Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $1568$ | ||
| Index: | $384$ | $\PSL_2$-index: | $384$ | ||||
| Genus: | $23 = 1 + \frac{ 384 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 20 }{2}$ | ||||||
| Cusps: | $20$ (of which $4$ are rational) | Cusp widths | $4^{8}\cdot8^{2}\cdot28^{8}\cdot56^{2}$ | Cusp orbits | $1^{4}\cdot2^{4}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 8$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 56P23 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.384.23.44 |
Level structure
Jacobian
| Conductor: | $2^{76}\cdot7^{35}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{11}\cdot2^{2}\cdot4^{2}$ |
| Newforms: | 14.2.a.a$^{3}$, 56.2.a.a, 56.2.a.b, 56.2.b.a, 56.2.b.b, 98.2.a.a, 392.2.a.b, 392.2.a.d, 784.2.a.b, 784.2.a.e, 784.2.a.i, 1568.2.b.a, 1568.2.b.d |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 28.192.11.e.1 | $28$ | $2$ | $2$ | $11$ | $2$ | $2^{2}\cdot4^{2}$ |
| 56.48.0.i.2 | $56$ | $8$ | $8$ | $0$ | $0$ | full Jacobian |
| 56.192.11.q.1 | $56$ | $2$ | $2$ | $11$ | $0$ | $1^{6}\cdot2\cdot4$ |
| 56.192.11.s.2 | $56$ | $2$ | $2$ | $11$ | $0$ | $1^{6}\cdot2\cdot4$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.768.45.bu.1 | $56$ | $2$ | $2$ | $45$ | $2$ | $2^{7}\cdot4^{2}$ |
| 56.768.45.bu.4 | $56$ | $2$ | $2$ | $45$ | $2$ | $2^{7}\cdot4^{2}$ |
| 56.768.45.bu.5 | $56$ | $2$ | $2$ | $45$ | $2$ | $2^{7}\cdot4^{2}$ |
| 56.768.45.bu.8 | $56$ | $2$ | $2$ | $45$ | $2$ | $2^{7}\cdot4^{2}$ |
| 56.768.45.bv.1 | $56$ | $2$ | $2$ | $45$ | $4$ | $2^{7}\cdot4^{2}$ |
| 56.768.45.bv.4 | $56$ | $2$ | $2$ | $45$ | $4$ | $2^{7}\cdot4^{2}$ |
| 56.768.45.bv.6 | $56$ | $2$ | $2$ | $45$ | $4$ | $2^{7}\cdot4^{2}$ |
| 56.768.45.bv.7 | $56$ | $2$ | $2$ | $45$ | $4$ | $2^{7}\cdot4^{2}$ |
| 56.768.49.dv.1 | $56$ | $2$ | $2$ | $49$ | $5$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.768.49.fx.2 | $56$ | $2$ | $2$ | $49$ | $5$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.768.49.gg.2 | $56$ | $2$ | $2$ | $49$ | $8$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.768.49.go.1 | $56$ | $2$ | $2$ | $49$ | $8$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.768.49.ke.1 | $56$ | $2$ | $2$ | $49$ | $2$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.768.49.ke.3 | $56$ | $2$ | $2$ | $49$ | $2$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.768.49.kf.1 | $56$ | $2$ | $2$ | $49$ | $4$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.768.49.kf.4 | $56$ | $2$ | $2$ | $49$ | $4$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.768.49.ma.1 | $56$ | $2$ | $2$ | $49$ | $2$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.768.49.ma.4 | $56$ | $2$ | $2$ | $49$ | $2$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.768.49.mb.1 | $56$ | $2$ | $2$ | $49$ | $4$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.768.49.mb.3 | $56$ | $2$ | $2$ | $49$ | $4$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.768.49.no.2 | $56$ | $2$ | $2$ | $49$ | $8$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.768.49.nw.1 | $56$ | $2$ | $2$ | $49$ | $8$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.768.49.nz.1 | $56$ | $2$ | $2$ | $49$ | $7$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.768.49.od.2 | $56$ | $2$ | $2$ | $49$ | $7$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.1152.67.bw.1 | $56$ | $3$ | $3$ | $67$ | $2$ | $2^{10}\cdot12^{2}$ |
| 56.1152.67.bw.2 | $56$ | $3$ | $3$ | $67$ | $2$ | $2^{10}\cdot12^{2}$ |
| 56.1152.67.cl.1 | $56$ | $3$ | $3$ | $67$ | $10$ | $1^{20}\cdot6^{4}$ |
| 56.2688.185.em.1 | $56$ | $7$ | $7$ | $185$ | $19$ | $1^{48}\cdot2^{21}\cdot4^{6}\cdot6^{4}\cdot12^{2}$ |