Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
| Index: | $504$ | $\PSL_2$-index: | $252$ | ||||
| Genus: | $16 = 1 + \frac{ 252 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $7^{6}\cdot14^{3}\cdot56^{3}$ | Cusp orbits | $3^{2}\cdot6$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $9$ | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 8$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4,-16$) |
Other labels
| Cummins and Pauli (CP) label: | 56B16 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.504.16.69 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}3&36\\36&39\end{bmatrix}$, $\begin{bmatrix}6&51\\37&36\end{bmatrix}$, $\begin{bmatrix}28&33\\31&14\end{bmatrix}$, $\begin{bmatrix}30&3\\15&50\end{bmatrix}$, $\begin{bmatrix}47&10\\8&25\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.252.16.cx.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $192$ |
| Full 56-torsion field degree: | $6144$ |
Jacobian
| Conductor: | $2^{64}\cdot7^{32}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}\cdot2^{6}$ |
| Newforms: | 98.2.a.b$^{2}$, 196.2.a.b, 196.2.a.c, 3136.2.a.b, 3136.2.a.bk, 3136.2.a.bn, 3136.2.a.br, 3136.2.a.h, 3136.2.a.u |
Rational points
This modular curve has 2 rational CM points 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(7)$ | $7$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 8.24.0-8.p.1.7 | $8$ | $21$ | $21$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.24.0-8.p.1.7 | $8$ | $21$ | $21$ | $0$ | $0$ | full Jacobian |
| 56.252.7-28.c.1.11 | $56$ | $2$ | $2$ | $7$ | $0$ | $1^{3}\cdot2^{3}$ |
| 56.252.7-28.c.1.21 | $56$ | $2$ | $2$ | $7$ | $0$ | $1^{3}\cdot2^{3}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.1008.31-56.or.1.7 | $56$ | $2$ | $2$ | $31$ | $10$ | $1^{13}\cdot2$ |
| 56.1008.31-56.ot.1.7 | $56$ | $2$ | $2$ | $31$ | $18$ | $1^{13}\cdot2$ |
| 56.1008.31-56.oz.1.7 | $56$ | $2$ | $2$ | $31$ | $24$ | $1^{13}\cdot2$ |
| 56.1008.31-56.pb.1.5 | $56$ | $2$ | $2$ | $31$ | $12$ | $1^{13}\cdot2$ |
| 56.1008.31-56.qb.1.7 | $56$ | $2$ | $2$ | $31$ | $13$ | $1^{13}\cdot2$ |
| 56.1008.31-56.qd.1.3 | $56$ | $2$ | $2$ | $31$ | $16$ | $1^{13}\cdot2$ |
| 56.1008.31-56.qj.1.3 | $56$ | $2$ | $2$ | $31$ | $16$ | $1^{13}\cdot2$ |
| 56.1008.31-56.ql.1.4 | $56$ | $2$ | $2$ | $31$ | $11$ | $1^{13}\cdot2$ |
| 56.1008.34-56.cg.1.8 | $56$ | $2$ | $2$ | $34$ | $10$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.cp.1.4 | $56$ | $2$ | $2$ | $34$ | $10$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.ev.1.3 | $56$ | $2$ | $2$ | $34$ | $22$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.ew.1.7 | $56$ | $2$ | $2$ | $34$ | $22$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.fp.1.4 | $56$ | $2$ | $2$ | $34$ | $14$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.fr.1.3 | $56$ | $2$ | $2$ | $34$ | $14$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.gb.1.4 | $56$ | $2$ | $2$ | $34$ | $18$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.gd.1.2 | $56$ | $2$ | $2$ | $34$ | $18$ | $1^{6}\cdot2^{6}$ |
| 56.1008.34-56.hh.1.2 | $56$ | $2$ | $2$ | $34$ | $17$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.hj.1.4 | $56$ | $2$ | $2$ | $34$ | $17$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.hp.1.3 | $56$ | $2$ | $2$ | $34$ | $14$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.hr.1.4 | $56$ | $2$ | $2$ | $34$ | $14$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.in.1.7 | $56$ | $2$ | $2$ | $34$ | $12$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.ip.1.3 | $56$ | $2$ | $2$ | $34$ | $12$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.iv.1.4 | $56$ | $2$ | $2$ | $34$ | $21$ | $1^{14}\cdot2^{2}$ |
| 56.1008.34-56.ix.1.1 | $56$ | $2$ | $2$ | $34$ | $21$ | $1^{14}\cdot2^{2}$ |