Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $1568$ | ||
| Index: | $1344$ | $\PSL_2$-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: | 56.1344.45.166 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}16&27\\31&12\end{bmatrix}$, $\begin{bmatrix}22&19\\5&48\end{bmatrix}$, $\begin{bmatrix}30&7\\21&44\end{bmatrix}$, $\begin{bmatrix}30&21\\49&2\end{bmatrix}$, $\begin{bmatrix}44&35\\3&12\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.672.45.gq.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $2$ |
| Cyclic 56-torsion field degree: | $48$ |
| Full 56-torsion field degree: | $2304$ |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(7)$ | $7$ | $48$ | $24$ | $0$ | $0$ | full Jacobian |
| 8.48.0-8.ba.2.3 | $8$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.0-8.ba.2.3 | $8$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
| 56.672.21-56.cj.1.20 | $56$ | $2$ | $2$ | $21$ | $1$ | $2\cdot4\cdot6\cdot12$ |
| 56.672.21-56.cj.1.30 | $56$ | $2$ | $2$ | $21$ | $1$ | $2\cdot4\cdot6\cdot12$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.2688.89-56.bbt.1.11 | $56$ | $2$ | $2$ | $89$ | $5$ | $1^{18}\cdot2^{4}\cdot4^{3}\cdot6$ |
| 56.2688.89-56.bby.1.6 | $56$ | $2$ | $2$ | $89$ | $17$ | $1^{18}\cdot2^{4}\cdot4^{3}\cdot6$ |
| 56.2688.89-56.bcc.2.12 | $56$ | $2$ | $2$ | $89$ | $11$ | $1^{18}\cdot2^{4}\cdot4^{3}\cdot6$ |
| 56.2688.89-56.bcg.2.10 | $56$ | $2$ | $2$ | $89$ | $2$ | $1^{18}\cdot2^{4}\cdot4^{3}\cdot6$ |
| 56.2688.89-56.bcl.1.12 | $56$ | $2$ | $2$ | $89$ | $3$ | $1^{18}\cdot2^{4}\cdot4^{3}\cdot6$ |
| 56.2688.89-56.bcq.1.7 | $56$ | $2$ | $2$ | $89$ | $13$ | $1^{18}\cdot2^{4}\cdot4^{3}\cdot6$ |
| 56.2688.89-56.bcu.2.16 | $56$ | $2$ | $2$ | $89$ | $11$ | $1^{18}\cdot2^{4}\cdot4^{3}\cdot6$ |
| 56.2688.89-56.bcy.2.15 | $56$ | $2$ | $2$ | $89$ | $6$ | $1^{18}\cdot2^{4}\cdot4^{3}\cdot6$ |
| 56.2688.93-56.fe.2.2 | $56$ | $2$ | $2$ | $93$ | $7$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.2688.93-56.ft.2.1 | $56$ | $2$ | $2$ | $93$ | $12$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.2688.93-56.fx.1.2 | $56$ | $2$ | $2$ | $93$ | $7$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.2688.93-56.fy.1.1 | $56$ | $2$ | $2$ | $93$ | $12$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.2688.93-56.gg.1.9 | $56$ | $2$ | $2$ | $93$ | $8$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.2688.93-56.gk.1.10 | $56$ | $2$ | $2$ | $93$ | $13$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.2688.93-56.go.1.9 | $56$ | $2$ | $2$ | $93$ | $8$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.2688.93-56.gs.1.10 | $56$ | $2$ | $2$ | $93$ | $13$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.2688.93-56.gw.1.14 | $56$ | $2$ | $2$ | $93$ | $12$ | $1^{20}\cdot2^{5}\cdot4^{3}\cdot6$ |
| 56.2688.93-56.ha.1.13 | $56$ | $2$ | $2$ | $93$ | $8$ | $1^{20}\cdot2^{5}\cdot4^{3}\cdot6$ |
| 56.2688.93-56.he.1.14 | $56$ | $2$ | $2$ | $93$ | $12$ | $1^{20}\cdot2^{5}\cdot4^{3}\cdot6$ |
| 56.2688.93-56.hi.1.13 | $56$ | $2$ | $2$ | $93$ | $8$ | $1^{20}\cdot2^{5}\cdot4^{3}\cdot6$ |
| 56.2688.93-56.ho.1.10 | $56$ | $2$ | $2$ | $93$ | $12$ | $1^{20}\cdot2^{5}\cdot4^{3}\cdot6$ |
| 56.2688.93-56.hs.1.9 | $56$ | $2$ | $2$ | $93$ | $8$ | $1^{20}\cdot2^{5}\cdot4^{3}\cdot6$ |
| 56.2688.93-56.hw.2.12 | $56$ | $2$ | $2$ | $93$ | $12$ | $1^{20}\cdot2^{5}\cdot4^{3}\cdot6$ |
| 56.2688.93-56.ia.2.11 | $56$ | $2$ | $2$ | $93$ | $8$ | $1^{20}\cdot2^{5}\cdot4^{3}\cdot6$ |
| 56.4032.133-56.pa.2.21 | $56$ | $3$ | $3$ | $133$ | $7$ | $1^{26}\cdot2^{10}\cdot4^{3}\cdot6^{3}\cdot12$ |