Properties

Label 256.a.512.1
Conductor $256$
Discriminant $-512$
Mordell-Weil group \(\Z/{2}\Z \times \Z/{10}\Z\)
Sato-Tate group $E_4$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\) \(\mathsf{CM}\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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

Minimal equation

Minimal equation

Simplified equation

$y^2 + y = 2x^5 - 3x^4 + x^3 + x^2 - x$ (homogenize, simplify)
$y^2 + z^3y = 2x^5z - 3x^4z^2 + x^3z^3 + x^2z^4 - xz^5$ (dehomogenize, simplify)
$y^2 = 8x^5 - 12x^4 + 4x^3 + 4x^2 - 4x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 1, 1, -3, 2]), R([1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 1, 1, -3, 2], R![1]);
 
sage: X = HyperellipticCurve(R([1, -4, 4, 4, -12, 8]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

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

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(26\) \(=\)  \( 2 \cdot 13 \)
\( I_4 \)  \(=\) \(-2\) \(=\)  \( -2 \)
\( I_6 \)  \(=\) \(40\) \(=\)  \( 2^{3} \cdot 5 \)
\( I_{10} \)  \(=\) \(2\) \(=\)  \( 2 \)
\( J_2 \)  \(=\) \(52\) \(=\)  \( 2^{2} \cdot 13 \)
\( J_4 \)  \(=\) \(118\) \(=\)  \( 2 \cdot 59 \)
\( J_6 \)  \(=\) \(-36\) \(=\)  \( - 2^{2} \cdot 3^{2} \)
\( J_8 \)  \(=\) \(-3949\) \(=\)  \( - 11 \cdot 359 \)
\( J_{10} \)  \(=\) \(512\) \(=\)  \( 2^{9} \)
\( g_1 \)  \(=\) \(742586\)
\( g_2 \)  \(=\) \(129623/4\)
\( g_3 \)  \(=\) \(-1521/8\)

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_4$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $D_4$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

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

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]];
 

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : -4 : 2) - (1 : 0 : 0)\) \(2x - z\) \(=\) \(0,\) \(2y\) \(=\) \(-z^3\) \(0\) \(2\)
\((1 : 0 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(10\)

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

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 26.84182 \)
Tamagawa product: \( 2 \)
Torsion order:\( 20 \)
Leading coefficient: \( 0.134209 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(8\) \(9\) \(2\) \(1 + 2 T + 2 T^{2}\)

Sato-Tate group

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

Decomposition of the Jacobian

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 the Jacobian

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