# Properties

 Label 256.a.512.1 Conductor 256 Discriminant -512 Sato-Tate group $E_4$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands for: Magma / SageMath

This is a model for the modular curve $X_1(16)$.

## Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 1, 1, -3, 2], R![1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 1, 1, -3, 2]), R([1]))

$y^2 + y = 2x^5 - 3x^4 + x^3 + x^2 - x$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$256$$ = $$2^{8}$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-512$$ = $$-1 \cdot 2^{9}$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-416$$ = $$-1 \cdot 2^{5} \cdot 13$$ $$I_4$$ = $$-512$$ = $$-1 \cdot 2^{9}$$ $$I_6$$ = $$-163840$$ = $$-1 \cdot 2^{15} \cdot 5$$ $$I_{10}$$ = $$-2097152$$ = $$-1 \cdot 2^{21}$$ $$J_2$$ = $$-52$$ = $$-1 \cdot 2^{2} \cdot 13$$ $$J_4$$ = $$118$$ = $$2 \cdot 59$$ $$J_6$$ = $$36$$ = $$2^{2} \cdot 3^{2}$$ $$J_8$$ = $$-3949$$ = $$-1 \cdot 11 \cdot 359$$ $$J_{10}$$ = $$-512$$ = $$-1 \cdot 2^{9}$$ $$g_1$$ = $$742586$$ $$g_2$$ = $$129623/4$$ $$g_3$$ = $$-1521/8$$
Alternative geometric invariants: G2

## Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$ $$C_4$$ (GAP id : [4,1]) magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $$D_4$$ (GAP id : [8,3])

## Rational points

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.

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

All rational points: (0 : -1 : 1), (0 : 0 : 1), (1 : -4 : 2), (1 : -1 : 1), (1 : 0 : 0), (1 : 0 : 1)

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

Number of rational Weierstrass points: $$2$$

## Invariants of the Jacobian:

Analytic rank: $$0$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$2$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Tamagawa numbers: 2 (p = 2) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{2}\Z \times \Z/{10}\Z$$

### Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $E_4$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)$$

### Decomposition

Splits over the number field $$\Q (b) \simeq$$ $$\Q(\zeta_{16})^+$$ with defining polynomial:
$$x^{4} - 4 x^{2} + 2$$

Decomposes up to isogeny as the square of the elliptic curve:
Elliptic curve 4.4.2048.1-1.1-a5

### Endomorphisms

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

Endomorphism ring over $$\Q$$:
 $$\End (J_{})$$ $$\simeq$$ $$\Z [\sqrt{-1}]$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-1})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\C$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\zeta_{16})^+$$ with defining polynomial $$x^{4} - 4 x^{2} + 2$$

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

Endomorphism ring over $$\overline{\Q}$$:
 $$\End (J_{\overline{\Q}})$$ $$\simeq$$ a non-Eichler order of index $$4$$ in a maximal order of $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q$$$$)$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\R)$$

### Remainder of the endomorphism lattice by field

Over subfield $$F \simeq$$ $$\Q(\sqrt{2})$$ with generator $$a^{2} - 2$$ with minimal polynomial $$x^{2} - 2$$:
 $$\End (J_{F})$$ $$\simeq$$ $$\Z [\sqrt{-1}]$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-1})$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\C$$
Sato Tate group: $E_2$
of $$\GL_2$$-type, simple