# Properties

 Label 8281.a.8281.1 Conductor $8281$ Discriminant $8281$ Mordell-Weil group $$\Z \times \Z \times \Z/{3}\Z$$ Sato-Tate group $G_{3,3}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands for: SageMath / Magma

This curve is isomorphic to the quotient of the modular curve $X_0(91)$ by its Fricke involution $w_{91}$.

## Simplified equation

 $y^2 + (x^3 + x^2 + x + 1)y = -x^4 - 3x^3 - x^2$ (homogenize, simplify) $y^2 + (x^3 + x^2z + xz^2 + z^3)y = -x^4z^2 - 3x^3z^3 - x^2z^4$ (dehomogenize, simplify) $y^2 = x^6 + 2x^5 - x^4 - 8x^3 - x^2 + 2x + 1$ (minimize, homogenize)

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

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

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

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$8281$$ $$=$$ $$7^{2} \cdot 13^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$8281$$ $$=$$ $$7^{2} \cdot 13^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$72$$ $$=$$ $$2^{3} \cdot 3^{2}$$ $$I_4$$ $$=$$ $$1236$$ $$=$$ $$2^{2} \cdot 3 \cdot 103$$ $$I_6$$ $$=$$ $$15984$$ $$=$$ $$2^{4} \cdot 3^{3} \cdot 37$$ $$I_{10}$$ $$=$$ $$33124$$ $$=$$ $$2^{2} \cdot 7^{2} \cdot 13^{2}$$ $$J_2$$ $$=$$ $$36$$ $$=$$ $$2^{2} \cdot 3^{2}$$ $$J_4$$ $$=$$ $$-152$$ $$=$$ $$- 2^{3} \cdot 19$$ $$J_6$$ $$=$$ $$392$$ $$=$$ $$2^{3} \cdot 7^{2}$$ $$J_8$$ $$=$$ $$-2248$$ $$=$$ $$- 2^{3} \cdot 281$$ $$J_{10}$$ $$=$$ $$8281$$ $$=$$ $$7^{2} \cdot 13^{2}$$ $$g_1$$ $$=$$ $$60466176/8281$$ $$g_2$$ $$=$$ $$-7091712/8281$$ $$g_3$$ $$=$$ $$10368/169$$

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

Known points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(-1 : -1 : 1)$$ $$(-1 : 1 : 1)$$
$$(1 : -4 : 2)$$ $$(2 : -4 : 1)$$ $$(1 : -11 : 2)$$ $$(2 : -11 : 1)$$

magma: [C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-11,2],C![1,-4,2],C![1,-1,0],C![1,0,0],C![2,-11,1],C![2,-4,1]];

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: $$\Z \times \Z \times \Z/{3}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : -1 : 1) - (1 : -1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.600818$$ $$\infty$$
$$D_0 - 2 \cdot(1 : -1 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.284784$$ $$\infty$$
$$(0 : -1 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$3$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.150828$$ Real period: $$23.53747$$ Tamagawa product: $$1$$ Torsion order: $$3$$ Leading coefficient: $$0.394457$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$7$$ $$2$$ $$2$$ $$1$$ $$( 1 - T )( 1 + T )$$
$$13$$ $$2$$ $$2$$ $$1$$ $$( 1 - T )( 1 + T )$$

## 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 91.b2
Elliptic curve 91.a1

## 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 $$\Q$$.