Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $1568$ | ||
| Index: | $672$ | $\PSL_2$-index: | $672$ | ||||
| Genus: | $45 = 1 + \frac{ 672 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (of which $2$ are rational) | Cusp widths | $7^{8}\cdot14^{4}\cdot28^{4}\cdot56^{8}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot3^{2}\cdot6^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $7 \le \gamma \le 24$ | ||||||
| $\overline{\Q}$-gonality: | $7 \le \gamma \le 24$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.672.45.32 |
Level structure
Jacobian
| Conductor: | $2^{159}\cdot7^{79}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{9}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| Newforms: | 14.2.a.a$^{3}$, 56.2.a.a, 56.2.a.b, 98.2.a.b$^{3}$, 196.2.a.b$^{2}$, 196.2.a.c$^{2}$, 224.2.b.a, 224.2.b.b, 392.2.a.c, 392.2.a.f, 392.2.a.g, 1568.2.b.f, 1568.2.b.g |
Rational points
This modular curve has 2 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(7)$ | $7$ | $24$ | $24$ | $0$ | $0$ | full Jacobian |
| 8.24.0.ba.1 | $8$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.24.0.ba.1 | $8$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
| 56.336.21.cj.1 | $56$ | $2$ | $2$ | $21$ | $1$ | $2\cdot4\cdot6\cdot12$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.1344.89.bbt.2 | $56$ | $2$ | $2$ | $89$ | $5$ | $1^{18}\cdot2^{4}\cdot4^{3}\cdot6$ |
| 56.1344.89.bby.2 | $56$ | $2$ | $2$ | $89$ | $17$ | $1^{18}\cdot2^{4}\cdot4^{3}\cdot6$ |
| 56.1344.89.bcc.1 | $56$ | $2$ | $2$ | $89$ | $11$ | $1^{18}\cdot2^{4}\cdot4^{3}\cdot6$ |
| 56.1344.89.bcg.1 | $56$ | $2$ | $2$ | $89$ | $2$ | $1^{18}\cdot2^{4}\cdot4^{3}\cdot6$ |
| 56.1344.89.bcl.2 | $56$ | $2$ | $2$ | $89$ | $3$ | $1^{18}\cdot2^{4}\cdot4^{3}\cdot6$ |
| 56.1344.89.bcq.2 | $56$ | $2$ | $2$ | $89$ | $13$ | $1^{18}\cdot2^{4}\cdot4^{3}\cdot6$ |
| 56.1344.89.bcu.1 | $56$ | $2$ | $2$ | $89$ | $11$ | $1^{18}\cdot2^{4}\cdot4^{3}\cdot6$ |
| 56.1344.89.bcy.1 | $56$ | $2$ | $2$ | $89$ | $6$ | $1^{18}\cdot2^{4}\cdot4^{3}\cdot6$ |
| 56.1344.93.fe.1 | $56$ | $2$ | $2$ | $93$ | $7$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.1344.93.ft.1 | $56$ | $2$ | $2$ | $93$ | $12$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.1344.93.fx.2 | $56$ | $2$ | $2$ | $93$ | $7$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.1344.93.fy.1 | $56$ | $2$ | $2$ | $93$ | $12$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.1344.93.gg.2 | $56$ | $2$ | $2$ | $93$ | $8$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.1344.93.gk.2 | $56$ | $2$ | $2$ | $93$ | $13$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.1344.93.go.2 | $56$ | $2$ | $2$ | $93$ | $8$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.1344.93.gs.2 | $56$ | $2$ | $2$ | $93$ | $13$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.1344.93.gw.2 | $56$ | $2$ | $2$ | $93$ | $12$ | $1^{20}\cdot2^{5}\cdot4^{3}\cdot6$ |
| 56.1344.93.ha.2 | $56$ | $2$ | $2$ | $93$ | $8$ | $1^{20}\cdot2^{5}\cdot4^{3}\cdot6$ |
| 56.1344.93.he.2 | $56$ | $2$ | $2$ | $93$ | $12$ | $1^{20}\cdot2^{5}\cdot4^{3}\cdot6$ |
| 56.1344.93.hi.2 | $56$ | $2$ | $2$ | $93$ | $8$ | $1^{20}\cdot2^{5}\cdot4^{3}\cdot6$ |
| 56.1344.93.ho.2 | $56$ | $2$ | $2$ | $93$ | $12$ | $1^{20}\cdot2^{5}\cdot4^{3}\cdot6$ |
| 56.1344.93.hs.2 | $56$ | $2$ | $2$ | $93$ | $8$ | $1^{20}\cdot2^{5}\cdot4^{3}\cdot6$ |
| 56.1344.93.hw.1 | $56$ | $2$ | $2$ | $93$ | $12$ | $1^{20}\cdot2^{5}\cdot4^{3}\cdot6$ |
| 56.1344.93.ia.1 | $56$ | $2$ | $2$ | $93$ | $8$ | $1^{20}\cdot2^{5}\cdot4^{3}\cdot6$ |
| 56.2016.133.pa.1 | $56$ | $3$ | $3$ | $133$ | $7$ | $1^{26}\cdot2^{10}\cdot4^{3}\cdot6^{3}\cdot12$ |