# Properties

 Label 686.a.686.1 Conductor 686 Discriminant 686 Mordell-Weil group $$\Z/{6}\Z$$ Sato-Tate group $N(G_{1,3})$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\C \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{CM} \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands for: Magma / SageMath

## Simplified equation

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

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

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 4, 5, 10, 5, 4]))

 $y^2 + (x^2 + x)y = x^5 + x^4 + 2x^3 + x^2 + x$ (homogenize, simplify) $y^2 + (x^2z + xz^2)y = x^5z + x^4z^2 + 2x^3z^3 + x^2z^4 + xz^5$ (dehomogenize, simplify) $y^2 = 4x^5 + 5x^4 + 10x^3 + 5x^2 + 4x$ (minimize, homogenize)

## Invariants

 $$N$$ = $$686$$ = $$2 \cdot 7^{3}$$ magma: Conductor(LSeries(C)); Factorization($1); $$\Delta$$ = $$686$$ = $$2 \cdot 7^{3}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$840$$ = $$2^{3} \cdot 3 \cdot 5 \cdot 7$$ $$I_4$$ = $$17220$$ = $$2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 41$$ $$I_6$$ = $$5121480$$ = $$2^{3} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13 \cdot 67$$ $$I_{10}$$ = $$2809856$$ = $$2^{13} \cdot 7^{3}$$ $$J_2$$ = $$105$$ = $$3 \cdot 5 \cdot 7$$ $$J_4$$ = $$280$$ = $$2^{3} \cdot 5 \cdot 7$$ $$J_6$$ = $$-980$$ = $$- 2^{2} \cdot 5 \cdot 7^{2}$$ $$J_8$$ = $$-45325$$ = $$- 5^{2} \cdot 7^{2} \cdot 37$$ $$J_{10}$$ = $$686$$ = $$2 \cdot 7^{3}$$ $$g_1$$ = $$37209375/2$$ $$g_2$$ = $$472500$$ $$g_3$$ = $$-15750$$

## Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$$C_2^2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $C_2^2$

## Rational points

magma: [C![0,0,1],C![1,0,0]];

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

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

Number of rational Weierstrass points: $$2$$

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);

This curve is locally solvable everywhere.

## Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));

Group structure: $$\Z/{6}\Z$$

Generator Height Order
$$x^2 + xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$z^3$$ $$0$$ $$6$$

## BSD invariants

 Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$11.49165$$ Tamagawa product: $$1$$ Torsion order: $$6$$ Leading coefficient: $$0.319212$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

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

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $N(G_{1,3})$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{U}(1)\times\mathrm{SU}(2)$$

## Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 14.a5
Elliptic curve 49.a4

## Endomorphisms of the Jacobian

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

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ an order of index $$2$$ in $$\Z \times \Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\sqrt{-7})$$ with defining polynomial $$x^{2} - x + 2$$

Not of $$\GL_2$$-type over $$\overline{\Q}$$

Endomorphism ring over $$\overline{\Q}$$:

 $$\End (J_{\overline{\Q}})$$ $$\simeq$$ an order of index $$4$$ in $$\Z \times \Z [\frac{1 + \sqrt{-7}}{2}]$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q(\sqrt{-7})$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\R \times \C$$