# Properties

 Label 604.a.9664.1 Conductor $604$ Discriminant $9664$ Mordell-Weil group trivial 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

 $y^2 + (x^2 + x + 1)y = 4x^5 + 9x^4 + 48x^3 - 4x^2 - 53x - 21$ (homogenize, simplify) $y^2 + (x^2z + xz^2 + z^3)y = 4x^5z + 9x^4z^2 + 48x^3z^3 - 4x^2z^4 - 53xz^5 - 21z^6$ (dehomogenize, simplify) $y^2 = 16x^5 + 37x^4 + 194x^3 - 13x^2 - 210x - 83$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-21, -53, -4, 48, 9, 4]), R([1, 1, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-21, -53, -4, 48, 9, 4], R![1, 1, 1]);

sage: X = HyperellipticCurve(R([-83, -210, -13, 194, 37, 16]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$604$$ $$=$$ $$2^{2} \cdot 151$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$9664$$ $$=$$ $$2^{6} \cdot 151$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$49556$$ $$=$$ $$2^{2} \cdot 13 \cdot 953$$ $$I_4$$ $$=$$ $$-797087975$$ $$=$$ $$- 5^{2} \cdot 457 \cdot 69767$$ $$I_6$$ $$=$$ $$-23996873337603$$ $$=$$ $$- 3 \cdot 20747 \cdot 385547683$$ $$I_{10}$$ $$=$$ $$1236992$$ $$=$$ $$2^{13} \cdot 151$$ $$J_2$$ $$=$$ $$12389$$ $$=$$ $$13 \cdot 953$$ $$J_4$$ $$=$$ $$39607304$$ $$=$$ $$2^{3} \cdot 11 \cdot 450083$$ $$J_6$$ $$=$$ $$223396249616$$ $$=$$ $$2^{4} \cdot 10289 \cdot 1357009$$ $$J_8$$ $$=$$ $$299729401586052$$ $$=$$ $$2^{2} \cdot 3 \cdot 2971 \cdot 8407085201$$ $$J_{10}$$ $$=$$ $$9664$$ $$=$$ $$2^{6} \cdot 151$$ $$g_1$$ $$=$$ $$291864493641401980949/9664$$ $$g_2$$ $$=$$ $$9414430497536890397/1208$$ $$g_3$$ $$=$$ $$2143030742187944921/604$$

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: $$(1 : 0 : 0)$$
All points: $$(1 : 0 : 0)$$
All points: $$(1 : 0 : 0)$$

magma: [C![1,0,0]]; // minimal model

magma: [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: trivial

magma: MordellWeilGroupGenus2(Jacobian(C));

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$0.291788$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.291788$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$2$$ $$6$$ $$1$$ $$( 1 - T )( 1 + T )$$
$$151$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 10 T + 151 T^{2} )$$

## 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?
$$2$$ 2.6.1 no
$$3$$ 3.720.5 no

## 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);