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) |
$y^2 = 8x^5 - 12x^4 + 4x^3 + 4x^2 - 4x + 1$ | (homogenize, minimize) |
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
\( I_2 \) | \(=\) | \(26\) | \(=\) | \( 2 \cdot 13 \) |
\( I_4 \) | \(=\) | \(-2\) | \(=\) | \( -2 \) |
\( I_6 \) | \(=\) | \(40\) | \(=\) | \( 2^{3} \cdot 5 \) |
\( I_{10} \) | \(=\) | \(2\) | \(=\) | \( 2 \) |
\( J_2 \) | \(=\) | \(52\) | \(=\) | \( 2^{2} \cdot 13 \) |
\( J_4 \) | \(=\) | \(118\) | \(=\) | \( 2 \cdot 59 \) |
\( J_6 \) | \(=\) | \(-36\) | \(=\) | \( - 2^{2} \cdot 3^{2} \) |
\( J_8 \) | \(=\) | \(-3949\) | \(=\) | \( - 11 \cdot 359 \) |
\( J_{10} \) | \(=\) | \(512\) | \(=\) | \( 2^{9} \) |
\( g_1 \) | \(=\) | \(742586\) | ||
\( g_2 \) | \(=\) | \(129623/4\) | ||
\( g_3 \) | \(=\) | \(-1521/8\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_4$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $D_4$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
Number of rational Weierstrass points: \(2\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z \oplus \Z/{10}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -4 : 2) - (1 : 0 : 0)\) | \(2x - z\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
\((1 : 0 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(10\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((1 : -4 : 2) - (1 : 0 : 0)\) | \(2x - z\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
\((1 : 0 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(10\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(2x - z\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
\((1 : 1 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(z^3\) | \(0\) | \(10\) |
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 |
---|---|---|---|---|---|
\(2\) | \(8\) | \(9\) | \(2\) | \(1 + 2 T + 2 T^{2}\) |
Galois representations
For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.
For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.
Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
---|---|---|
\(2\) | 2.180.3 | yes |
\(3\) | 3.540.6 | no |
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 isogeny class:
Elliptic curve isogeny class 4.4.2048.1-1.1-a
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 \(x^{2} - 2\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{-1}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-1}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple