# Properties

 Label 5329.a.5329.1 Conductor $5329$ Discriminant $5329$ Mordell-Weil group trivial 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

 $y^2 + (x^3 + x + 1)y = -x^6 - 8x^5 - 16x^4 + 25x^3 - 40x^2 + 31x - 8$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = -x^6 - 8x^5z - 16x^4z^2 + 25x^3z^3 - 40x^2z^4 + 31xz^5 - 8z^6$ (dehomogenize, simplify) $y^2 = -3x^6 - 32x^5 - 62x^4 + 102x^3 - 159x^2 + 126x - 31$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-8, 31, -40, 25, -16, -8, -1]), R([1, 1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-8, 31, -40, 25, -16, -8, -1], R![1, 1, 0, 1]);

sage: X = HyperellipticCurve(R([-31, 126, -159, 102, -62, -32, -3]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$5329$$ $$=$$ $$73^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$5329$$ $$=$$ $$73^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$139452$$ $$=$$ $$2^{2} \cdot 3 \cdot 11621$$ $$I_4$$ $$=$$ $$310895097$$ $$=$$ $$3 \cdot 103 \cdot 1006133$$ $$I_6$$ $$=$$ $$16328330150691$$ $$=$$ $$3 \cdot 13 \cdot 257 \cdot 1629086117$$ $$I_{10}$$ $$=$$ $$-682112$$ $$=$$ $$- 2^{7} \cdot 73^{2}$$ $$J_2$$ $$=$$ $$34863$$ $$=$$ $$3 \cdot 11621$$ $$J_4$$ $$=$$ $$37688903$$ $$=$$ $$7 \cdot 37 \cdot 145517$$ $$J_6$$ $$=$$ $$-3247242817$$ $$=$$ $$- 7 \cdot 13 \cdot 379 \cdot 94153$$ $$J_8$$ $$=$$ $$-383415508918120$$ $$=$$ $$- 2^{3} \cdot 5 \cdot 7 \cdot 1369341103279$$ $$J_{10}$$ $$=$$ $$-5329$$ $$=$$ $$- 73^{2}$$ $$g_1$$ $$=$$ $$-51501962646275676450543/5329$$ $$g_2$$ $$=$$ $$-1597010473992743939241/5329$$ $$g_3$$ $$=$$ $$3946792339710402273/5329$$

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

This curve has no rational points.
This curve has no rational points.
This curve has no rational points.

magma: []; // minimal model

magma: []; // 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: trivial

magma: MordellWeilGroupGenus2(Jacobian(C));

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$0.296753$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$1.187013$$ Analytic order of Ш: $$4$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$73$$ $$2$$ $$2$$ $$1$$ $$( 1 - T )^{2}$$

## 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?
$$2$$ 2.12.2 no
$$3$$ 3.2160.23 no

## 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 $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ $$\Z [\frac{1 + \sqrt{13}}{2}]$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{13})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

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);