# Properties

 Label 686.a.686.1 Conductor $686$ Discriminant $686$ Mordell-Weil group $$\Z/{6}\Z$$ Sato-Tate group $N(\mathrm{U}(1)\times\mathrm{SU}(2))$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\C \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathsf{CM} \times \Q$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands for: SageMath / Magma

## Simplified equation

 $y^2 + (x^2 + x)y = x^5 + x^4 + 2x^3 + x^2 + x$ (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)

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: X = HyperellipticCurve(R([0, 4, 5, 10, 5, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

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

### G2 invariants

 $$I_2$$ $$=$$ $$420$$ $$=$$ $$2^{2} \cdot 3 \cdot 5 \cdot 7$$ $$I_4$$ $$=$$ $$4305$$ $$=$$ $$3 \cdot 5 \cdot 7 \cdot 41$$ $$I_6$$ $$=$$ $$640185$$ $$=$$ $$3 \cdot 5 \cdot 7^{2} \cdot 13 \cdot 67$$ $$I_{10}$$ $$=$$ $$87808$$ $$=$$ $$2^{8} \cdot 7^{3}$$ $$J_2$$ $$=$$ $$105$$ $$=$$ $$3 \cdot 5 \cdot 7$$ $$J_4$$ $$=$$ $$280$$ $$=$$ $$2^{3} \cdot 5 \cdot 7$$ $$J_6$$ $$=$$ $$-980$$ $$=$$ $$- 2^{2} \cdot 5 \cdot 7^{2}$$ $$J_8$$ $$=$$ $$-45325$$ $$=$$ $$- 5^{2} \cdot 7^{2} \cdot 37$$ $$J_{10}$$ $$=$$ $$686$$ $$=$$ $$2 \cdot 7^{3}$$ $$g_1$$ $$=$$ $$37209375/2$$ $$g_2$$ $$=$$ $$472500$$ $$g_3$$ $$=$$ $$-15750$$

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

 All points: $$(1 : 0 : 0),\, (0 : 0 : 1)$$ All points: $$(1 : 0 : 0),\, (0 : 0 : 1)$$ All points: $$(1 : 0 : 0),\, (0 : 0 : 1)$$

magma: [C![0,0,1],C![1,0,0]]; // minimal model

magma: [C![0,0,1],C![1,0,0]]; // simplified model

Number of rational Weierstrass points: $$2$$

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

magma: MordellWeilGroupGenus2(Jacobian(C));

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

## BSD invariants

 Hasse-Weil conjecture: verified 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 Cluster picture
$$2$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - T + 2 T^{2} )$$
$$7$$ $$3$$ $$3$$ $$1$$ $$1 - T$$

## Sato-Tate group

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

## Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 14.a
Elliptic curve isogeny class 49.a

## 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 $$\Z \times \Z [\frac{1 + \sqrt{-7}}{2}]$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q(\sqrt{-7})$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\R \times \C$$