# Properties

 Label 990.a.240570.1 Conductor 990 Discriminant 240570 Mordell-Weil group $$\Z/{2}\Z \times \Z/{2}\Z$$ Sato-Tate group $G_{3,3}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands for: Magma / SageMath

## Simplified equation

 $y^2 + (x^2 + x)y = 3x^5 + 28x^4 + 72x^3 + 28x^2 + 3x$ (homogenize, simplify) $y^2 + (x^2z + xz^2)y = 3x^5z + 28x^4z^2 + 72x^3z^3 + 28x^2z^4 + 3xz^5$ (dehomogenize, simplify) $y^2 = 12x^5 + 113x^4 + 290x^3 + 113x^2 + 12x$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 3, 28, 72, 28, 3], R![0, 1, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 3, 28, 72, 28, 3]), R([0, 1, 1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([0, 12, 113, 290, 113, 12]))

## Invariants

 Conductor: $$N$$ = $$990$$ = $$2 \cdot 3^{2} \cdot 5 \cdot 11$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$240570$$ = $$2 \cdot 3^{7} \cdot 5 \cdot 11$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$306056$$ = $$2^{3} \cdot 67 \cdot 571$$ $$I_4$$ = $$27393028$$ = $$2^{2} \cdot 13 \cdot 263 \cdot 2003$$ $$I_6$$ = $$2746933168904$$ = $$2^{3} \cdot 31 \cdot 11076343423$$ $$I_{10}$$ = $$985374720$$ = $$2^{13} \cdot 3^{7} \cdot 5 \cdot 11$$ $$J_2$$ = $$38257$$ = $$67 \cdot 571$$ $$J_4$$ = $$60697908$$ = $$2^{2} \cdot 3^{2} \cdot 37 \cdot 45569$$ $$J_6$$ = $$127876480380$$ = $$2^{2} \cdot 3^{5} \cdot 5 \cdot 11 \cdot 2392003$$ $$J_8$$ = $$301983618580299$$ = $$3^{4} \cdot 19^{2} \cdot 20117 \cdot 513367$$ $$J_{10}$$ = $$240570$$ = $$2 \cdot 3^{7} \cdot 5 \cdot 11$$ $$g_1$$ = $$81951056110393451083057/240570$$ $$g_2$$ = $$188813894774599018858/13365$$ $$g_3$$ = $$7001861848004294/9$$

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2^2$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$C_2^2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

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

magma: [C![0,0,1],C![1,0,0]];

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: $$\Z/{2}\Z \times \Z/{2}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$4x^2 + 19xz + 4z^2$$ $$=$$ $$0,$$ $$8y$$ $$=$$ $$15xz^2 + 4z^3$$ $$0$$ $$2$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$3x^2 + 14xz + 3z^2$$ $$=$$ $$0,$$ $$6y$$ $$=$$ $$11xz^2 + 3z^3$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$3.087468$$ Tamagawa product: $$2$$ Torsion order: $$4$$ Leading coefficient: $$0.385933$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + T + 2 T^{2} )$$
$$3$$ $$7$$ $$2$$ $$2$$ $$( 1 + T )^{2}$$
$$5$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 2 T + 5 T^{2} )$$
$$11$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 4 T + 11 T^{2} )$$

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $G_{3,3}$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)\times\mathrm{SU}(2)$$

## Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 15.a2
Elliptic curve 66.b2

## Endomorphisms of the Jacobian

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

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ an order of index $$2$$ 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 $$\Q$$.