# Properties

 Label 600.a.96000.1 Conductor $600$ Discriminant $96000$ Mordell-Weil group $$\Z/{2}\Z \oplus \Z/{6}\Z$$ Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands: Magma / SageMath

## Simplified equation

 $y^2 + (x + 1)y = 4x^5 + 5x^4 + 3x^3 + 2x^2$ (homogenize, simplify) $y^2 + (xz^2 + z^3)y = 4x^5z + 5x^4z^2 + 3x^3z^3 + 2x^2z^4$ (dehomogenize, simplify) $y^2 = 16x^5 + 20x^4 + 12x^3 + 9x^2 + 2x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 2, 3, 5, 4]), R([1, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 2, 3, 5, 4], R![1, 1]);

sage: X = HyperellipticCurve(R([1, 2, 9, 12, 20, 16]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$600$$ $$=$$ $$2^{3} \cdot 3 \cdot 5^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$96000$$ $$=$$ $$2^{8} \cdot 3 \cdot 5^{3}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$92$$ $$=$$ $$2^{2} \cdot 23$$ $$I_4$$ $$=$$ $$4981$$ $$=$$ $$17 \cdot 293$$ $$I_6$$ $$=$$ $$43947$$ $$=$$ $$3^{2} \cdot 19 \cdot 257$$ $$I_{10}$$ $$=$$ $$-12000$$ $$=$$ $$- 2^{5} \cdot 3 \cdot 5^{3}$$ $$J_2$$ $$=$$ $$92$$ $$=$$ $$2^{2} \cdot 23$$ $$J_4$$ $$=$$ $$-2968$$ $$=$$ $$- 2^{3} \cdot 7 \cdot 53$$ $$J_6$$ $$=$$ $$47600$$ $$=$$ $$2^{4} \cdot 5^{2} \cdot 7 \cdot 17$$ $$J_8$$ $$=$$ $$-1107456$$ $$=$$ $$- 2^{9} \cdot 3 \cdot 7 \cdot 103$$ $$J_{10}$$ $$=$$ $$-96000$$ $$=$$ $$- 2^{8} \cdot 3 \cdot 5^{3}$$ $$g_1$$ $$=$$ $$-25745372/375$$ $$g_2$$ $$=$$ $$9027914/375$$ $$g_3$$ $$=$$ $$-62951/15$$

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

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

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

Number of rational Weierstrass points: $$2$$

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

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$9.467159$$ Tamagawa product: $$4$$ Torsion order: $$12$$ Leading coefficient: $$0.262976$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$3$$ $$8$$ $$2$$ $$1 + T$$
$$3$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 2 T + 3 T^{2} )$$
$$5$$ $$2$$ $$3$$ $$2$$ $$( 1 + T )^{2}$$

## 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?
$$2$$ 2.180.3 yes
$$3$$ 3.640.2 yes

## 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

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 30.a
Elliptic curve isogeny class 20.a

magma: HeuristicDecompositionFactors(C);

## Endomorphisms of the Jacobian

Of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ an order of index $$3$$ in $$\Z \times \Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

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);