Invariants
Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
Index: | $1008$ | $\PSL_2$-index: | $504$ | ||||
Genus: | $34 = 1 + \frac{ 504 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 18 }{2}$ | ||||||
Cusps: | $18$ (none of which are rational) | Cusp widths | $14^{12}\cdot56^{6}$ | Cusp orbits | $6^{3}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $22$ | ||||||
$\Q$-gonality: | $9 \le \gamma \le 16$ | ||||||
$\overline{\Q}$-gonality: | $9 \le \gamma \le 16$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | yes $\quad(D =$ $-16$) |
Other labels
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.1008.34.152 |
Level structure
$\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}8&23\\45&54\end{bmatrix}$, $\begin{bmatrix}17&4\\14&39\end{bmatrix}$, $\begin{bmatrix}23&14\\18&19\end{bmatrix}$, $\begin{bmatrix}38&21\\7&24\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 56.504.34.ew.1 for the level structure with $-I$) |
Cyclic 56-isogeny field degree: | $16$ |
Cyclic 56-torsion field degree: | $192$ |
Full 56-torsion field degree: | $3072$ |
Jacobian
Conductor: | $2^{154}\cdot7^{68}$ |
Simple: | no |
Squarefree: | no |
Decomposition: | $1^{10}\cdot2^{12}$ |
Newforms: | 98.2.a.b$^{2}$, 196.2.a.b, 196.2.a.c, 784.2.a.a, 784.2.a.d, 784.2.a.h, 784.2.a.k, 784.2.a.l, 784.2.a.m, 3136.2.a.b$^{2}$, 3136.2.a.bk$^{2}$, 3136.2.a.bn$^{2}$, 3136.2.a.br$^{2}$, 3136.2.a.h$^{2}$, 3136.2.a.u$^{2}$ |
Rational points
This modular curve has 1 rational CM point but no rational cusps or other known rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
$X_{\mathrm{ns}}^+(7)$ | $7$ | $48$ | $24$ | $0$ | $0$ | full Jacobian |
8.48.0-8.y.1.3 | $8$ | $21$ | $21$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
8.48.0-8.y.1.3 | $8$ | $21$ | $21$ | $0$ | $0$ | full Jacobian |
56.504.16-56.bu.1.2 | $56$ | $2$ | $2$ | $16$ | $9$ | $1^{6}\cdot2^{6}$ |
56.504.16-56.bu.1.14 | $56$ | $2$ | $2$ | $16$ | $9$ | $1^{6}\cdot2^{6}$ |
56.504.16-56.cw.1.2 | $56$ | $2$ | $2$ | $16$ | $4$ | $1^{6}\cdot2^{6}$ |
56.504.16-56.cw.1.28 | $56$ | $2$ | $2$ | $16$ | $4$ | $1^{6}\cdot2^{6}$ |
56.504.16-56.cx.1.7 | $56$ | $2$ | $2$ | $16$ | $9$ | $1^{6}\cdot2^{6}$ |
56.504.16-56.cx.1.32 | $56$ | $2$ | $2$ | $16$ | $9$ | $1^{6}\cdot2^{6}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
56.2016.67-56.rr.1.7 | $56$ | $2$ | $2$ | $67$ | $26$ | $1^{27}\cdot2^{3}$ |
56.2016.67-56.rs.1.7 | $56$ | $2$ | $2$ | $67$ | $34$ | $1^{27}\cdot2^{3}$ |
56.2016.67-56.rz.1.8 | $56$ | $2$ | $2$ | $67$ | $34$ | $1^{27}\cdot2^{3}$ |
56.2016.67-56.sa.1.7 | $56$ | $2$ | $2$ | $67$ | $37$ | $1^{27}\cdot2^{3}$ |
56.2016.67-56.uc.1.8 | $56$ | $2$ | $2$ | $67$ | $34$ | $1^{27}\cdot2^{3}$ |
56.2016.67-56.ud.1.8 | $56$ | $2$ | $2$ | $67$ | $29$ | $1^{27}\cdot2^{3}$ |
56.2016.67-56.uk.1.3 | $56$ | $2$ | $2$ | $67$ | $49$ | $1^{27}\cdot2^{3}$ |
56.2016.67-56.ul.1.5 | $56$ | $2$ | $2$ | $67$ | $37$ | $1^{27}\cdot2^{3}$ |