# Properties

 Label 65536.a.65536.1 Conductor $65536$ Discriminant $65536$ Mordell-Weil group $$\Z \times \Z/{2}\Z$$ Sato-Tate group $D_{2,1}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\C)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\mathsf{CM})$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands: SageMath / Magma

## Simplified equation

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

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

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

sage: X = HyperellipticCurve(R([0, 1, 0, 0, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$65536$$ $$=$$ $$2^{16}$$ magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(65536,2),R![1]>*])); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$65536$$ $$=$$ $$2^{16}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$20$$ $$=$$ $$2^{2} \cdot 5$$ $$I_4$$ $$=$$ $$-20$$ $$=$$ $$- 2^{2} \cdot 5$$ $$I_6$$ $$=$$ $$-40$$ $$=$$ $$- 2^{3} \cdot 5$$ $$I_{10}$$ $$=$$ $$8$$ $$=$$ $$2^{3}$$ $$J_2$$ $$=$$ $$80$$ $$=$$ $$2^{4} \cdot 5$$ $$J_4$$ $$=$$ $$480$$ $$=$$ $$2^{5} \cdot 3 \cdot 5$$ $$J_6$$ $$=$$ $$-1280$$ $$=$$ $$- 2^{8} \cdot 5$$ $$J_8$$ $$=$$ $$-83200$$ $$=$$ $$- 2^{8} \cdot 5^{2} \cdot 13$$ $$J_{10}$$ $$=$$ $$65536$$ $$=$$ $$2^{16}$$ $$g_1$$ $$=$$ $$50000$$ $$g_2$$ $$=$$ $$3750$$ $$g_3$$ $$=$$ $$-125$$

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); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$GL(2,3)$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: $$\Z \times \Z/{2}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2 + z^3$$ $$0.960251$$ $$\infty$$
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2 + z^3$$ $$0.960251$$ $$\infty$$
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$1/2xz^2 + 1/2z^3$$ $$0.960251$$ $$\infty$$
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$2$$ Regulator: $$0.960251$$ Real period: $$8.973175$$ Tamagawa product: $$1$$ Torsion order: $$2$$ Leading coefficient: $$2.154126$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$16$$ $$16$$ $$1$$ $$1$$

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $D_{2,1}$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{U}(1)$$

## 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 256.d
Elliptic curve isogeny class 256.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 $$K = \Q (a) \simeq$$ $$\Q(\zeta_{8})$$ with defining polynomial $$x^{4} + 1$$

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

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

 $$\End (J_{\overline{\Q}})$$ $$\simeq$$ a maximal order of $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q(\sqrt{-2})$$$$)$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\C)$$

### Remainder of the endomorphism lattice by field

Over subfield $$F \simeq$$ $$\Q(\sqrt{-2})$$ with generator $$-a^{3} - a$$ with minimal polynomial $$x^{2} + 2$$:

 $$\End (J_{F})$$ $$\simeq$$ an order of index $$4$$ in $$\Z [\sqrt{-2}] \times \Z [\sqrt{-2}]$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-2})$$ $$\times$$ $$\Q(\sqrt{-2})$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\C \times \C$$
Sato Tate group: $C_2$
Not of $$\GL_2$$-type, not simple

Over subfield $$F \simeq$$ $$\Q(\sqrt{2})$$ with generator $$a^{3} - a$$ with minimal polynomial $$x^{2} - 2$$:

 $$\End (J_{F})$$ $$\simeq$$ a non-Eichler order of index $$8$$ in a maximal order of $$\End (J_{F}) \otimes \Q$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q$$$$)$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\R)$$
Sato Tate group: $C_{2,1}$
Not of $$\GL_2$$-type, not simple

Over subfield $$F \simeq$$ $$\Q(\sqrt{-1})$$ with generator $$-a^{2}$$ with minimal polynomial $$x^{2} + 1$$:

 $$\End (J_{F})$$ $$\simeq$$ a non-Eichler order of index $$4$$ in a maximal order of $$\End (J_{F}) \otimes \Q$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q$$$$)$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\R)$$
Sato Tate group: $C_{2,1}$
Not of $$\GL_2$$-type, not simple