# Properties

 Label 131040.a.131040.1 Conductor 131040 Discriminant 131040 Mordell-Weil group $$\Z \times \Z/{2}\Z \times \Z/{2}\Z$$ 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

 $y^2 + (x^3 + x)y = -10x^6 + 81x^4 - 227x^2 + 210$ (homogenize, simplify) $y^2 + (x^3 + xz^2)y = -10x^6 + 81x^4z^2 - 227x^2z^4 + 210z^6$ (dehomogenize, simplify) $y^2 = -39x^6 + 326x^4 - 907x^2 + 840$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![210, 0, -227, 0, 81, 0, -10], R![0, 1, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([210, 0, -227, 0, 81, 0, -10]), R([0, 1, 0, 1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([840, 0, -907, 0, 326, 0, -39]))

## Invariants

 Conductor: $$N$$ = $$131040$$ = $$2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$131040$$ = $$2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$12593312$$ = $$2^{5} \cdot 393541$$ $$I_4$$ = $$11343232$$ = $$2^{7} \cdot 23 \cdot 3853$$ $$I_6$$ = $$47604155826432$$ = $$2^{8} \cdot 3 \cdot 97 \cdot 9431 \cdot 67757$$ $$I_{10}$$ = $$536739840$$ = $$2^{17} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13$$ $$J_2$$ = $$1574164$$ = $$2^{2} \cdot 393541$$ $$J_4$$ = $$103249560962$$ = $$2 \cdot 17 \cdot 83 \cdot 5927 \cdot 6173$$ $$J_6$$ = $$9029520569946240$$ = $$2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \cdot 31 \cdot 555698369$$ $$J_8$$ = $$888368594905774644479$$ = $$888368594905774644479$$ $$J_{10}$$ = $$131040$$ = $$2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13$$ $$g_1$$ = $$302064649214662608101539958432/4095$$ $$g_2$$ = $$12586012647194024913614166004/4095$$ $$g_3$$ = $$170750018582492394877376$$

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

No rational points are known for this curve.

magma: [];

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

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$8x^2 - 21z^2$$ $$=$$ $$0,$$ $$4y$$ $$=$$ $$-7xz^2$$ $$0.532646$$ $$\infty$$
$$D_0 - D_\infty$$ $$13x^2 - 35z^2$$ $$=$$ $$0,$$ $$13y$$ $$=$$ $$-24xz^2$$ $$0$$ $$2$$
$$D_0 - D_\infty$$ $$x^2 - 3z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2xz^2$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$5$$ Regulator: $$0.532646$$ Real period: $$10.74591$$ Tamagawa product: $$1$$ Torsion order: $$4$$ Leading coefficient: $$1.430943$$ Analytic order of Ш: $$4$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$5$$ $$5$$ $$1$$ $$1 + T$$
$$3$$ $$2$$ $$2$$ $$1$$ $$( 1 + T )^{2}$$
$$5$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 2 T + 5 T^{2} )$$
$$7$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 7 T^{2} )$$
$$13$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 2 T + 13 T^{2} )$$

## 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 210.a2
Elliptic curve 624.d2

## 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 $$\Q$$.