# Properties

 Label 704.a.45056.1 Conductor $704$ Discriminant $-45056$ Mordell-Weil group $$\Z/{2}\Z \oplus \Z/{6}\Z$$ 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

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

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

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

sage: X = HyperellipticCurve(R([1, 0, -8, -4, 16, 16]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$704$$ $$=$$ $$2^{6} \cdot 11$$ magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(704,2),R![1, 0, 2]>*])); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-45056$$ $$=$$ $$- 2^{12} \cdot 11$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$134$$ $$=$$ $$2 \cdot 67$$ $$I_4$$ $$=$$ $$-464$$ $$=$$ $$- 2^{4} \cdot 29$$ $$I_6$$ $$=$$ $$-15328$$ $$=$$ $$- 2^{5} \cdot 479$$ $$I_{10}$$ $$=$$ $$-176$$ $$=$$ $$- 2^{4} \cdot 11$$ $$J_2$$ $$=$$ $$268$$ $$=$$ $$2^{2} \cdot 67$$ $$J_4$$ $$=$$ $$4230$$ $$=$$ $$2 \cdot 3^{2} \cdot 5 \cdot 47$$ $$J_6$$ $$=$$ $$61444$$ $$=$$ $$2^{2} \cdot 15361$$ $$J_8$$ $$=$$ $$-356477$$ $$=$$ $$- 11 \cdot 23 \cdot 1409$$ $$J_{10}$$ $$=$$ $$-45056$$ $$=$$ $$- 2^{12} \cdot 11$$ $$g_1$$ $$=$$ $$-1350125107/44$$ $$g_2$$ $$=$$ $$-636113745/352$$ $$g_3$$ $$=$$ $$-68955529/704$$

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: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (-1 : -4 : 2),\, (1 : -4 : 2)$$
All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (-1 : -4 : 2),\, (1 : -4 : 2)$$
All points: $$(1 : 0 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (-1 : 0 : 2),\, (1 : 0 : 2)$$

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

magma: [C![-1,0,2],C![0,-1,1],C![0,1,1],C![1,0,2],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: $$\Z/{2}\Z \oplus \Z/{6}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

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

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$11.97602$$ Tamagawa product: $$4$$ Torsion order: $$12$$ Leading coefficient: $$0.332667$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$6$$ $$12$$ $$4$$ $$1 + 2 T^{2}$$
$$11$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 11 T^{2} )$$

## 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
$$3$$ 3.80.1 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);