# Properties

 Label 1225.a.6125.1 Conductor 1225 Discriminant 6125 Mordell-Weil group $$\Z/{8}\Z$$ Sato-Tate group $G_{3,3}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{RM}$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands for: SageMath / Magma

The Jacobian of this curve is isogenous to that of the quotient of the modular curve $X_0(35)$ by the involution $W_7$ (see [MR:1373390]), which has discriminant $5^77^3$.

## Simplified equation

 $y^2 + (x^3 + x^2)y = 2x^3 + x^2 + x + 2$ (homogenize, simplify) $y^2 + (x^3 + x^2z)y = 2x^3z^3 + x^2z^4 + xz^5 + 2z^6$ (dehomogenize, simplify) $y^2 = x^6 + 2x^5 + x^4 + 8x^3 + 4x^2 + 4x + 8$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 1, 1, 2]), R([0, 0, 1, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, 1, 1, 2], R![0, 0, 1, 1]);

sage: X = HyperellipticCurve(R([8, 4, 4, 8, 1, 2, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$1225$$ $$=$$ $$5^{2} \cdot 7^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$6125$$ $$=$$ $$5^{3} \cdot 7^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$-1280$$ $$=$$ $$- 2^{8} \cdot 5$$ $$I_4$$ $$=$$ $$229504$$ $$=$$ $$2^{7} \cdot 11 \cdot 163$$ $$I_6$$ $$=$$ $$-61598784$$ $$=$$ $$- 2^{6} \cdot 3 \cdot 13 \cdot 23 \cdot 29 \cdot 37$$ $$I_{10}$$ $$=$$ $$25088000$$ $$=$$ $$2^{12} \cdot 5^{3} \cdot 7^{2}$$ $$J_2$$ $$=$$ $$-160$$ $$=$$ $$- 2^{5} \cdot 5$$ $$J_4$$ $$=$$ $$-1324$$ $$=$$ $$- 2^{2} \cdot 331$$ $$J_6$$ $$=$$ $$-8791$$ $$=$$ $$- 59 \cdot 149$$ $$J_8$$ $$=$$ $$-86604$$ $$=$$ $$- 2^{2} \cdot 3 \cdot 7 \cdot 1031$$ $$J_{10}$$ $$=$$ $$6125$$ $$=$$ $$5^{3} \cdot 7^{2}$$ $$g_1$$ $$=$$ $$-838860800/49$$ $$g_2$$ $$=$$ $$43384832/49$$ $$g_3$$ $$=$$ $$-9001984/245$$

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

magma: [C![-1,0,1],C![1,-1,0],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: $$\Z/{8}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

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

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$11.92789$$ Tamagawa product: $$2$$ Torsion order: $$8$$ Leading coefficient: $$0.372746$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

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

## Sato-Tate group

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

## Decomposition of the Jacobian

Simple over $$\overline{\Q}$$

## Endomorphisms of the Jacobian

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

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ $$\Z [\frac{1 + \sqrt{17}}{2}]$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{17})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.