# Properties

 Label 1200.a.450000.1 Conductor 1200 Discriminant -450000 Mordell-Weil group $$\Z/{2}\Z \times \Z/{8}\Z$$ 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

 $y^2 + (x^3 + x)y = 4x^4 + 25x^2 + 45$ (homogenize, simplify) $y^2 + (x^3 + xz^2)y = 4x^4z^2 + 25x^2z^4 + 45z^6$ (dehomogenize, simplify) $y^2 = x^6 + 18x^4 + 101x^2 + 180$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![45, 0, 25, 0, 4], R![0, 1, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([45, 0, 25, 0, 4]), R([0, 1, 0, 1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([180, 0, 101, 0, 18, 0, 1]))

## Invariants

 Conductor: $$N$$ = $$1200$$ = $$2^{4} \cdot 3 \cdot 5^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-450000$$ = $$- 2^{4} \cdot 3^{2} \cdot 5^{5}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$-72288$$ = $$- 2^{5} \cdot 3^{2} \cdot 251$$ $$I_4$$ = $$622464$$ = $$2^{7} \cdot 3 \cdot 1621$$ $$I_6$$ = $$-14918139648$$ = $$- 2^{8} \cdot 3^{2} \cdot 6474887$$ $$I_{10}$$ = $$-1843200000$$ = $$- 2^{16} \cdot 3^{2} \cdot 5^{5}$$ $$J_2$$ = $$-9036$$ = $$- 2^{2} \cdot 3^{2} \cdot 251$$ $$J_4$$ = $$3395570$$ = $$2 \cdot 5 \cdot 339557$$ $$J_6$$ = $$-1698206400$$ = $$- 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 11 \cdot 71 \cdot 151$$ $$J_8$$ = $$953774351375$$ = $$5^{3} \cdot 227 \cdot 33613193$$ $$J_{10}$$ = $$-450000$$ = $$- 2^{4} \cdot 3^{2} \cdot 5^{5}$$ $$g_1$$ = $$418329622965299904/3125$$ $$g_2$$ = $$3479436045234936/625$$ $$g_3$$ = $$38515932506304/125$$

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

magma: [C![1,-1,0],C![1,0,0]];

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: $$\Z/{2}\Z \times \Z/{8}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 + 4z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$3xz^2$$ $$0$$ $$2$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$2x^2 + xz + 12z^2$$ $$=$$ $$0,$$ $$4y$$ $$=$$ $$11xz^2$$ $$0$$ $$8$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$5.655371$$ Tamagawa product: $$16$$ Torsion order: $$16$$ Leading coefficient: $$0.353460$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$4$$ $$4$$ $$1$$ $$1 + T + 2 T^{2}$$
$$3$$ $$2$$ $$1$$ $$2$$ $$( 1 + T )( 1 + 3 T^{2} )$$
$$5$$ $$5$$ $$2$$ $$8$$ $$( 1 - T )^{2}$$

## 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 80.a2
Elliptic curve 15.a5

## 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 $$\Q$$.