# Properties

 Label 1125.a.151875.1 Conductor $1125$ Discriminant $-151875$ Mordell-Weil group $$\Z/{2}\Z \times \Z/{2}\Z$$ 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: SageMath / Magma

## Simplified equation

 $y^2 + xy = 15x^5 + 50x^4 + 55x^3 + 22x^2 + 3x$ (homogenize, simplify) $y^2 + xz^2y = 15x^5z + 50x^4z^2 + 55x^3z^3 + 22x^2z^4 + 3xz^5$ (dehomogenize, simplify) $y^2 = 60x^5 + 200x^4 + 220x^3 + 89x^2 + 12x$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 3, 22, 55, 50, 15]), R([0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 3, 22, 55, 50, 15], R![0, 1]);

sage: X = HyperellipticCurve(R([0, 12, 89, 220, 200, 60]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$1125$$ $$=$$ $$3^{2} \cdot 5^{3}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-151875$$ $$=$$ $$- 3^{5} \cdot 5^{4}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$8600$$ $$=$$ $$2^{3} \cdot 5^{2} \cdot 43$$ $$I_4$$ $$=$$ $$612100$$ $$=$$ $$2^{2} \cdot 5^{2} \cdot 6121$$ $$I_6$$ $$=$$ $$1556297975$$ $$=$$ $$5^{2} \cdot 62251919$$ $$I_{10}$$ $$=$$ $$-607500$$ $$=$$ $$- 2^{2} \cdot 3^{5} \cdot 5^{4}$$ $$J_2$$ $$=$$ $$4300$$ $$=$$ $$2^{2} \cdot 5^{2} \cdot 43$$ $$J_4$$ $$=$$ $$668400$$ $$=$$ $$2^{4} \cdot 3 \cdot 5^{2} \cdot 557$$ $$J_6$$ $$=$$ $$132975225$$ $$=$$ $$3^{2} \cdot 5^{2} \cdot 37 \cdot 15973$$ $$J_8$$ $$=$$ $$31258726875$$ $$=$$ $$3^{3} \cdot 5^{4} \cdot 211 \cdot 8779$$ $$J_{10}$$ $$=$$ $$-151875$$ $$=$$ $$- 3^{5} \cdot 5^{4}$$ $$g_1$$ $$=$$ $$-2352135088000000/243$$ $$g_2$$ $$=$$ $$-28342655360000/81$$ $$g_3$$ $$=$$ $$-437104339600/27$$

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: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (-4 : 18 : 3)$$ All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (-4 : 18 : 3)$$ All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (-4 : 0 : 3)$$

magma: [C![-4,18,3],C![0,0,1],C![1,0,0]]; // minimal model

magma: [C![-4,0,3],C![0,0,1],C![1,0,0]]; // simplified model

Number of rational Weierstrass points: $$3$$

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
$$(-4 : 18 : 3) - (1 : 0 : 0)$$ $$3x + 4z$$ $$=$$ $$0,$$ $$3y$$ $$=$$ $$2z^3$$ $$0$$ $$2$$
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$(-4 : 18 : 3) - (1 : 0 : 0)$$ $$3x + 4z$$ $$=$$ $$0,$$ $$3y$$ $$=$$ $$2z^3$$ $$0$$ $$2$$
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$3x + 4z$$ $$=$$ $$0,$$ $$3y$$ $$=$$ $$xz^2 + 4z^3$$ $$0$$ $$2$$
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$1.964401$$ Tamagawa product: $$4$$ Torsion order: $$4$$ Leading coefficient: $$0.491100$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

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

## 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 15.a
Elliptic curve isogeny class 75.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 $$\Q$$.