1],C![1,0,1],C![1,0,0]]; // simplified model

Number of rational Weierstrass points: $$3$$

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: $$\Z/{2}\Z \oplus \Z/{10}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : -9 : 3) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)$$ $$(x - z) (3x + z)$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-xz^2 - z^3$$ $$0$$ $$2$$
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$10$$
Generator $D_0$ Height Order
$$(-1 : -9 : 3) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)$$ $$(x - z) (3x + z)$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-xz^2 - z^3$$ $$0$$ $$2$$
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$10$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$(x - z) (3x + z)$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-xz^2 - z^3$$ $$0$$ $$2$$
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2 - z^3$$ $$0$$ $$10$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$25.69247$$ Tamagawa product: $$4$$ Torsion order: $$20$$ Leading coefficient: $$0.256924$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$3$$ $$1$$ $$2$$ $$2$$ $$( 1 - T )( 1 + T + 3 T^{2} )$$
$$5$$ $$1$$ $$2$$ $$2$$ $$( 1 - T )( 1 + 4 T + 5 T^{2} )$$
$$37$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 2 T + 37 T^{2} )$$

## Galois representations

The mod-$\ell$ Galois representation has maximal image $$\GSp(4,\F_\ell)$$ for all primes $$\ell$$ except those listed.

Prime $$\ell$$ mod-$$\ell$$ image Is torsion prime?
$$2$$ 2.120.3 yes
$$5$$ not computed yes

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{USp}(4)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{USp}(4)$$

## Decomposition of the Jacobian

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

magma: HeuristicDecompositionFactors(C);

## Endomorphisms of the Jacobian

Not of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ $$\Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R$$

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);