Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $112$ | ||
| 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 $8$ are rational) | Cusp widths | $4^{8}\cdot8^{2}\cdot28^{8}\cdot56^{2}$ | Cusp orbits | $1^{8}\cdot2^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 8$ | ||||||
| Rational cusps: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 56P23 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.384.23.2 |
Level structure
Jacobian
| Conductor: | $2^{64}\cdot7^{23}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{11}\cdot2^{2}\cdot4^{2}$ |
| Newforms: | 14.2.a.a$^{4}$, 56.2.a.a$^{2}$, 56.2.a.b$^{2}$, 56.2.b.a$^{2}$, 56.2.b.b$^{2}$, 112.2.a.a, 112.2.a.b, 112.2.a.c |
Rational points
This modular curve has 8 rational cusps but no known non-cuspidal 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 |
|---|---|---|---|---|---|---|
| $X_0(7)$ | $7$ | $48$ | $48$ | $0$ | $0$ | full Jacobian |
| 8.48.0.c.1 | $8$ | $8$ | $8$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.0.c.1 | $8$ | $8$ | $8$ | $0$ | $0$ | full Jacobian |
| 28.192.11.b.1 | $28$ | $2$ | $2$ | $11$ | $1$ | $2^{2}\cdot4^{2}$ |
| 56.192.11.s.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.g.1 | $56$ | $2$ | $2$ | $45$ | $1$ | $2^{7}\cdot4^{2}$ |
| 56.768.45.g.2 | $56$ | $2$ | $2$ | $45$ | $1$ | $2^{7}\cdot4^{2}$ |
| 56.768.45.h.1 | $56$ | $2$ | $2$ | $45$ | $3$ | $2^{7}\cdot4^{2}$ |
| 56.768.45.h.2 | $56$ | $2$ | $2$ | $45$ | $3$ | $2^{7}\cdot4^{2}$ |
| 56.768.45.be.1 | $56$ | $2$ | $2$ | $45$ | $1$ | $2^{7}\cdot4^{2}$ |
| 56.768.45.be.2 | $56$ | $2$ | $2$ | $45$ | $1$ | $2^{7}\cdot4^{2}$ |
| 56.768.45.bf.1 | $56$ | $2$ | $2$ | $45$ | $3$ | $2^{7}\cdot4^{2}$ |
| 56.768.45.bf.2 | $56$ | $2$ | $2$ | $45$ | $3$ | $2^{7}\cdot4^{2}$ |
| 56.768.49.fr.1 | $56$ | $2$ | $2$ | $49$ | $6$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.768.49.fr.2 | $56$ | $2$ | $2$ | $49$ | $6$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.768.49.fs.1 | $56$ | $2$ | $2$ | $49$ | $8$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.768.49.fs.2 | $56$ | $2$ | $2$ | $49$ | $8$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.768.49.ft.1 | $56$ | $2$ | $2$ | $49$ | $1$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.768.49.ft.2 | $56$ | $2$ | $2$ | $49$ | $1$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.768.49.ft.3 | $56$ | $2$ | $2$ | $49$ | $1$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.768.49.ft.4 | $56$ | $2$ | $2$ | $49$ | $1$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.768.49.fu.1 | $56$ | $2$ | $2$ | $49$ | $3$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.768.49.fu.2 | $56$ | $2$ | $2$ | $49$ | $3$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.768.49.fu.3 | $56$ | $2$ | $2$ | $49$ | $3$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.768.49.fu.4 | $56$ | $2$ | $2$ | $49$ | $3$ | $2^{5}\cdot4^{2}\cdot8$ |
| 56.768.49.fv.1 | $56$ | $2$ | $2$ | $49$ | $7$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.768.49.fv.2 | $56$ | $2$ | $2$ | $49$ | $7$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.768.49.fw.1 | $56$ | $2$ | $2$ | $49$ | $3$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.768.49.fw.2 | $56$ | $2$ | $2$ | $49$ | $3$ | $1^{10}\cdot2^{4}\cdot4^{2}$ |
| 56.768.53.cu.1 | $56$ | $2$ | $2$ | $53$ | $6$ | $1^{10}\cdot2^{6}\cdot4^{2}$ |
| 56.768.53.cv.1 | $56$ | $2$ | $2$ | $53$ | $3$ | $1^{10}\cdot2^{6}\cdot4^{2}$ |
| 56.768.53.cw.1 | $56$ | $2$ | $2$ | $53$ | $8$ | $1^{10}\cdot2^{6}\cdot4^{2}$ |
| 56.768.53.cx.1 | $56$ | $2$ | $2$ | $53$ | $7$ | $1^{10}\cdot2^{6}\cdot4^{2}$ |
| 56.768.53.cy.1 | $56$ | $2$ | $2$ | $53$ | $1$ | $2^{3}\cdot8^{3}$ |
| 56.768.53.cy.2 | $56$ | $2$ | $2$ | $53$ | $1$ | $2^{3}\cdot8^{3}$ |
| 56.768.53.cz.1 | $56$ | $2$ | $2$ | $53$ | $3$ | $2^{3}\cdot8^{3}$ |
| 56.768.53.cz.2 | $56$ | $2$ | $2$ | $53$ | $3$ | $2^{3}\cdot8^{3}$ |
| 56.1152.67.n.1 | $56$ | $3$ | $3$ | $67$ | $1$ | $2^{10}\cdot12^{2}$ |
| 56.1152.67.n.2 | $56$ | $3$ | $3$ | $67$ | $1$ | $2^{10}\cdot12^{2}$ |
| 56.1152.67.bc.1 | $56$ | $3$ | $3$ | $67$ | $9$ | $1^{20}\cdot6^{4}$ |
| 56.2688.185.m.1 | $56$ | $7$ | $7$ | $185$ | $18$ | $1^{48}\cdot2^{21}\cdot4^{6}\cdot6^{4}\cdot12^{2}$ |