# Properties

 Label 708.a.2832.1 Conductor $708$ Discriminant $2832$ Mordell-Weil group $$\Z/{10}\Z$$ 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

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

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

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

sage: X = HyperellipticCurve(R([1, 2, 3, 2, 1, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$708$$ $$=$$ $$2^{2} \cdot 3 \cdot 59$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$2832$$ $$=$$ $$2^{4} \cdot 3 \cdot 59$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$148$$ $$=$$ $$2^{2} \cdot 37$$ $$I_4$$ $$=$$ $$2065$$ $$=$$ $$5 \cdot 7 \cdot 59$$ $$I_6$$ $$=$$ $$76361$$ $$=$$ $$19 \cdot 4019$$ $$I_{10}$$ $$=$$ $$362496$$ $$=$$ $$2^{11} \cdot 3 \cdot 59$$ $$J_2$$ $$=$$ $$37$$ $$=$$ $$37$$ $$J_4$$ $$=$$ $$-29$$ $$=$$ $$-29$$ $$J_6$$ $$=$$ $$-59$$ $$=$$ $$-59$$ $$J_8$$ $$=$$ $$-756$$ $$=$$ $$- 2^{2} \cdot 3^{3} \cdot 7$$ $$J_{10}$$ $$=$$ $$2832$$ $$=$$ $$2^{4} \cdot 3 \cdot 59$$ $$g_1$$ $$=$$ $$69343957/2832$$ $$g_2$$ $$=$$ $$-1468937/2832$$ $$g_3$$ $$=$$ $$-1369/48$$

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

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

magma: [C![0,-1,1],C![0,1,1],C![1,0,0]]; // 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: $$\Z/{10}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 - xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-3xz^2 - 2z^3$$ $$0$$ $$10$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 - xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-3xz^2 - 2z^3$$ $$0$$ $$10$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 - xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^2z - 5xz^2 - 3z^3$$ $$0$$ $$10$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$16.26718$$ Tamagawa product: $$2$$ Torsion order: $$10$$ Leading coefficient: $$0.325343$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$2$$ $$4$$ $$2$$ $$1 + T^{2}$$
$$3$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + T + 3 T^{2} )$$
$$59$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 59 T^{2} )$$

## 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?
$$2$$ 2.60.1 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);