# Properties

 Label 450.a.36450.1 Conductor $450$ Discriminant $36450$ Mordell-Weil group $$\Z/{2}\Z \times \Z/{12}\Z$$ 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

This curve is isomorphic to the quotient of the modular curve $X_0(30)$ by the Atkin-Lehner involution $w_{10}$.

## Simplified equation

 $y^2 + (x^3 + 1)y = x^5 - 4x^4 - 9x^3 + 28x^2 - 6x - 16$ (homogenize, simplify) $y^2 + (x^3 + z^3)y = x^5z - 4x^4z^2 - 9x^3z^3 + 28x^2z^4 - 6xz^5 - 16z^6$ (dehomogenize, simplify) $y^2 = x^6 + 4x^5 - 16x^4 - 34x^3 + 112x^2 - 24x - 63$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-16, -6, 28, -9, -4, 1]), R([1, 0, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-16, -6, 28, -9, -4, 1], R![1, 0, 0, 1]);

sage: X = HyperellipticCurve(R([-63, -24, 112, -34, -16, 4, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$450$$ $$=$$ $$2 \cdot 3^{2} \cdot 5^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$36450$$ $$=$$ $$2 \cdot 3^{6} \cdot 5^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$23444$$ $$=$$ $$2^{2} \cdot 5861$$ $$I_4$$ $$=$$ $$212089$$ $$=$$ $$131 \cdot 1619$$ $$I_6$$ $$=$$ $$1627179821$$ $$=$$ $$29 \cdot 37 \cdot 59 \cdot 25703$$ $$I_{10}$$ $$=$$ $$4665600$$ $$=$$ $$2^{8} \cdot 3^{6} \cdot 5^{2}$$ $$J_2$$ $$=$$ $$5861$$ $$=$$ $$5861$$ $$J_4$$ $$=$$ $$1422468$$ $$=$$ $$2^{2} \cdot 3^{3} \cdot 13171$$ $$J_6$$ $$=$$ $$457836300$$ $$=$$ $$2^{2} \cdot 3^{5} \cdot 5^{2} \cdot 83 \cdot 227$$ $$J_8$$ $$=$$ $$164990835819$$ $$=$$ $$3^{5} \cdot 19 \cdot 71 \cdot 503317$$ $$J_{10}$$ $$=$$ $$36450$$ $$=$$ $$2 \cdot 3^{6} \cdot 5^{2}$$ $$g_1$$ $$=$$ $$6916057684302385301/36450$$ $$g_2$$ $$=$$ $$5303516319500302/675$$ $$g_3$$ $$=$$ $$1294426477922/3$$

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

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

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2^2$ 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: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (2 : -4 : 1),\, (2 : -5 : 1)$$ All points: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (2 : -4 : 1),\, (2 : -5 : 1)$$ All points: $$(1 : -1 : 0),\, (1 : 1 : 0),\, (2 : -1 : 1),\, (2 : 1 : 1)$$

magma: [C![1,-1,0],C![1,0,0],C![2,-5,1],C![2,-4,1]]; // minimal model

magma: [C![1,-1,0],C![1,1,0],C![2,-1,1],C![2,1,1]]; // simplified model

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: $$\Z/{2}\Z \times \Z/{12}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 - xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 - z^3$$ $$0$$ $$2$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$2x^2 - 2xz - 3z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-5xz^2 + 2z^3$$ $$0$$ $$12$$
Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 - xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 - z^3$$ $$0$$ $$2$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$2x^2 - 2xz - 3z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-5xz^2 + 2z^3$$ $$0$$ $$12$$
Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x^2 - xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - 2xz^2 - z^3$$ $$0$$ $$2$$
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$2x^2 - 2xz - 3z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$x^3 - 10xz^2 + 5z^3$$ $$0$$ $$12$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$18.77899$$ Tamagawa product: $$6$$ Torsion order: $$24$$ Leading coefficient: $$0.195614$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + T + 2 T^{2} )$$
$$3$$ $$2$$ $$6$$ $$6$$ $$( 1 - T )( 1 + T )$$
$$5$$ $$2$$ $$2$$ $$1$$ $$( 1 - T )( 1 + T )$$

## 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 30.a

## 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 $$\Q$$.