# Properties

 Label 880.a.225280.1 Conductor 880 Discriminant -225280 Mordell-Weil group $$\Z/{2}\Z$$ 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

 $y^2 = x^5 + 13x^4 + 55x^3 + 76x^2 - 44$ (homogenize, simplify) $y^2 = x^5z + 13x^4z^2 + 55x^3z^3 + 76x^2z^4 - 44z^6$ (dehomogenize, simplify) $y^2 = x^5 + 13x^4 + 55x^3 + 76x^2 - 44$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-44, 0, 76, 55, 13, 1], R![]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-44, 0, 76, 55, 13, 1]), R([]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([-44, 0, 76, 55, 13, 1]))

## Invariants

 Conductor: $$N$$ = $$880$$ = $$2^{4} \cdot 5 \cdot 11$$ magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(880,2),R![1]>*])); Factorization($1); Discriminant: $$\Delta$$ = $$-225280$$ = $$- 2^{12} \cdot 5 \cdot 11$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$37472$$ = $$2^{5} \cdot 1171$$ $$I_4$$ = $$28659712$$ = $$2^{12} \cdot 6997$$ $$I_6$$ = $$301361299456$$ = $$2^{15} \cdot 7 \cdot 19 \cdot 69149$$ $$I_{10}$$ = $$-922746880$$ = $$- 2^{24} \cdot 5 \cdot 11$$ $$J_2$$ = $$4684$$ = $$2^{2} \cdot 1171$$ $$J_4$$ = $$615622$$ = $$2 \cdot 7 \cdot 43973$$ $$J_6$$ = $$103120196$$ = $$2^{2} \cdot 83 \cdot 263 \cdot 1181$$ $$J_8$$ = $$26006137795$$ = $$5 \cdot 11 \cdot 541 \cdot 874009$$ $$J_{10}$$ = $$-225280$$ = $$- 2^{12} \cdot 5 \cdot 11$$ $$g_1$$ = $$-2201833501574851/220$$ $$g_2$$ = $$-494259267301121/1760$$ $$g_3$$ = $$-35350660170809/3520$$

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

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

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2$ 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: $$(1 : 0 : 0)$$

magma: [C![1,0,0]];

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

magma: MordellWeilGroupGenus2(Jacobian(C));

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

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$1.515081$$ Tamagawa product: $$1$$ Torsion order: $$2$$ Leading coefficient: $$0.378770$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$12$$ $$4$$ $$1$$ $$1$$
$$5$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 3 T + 5 T^{2} )$$
$$11$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 11 T^{2} )$$

## 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 20.a1
Elliptic curve 44.a1

## 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 $$\Q$$.