Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
| Index: | $1008$ | $\PSL_2$-index: | $504$ | ||||
| Genus: | $37 = 1 + \frac{ 504 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $28^{6}\cdot56^{6}$ | Cusp orbits | $3^{2}\cdot6$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $12$ | ||||||
| $\Q$-gonality: | $10 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $10 \le \gamma \le 16$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.1008.37.12 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}6&23\\51&22\end{bmatrix}$, $\begin{bmatrix}22&31\\3&38\end{bmatrix}$, $\begin{bmatrix}24&7\\35&24\end{bmatrix}$, $\begin{bmatrix}28&51\\19&28\end{bmatrix}$, $\begin{bmatrix}46&31\\23&38\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.504.37.bx.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $384$ |
| Full 56-torsion field degree: | $3072$ |
Jacobian
| Conductor: | $2^{172}\cdot7^{72}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}\cdot2^{14}\cdot4$ |
| Newforms: | 64.2.a.a, 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.bd, 3136.2.a.bh, 3136.2.a.bi, 3136.2.a.bj, 3136.2.a.bl, 3136.2.a.bo, 3136.2.a.bv, 3136.2.a.bw, 3136.2.a.bz |
Rational points
This modular curve has no $\Q_p$ points for $p=5,37,149$, and therefore no 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 |
| 8.48.1-8.n.1.4 | $8$ | $21$ | $21$ | $1$ | $0$ | $1^{4}\cdot2^{14}\cdot4$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.1-8.n.1.4 | $8$ | $21$ | $21$ | $1$ | $0$ | $1^{4}\cdot2^{14}\cdot4$ |
| 28.504.16-28.p.1.7 | $28$ | $2$ | $2$ | $16$ | $4$ | $1\cdot2^{8}\cdot4$ |
| 56.504.16-28.p.1.22 | $56$ | $2$ | $2$ | $16$ | $4$ | $1\cdot2^{8}\cdot4$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.2016.73-56.um.1.1 | $56$ | $2$ | $2$ | $73$ | $17$ | $1^{12}\cdot2^{12}$ |
| 56.2016.73-56.uv.1.6 | $56$ | $2$ | $2$ | $73$ | $23$ | $1^{12}\cdot2^{12}$ |
| 56.2016.73-56.za.1.4 | $56$ | $2$ | $2$ | $73$ | $23$ | $1^{12}\cdot2^{12}$ |
| 56.2016.73-56.zi.1.3 | $56$ | $2$ | $2$ | $73$ | $25$ | $1^{12}\cdot2^{12}$ |
| 56.2016.73-56.baf.1.6 | $56$ | $2$ | $2$ | $73$ | $27$ | $1^{22}\cdot2^{7}$ |
| 56.2016.73-56.baj.1.6 | $56$ | $2$ | $2$ | $73$ | $27$ | $1^{22}\cdot2^{7}$ |
| 56.2016.73-56.baw.1.5 | $56$ | $2$ | $2$ | $73$ | $18$ | $1^{28}\cdot2^{4}$ |
| 56.2016.73-56.bbe.1.8 | $56$ | $2$ | $2$ | $73$ | $27$ | $1^{28}\cdot2^{4}$ |
| 56.2016.73-56.bcb.1.7 | $56$ | $2$ | $2$ | $73$ | $27$ | $1^{22}\cdot2^{7}$ |
| 56.2016.73-56.bch.1.8 | $56$ | $2$ | $2$ | $73$ | $26$ | $1^{22}\cdot2^{7}$ |
| 56.2016.73-56.bcw.1.8 | $56$ | $2$ | $2$ | $73$ | $25$ | $1^{28}\cdot2^{4}$ |
| 56.2016.73-56.bde.1.7 | $56$ | $2$ | $2$ | $73$ | $28$ | $1^{28}\cdot2^{4}$ |
| 56.2016.73-56.beg.1.7 | $56$ | $2$ | $2$ | $73$ | $22$ | $1^{28}\cdot2^{4}$ |
| 56.2016.73-56.beo.1.8 | $56$ | $2$ | $2$ | $73$ | $25$ | $1^{28}\cdot2^{4}$ |
| 56.2016.73-56.bfl.1.6 | $56$ | $2$ | $2$ | $73$ | $24$ | $1^{22}\cdot2^{7}$ |
| 56.2016.73-56.bfp.1.8 | $56$ | $2$ | $2$ | $73$ | $27$ | $1^{22}\cdot2^{7}$ |
| 56.2016.73-56.bgc.1.8 | $56$ | $2$ | $2$ | $73$ | $27$ | $1^{28}\cdot2^{4}$ |
| 56.2016.73-56.bgk.1.5 | $56$ | $2$ | $2$ | $73$ | $36$ | $1^{28}\cdot2^{4}$ |
| 56.2016.73-56.bhh.1.8 | $56$ | $2$ | $2$ | $73$ | $30$ | $1^{22}\cdot2^{7}$ |
| 56.2016.73-56.bhl.1.8 | $56$ | $2$ | $2$ | $73$ | $32$ | $1^{22}\cdot2^{7}$ |
| 56.2016.73-56.bhy.1.3 | $56$ | $2$ | $2$ | $73$ | $27$ | $1^{12}\cdot2^{12}$ |
| 56.2016.73-56.big.1.4 | $56$ | $2$ | $2$ | $73$ | $29$ | $1^{12}\cdot2^{12}$ |
| 56.2016.73-56.bjd.1.4 | $56$ | $2$ | $2$ | $73$ | $29$ | $1^{12}\cdot2^{12}$ |
| 56.2016.73-56.bjk.1.1 | $56$ | $2$ | $2$ | $73$ | $35$ | $1^{12}\cdot2^{12}$ |