Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $1568$ | ||
| Index: | $2688$ | $\PSL_2$-index: | $1344$ | ||||
| Genus: | $93 = 1 + \frac{ 1344 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 40 }{2}$ | ||||||
| Cusps: | $40$ (none of which are rational) | Cusp widths | $14^{16}\cdot28^{8}\cdot56^{16}$ | Cusp orbits | $2^{3}\cdot4\cdot6^{3}\cdot12$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $8$ | ||||||
| $\Q$-gonality: | $14 \le \gamma \le 48$ | ||||||
| $\overline{\Q}$-gonality: | $14 \le \gamma \le 28$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.2688.93.1695 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}2&25\\29&54\end{bmatrix}$, $\begin{bmatrix}11&20\\34&17\end{bmatrix}$, $\begin{bmatrix}49&22\\12&7\end{bmatrix}$, $\begin{bmatrix}55&24\\6&37\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.1344.93.gg.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $2$ |
| Cyclic 56-torsion field degree: | $48$ |
| Full 56-torsion field degree: | $1152$ |
Jacobian
| Conductor: | $2^{354}\cdot7^{175}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{21}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| Newforms: | 14.2.a.a$^{3}$, 56.2.a.a, 56.2.a.b, 98.2.a.a, 98.2.a.b$^{4}$, 196.2.a.a, 196.2.a.b$^{2}$, 196.2.a.c$^{3}$, 224.2.b.a, 224.2.b.b, 392.2.a.a, 392.2.a.b, 392.2.a.c, 392.2.a.d, 392.2.a.e, 392.2.a.f, 392.2.a.g$^{2}$, 784.2.a.b, 784.2.a.c, 784.2.a.e, 784.2.a.g, 784.2.a.i, 784.2.a.j, 784.2.a.k, 784.2.a.l, 784.2.a.m, 1568.2.b.a, 1568.2.b.d, 1568.2.b.e, 1568.2.b.f, 1568.2.b.g$^{2}$ |
Rational points
This modular curve has no $\Q_p$ points for $p=79$, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.96.0-56.bg.2.5 | $56$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
| 56.1344.45-56.fd.1.10 | $56$ | $2$ | $2$ | $45$ | $8$ | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.1344.45-56.fd.1.17 | $56$ | $2$ | $2$ | $45$ | $8$ | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.1344.45-56.gq.1.16 | $56$ | $2$ | $2$ | $45$ | $1$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.1344.45-56.gq.1.21 | $56$ | $2$ | $2$ | $45$ | $1$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.1344.45-56.gr.1.16 | $56$ | $2$ | $2$ | $45$ | $1$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.1344.45-56.gr.1.21 | $56$ | $2$ | $2$ | $45$ | $1$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.5376.185-56.uf.1.5 | $56$ | $2$ | $2$ | $185$ | $19$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.185-56.ui.1.5 | $56$ | $2$ | $2$ | $185$ | $35$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.185-56.us.2.7 | $56$ | $2$ | $2$ | $185$ | $25$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.185-56.uu.2.7 | $56$ | $2$ | $2$ | $185$ | $20$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.185-56.vu.2.5 | $56$ | $2$ | $2$ | $185$ | $24$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.185-56.vw.2.5 | $56$ | $2$ | $2$ | $185$ | $25$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.185-56.wc.1.3 | $56$ | $2$ | $2$ | $185$ | $31$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.185-56.wf.1.3 | $56$ | $2$ | $2$ | $185$ | $17$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.8064.277-56.kk.1.5 | $56$ | $3$ | $3$ | $277$ | $29$ | $1^{58}\cdot2^{21}\cdot4^{6}\cdot6^{6}\cdot12^{2}$ |