# Properties

 Label 10098.a.272646.1 Conductor $10098$ Discriminant $-272646$ 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

 $y^2 + (x^3 + 1)y = -9x^6 + 16x^5 - 35x^4 + 33x^3 - 35x^2 + 16x - 9$ (homogenize, simplify) $y^2 + (x^3 + z^3)y = -9x^6 + 16x^5z - 35x^4z^2 + 33x^3z^3 - 35x^2z^4 + 16xz^5 - 9z^6$ (dehomogenize, simplify) $y^2 = -35x^6 + 64x^5 - 140x^4 + 134x^3 - 140x^2 + 64x - 35$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-9, 16, -35, 33, -35, 16, -9]), R([1, 0, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-9, 16, -35, 33, -35, 16, -9], R![1, 0, 0, 1]);

sage: X = HyperellipticCurve(R([-35, 64, -140, 134, -140, 64, -35]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$10098$$ $$=$$ $$2 \cdot 3^{3} \cdot 11 \cdot 17$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-272646$$ $$=$$ $$- 2 \cdot 3^{6} \cdot 11 \cdot 17$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$56004$$ $$=$$ $$2^{2} \cdot 3 \cdot 13 \cdot 359$$ $$I_4$$ $$=$$ $$288321$$ $$=$$ $$3 \cdot 11 \cdot 8737$$ $$I_6$$ $$=$$ $$5331417537$$ $$=$$ $$3 \cdot 19 \cdot 93533641$$ $$I_{10}$$ $$=$$ $$143616$$ $$=$$ $$2^{8} \cdot 3 \cdot 11 \cdot 17$$ $$J_2$$ $$=$$ $$42003$$ $$=$$ $$3^{2} \cdot 13 \cdot 359$$ $$J_4$$ $$=$$ $$73402380$$ $$=$$ $$2^{2} \cdot 3^{2} \cdot 5 \cdot 407791$$ $$J_6$$ $$=$$ $$170798965524$$ $$=$$ $$2^{2} \cdot 3^{4} \cdot 11 \cdot 17 \cdot 2819023$$ $$J_8$$ $$=$$ $$446539889810043$$ $$=$$ $$3^{4} \cdot 198173 \cdot 27818311$$ $$J_{10}$$ $$=$$ $$272646$$ $$=$$ $$2 \cdot 3^{6} \cdot 11 \cdot 17$$ $$g_1$$ $$=$$ $$179338702480653356667/374$$ $$g_2$$ $$=$$ $$3730727674118765970/187$$ $$g_3$$ $$=$$ $$1105214886926046$$

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

 This curve has no rational points. This curve has no rational points. This curve has no rational points.

magma: []; // minimal model

magma: []; // simplified model

Number of rational Weierstrass points: $$0$$

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

This curve is locally solvable except over $\R$ and $\Q_{2}$.

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
$$D_0 - D_\infty$$ $$5x^2 - 2xz + 5z^2$$ $$=$$ $$0,$$ $$50y$$ $$=$$ $$21xz^2 - 15z^3$$ $$0$$ $$2$$
$$D_0 - D_\infty$$ $$x^2 - xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$5x^2 - 2xz + 5z^2$$ $$=$$ $$0,$$ $$50y$$ $$=$$ $$21xz^2 - 15z^3$$ $$0$$ $$2$$
$$D_0 - D_\infty$$ $$x^2 - xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$5x^2 - 2xz + 5z^2$$ $$=$$ $$0,$$ $$50y$$ $$=$$ $$x^3 + 42xz^2 - 29z^3$$ $$0$$ $$2$$
$$D_0 - D_\infty$$ $$x^2 - xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 + z^3$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$3$$ Regulator: $$1$$ Real period: $$3.494508$$ Tamagawa product: $$4$$ Torsion order: $$4$$ Leading coefficient: $$1.747254$$ Analytic order of Ш: $$2$$   (rounded) Order of Ш: twice a square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - T + 2 T^{2} )$$
$$3$$ $$3$$ $$6$$ $$4$$ $$1 + T$$
$$11$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 11 T^{2} )$$
$$17$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 2 T + 17 T^{2} )$$

## 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 153.c
Elliptic curve isogeny class 66.b

## 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 $$\Q$$.