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

Learn more about

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);
 
\( \Delta \)  =  \(-512\)  =  \( -1 \cdot 2^{9} \)

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
 

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: Igusa-Clebsch, Igusa, 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

Regulator: 1.0

Real period: 26.841829436926411242441864998

Tamagawa numbers: 2 (p = 2)

magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);
 

Torsion: \(\Z/{2}\Z \times \Z/{10}\Z\)

2-torsion field: \(\Q(\zeta_{8})\)

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