Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $392$ | ||
| Index: | $672$ | $\PSL_2$-index: | $336$ | ||||
| Genus: | $21 = 1 + \frac{ 336 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $4$ are rational) | Cusp widths | $7^{8}\cdot14^{4}\cdot56^{4}$ | Cusp orbits | $1^{4}\cdot3^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $6 \le \gamma \le 12$ | ||||||
| $\overline{\Q}$-gonality: | $6 \le \gamma \le 12$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 56F21 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.672.21.66 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}9&52\\28&17\end{bmatrix}$, $\begin{bmatrix}13&2\\2&29\end{bmatrix}$, $\begin{bmatrix}18&21\\13&10\end{bmatrix}$, $\begin{bmatrix}38&7\\35&10\end{bmatrix}$, $\begin{bmatrix}50&21\\21&50\end{bmatrix}$, $\begin{bmatrix}55&12\\42&45\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.336.21.cj.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $2$ |
| Cyclic 56-torsion field degree: | $48$ |
| Full 56-torsion field degree: | $4608$ |
Jacobian
| Conductor: | $2^{39}\cdot7^{37}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{9}\cdot2^{6}$ |
| 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$^{2}$, 392.2.a.c, 392.2.a.f, 392.2.a.g |
Rational points
This modular curve has 4 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(7)$ | $7$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 8.24.0-8.n.1.4 | $8$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.24.0-8.n.1.4 | $8$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
| 56.336.9-28.c.1.2 | $56$ | $2$ | $2$ | $9$ | $0$ | $1^{6}\cdot2^{3}$ |
| 56.336.9-28.c.1.14 | $56$ | $2$ | $2$ | $9$ | $0$ | $1^{6}\cdot2^{3}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.1344.41-56.mz.1.23 | $56$ | $2$ | $2$ | $41$ | $5$ | $1^{18}\cdot2$ |
| 56.1344.41-56.nb.1.16 | $56$ | $2$ | $2$ | $41$ | $17$ | $1^{18}\cdot2$ |
| 56.1344.41-56.nd.1.20 | $56$ | $2$ | $2$ | $41$ | $11$ | $1^{18}\cdot2$ |
| 56.1344.41-56.nf.1.20 | $56$ | $2$ | $2$ | $41$ | $2$ | $1^{18}\cdot2$ |
| 56.1344.41-56.of.1.24 | $56$ | $2$ | $2$ | $41$ | $3$ | $1^{18}\cdot2$ |
| 56.1344.41-56.oh.1.16 | $56$ | $2$ | $2$ | $41$ | $13$ | $1^{18}\cdot2$ |
| 56.1344.41-56.oj.1.24 | $56$ | $2$ | $2$ | $41$ | $11$ | $1^{18}\cdot2$ |
| 56.1344.41-56.ol.1.24 | $56$ | $2$ | $2$ | $41$ | $6$ | $1^{18}\cdot2$ |
| 56.1344.45-56.cj.1.1 | $56$ | $2$ | $2$ | $45$ | $7$ | $1^{12}\cdot2^{6}$ |
| 56.1344.45-56.cr.1.14 | $56$ | $2$ | $2$ | $45$ | $12$ | $1^{12}\cdot2^{6}$ |
| 56.1344.45-56.dj.1.2 | $56$ | $2$ | $2$ | $45$ | $7$ | $1^{12}\cdot2^{6}$ |
| 56.1344.45-56.dk.1.15 | $56$ | $2$ | $2$ | $45$ | $12$ | $1^{12}\cdot2^{6}$ |
| 56.1344.45-56.fd.1.18 | $56$ | $2$ | $2$ | $45$ | $8$ | $1^{12}\cdot2^{6}$ |
| 56.1344.45-56.ff.1.18 | $56$ | $2$ | $2$ | $45$ | $13$ | $1^{12}\cdot2^{6}$ |
| 56.1344.45-56.fh.1.18 | $56$ | $2$ | $2$ | $45$ | $8$ | $1^{12}\cdot2^{6}$ |
| 56.1344.45-56.fj.1.18 | $56$ | $2$ | $2$ | $45$ | $13$ | $1^{12}\cdot2^{6}$ |
| 56.1344.45-56.gq.1.14 | $56$ | $2$ | $2$ | $45$ | $1$ | $2\cdot4\cdot6\cdot12$ |
| 56.1344.45-56.gq.2.32 | $56$ | $2$ | $2$ | $45$ | $1$ | $2\cdot4\cdot6\cdot12$ |
| 56.1344.45-56.gr.1.14 | $56$ | $2$ | $2$ | $45$ | $1$ | $2\cdot4\cdot6\cdot12$ |
| 56.1344.45-56.gr.2.32 | $56$ | $2$ | $2$ | $45$ | $1$ | $2\cdot4\cdot6\cdot12$ |
| 56.1344.45-56.gs.1.16 | $56$ | $2$ | $2$ | $45$ | $1$ | $2^{3}\cdot4^{3}\cdot6$ |
| 56.1344.45-56.gs.2.28 | $56$ | $2$ | $2$ | $45$ | $1$ | $2^{3}\cdot4^{3}\cdot6$ |
| 56.1344.45-56.gt.1.16 | $56$ | $2$ | $2$ | $45$ | $1$ | $2^{3}\cdot4^{3}\cdot6$ |
| 56.1344.45-56.gt.2.28 | $56$ | $2$ | $2$ | $45$ | $1$ | $2^{3}\cdot4^{3}\cdot6$ |
| 56.1344.45-56.gu.1.14 | $56$ | $2$ | $2$ | $45$ | $1$ | $2^{3}\cdot4^{3}\cdot6$ |
| 56.1344.45-56.gu.2.32 | $56$ | $2$ | $2$ | $45$ | $1$ | $2^{3}\cdot4^{3}\cdot6$ |
| 56.1344.45-56.gv.1.14 | $56$ | $2$ | $2$ | $45$ | $1$ | $2^{3}\cdot4^{3}\cdot6$ |
| 56.1344.45-56.gv.2.32 | $56$ | $2$ | $2$ | $45$ | $1$ | $2^{3}\cdot4^{3}\cdot6$ |
| 56.1344.45-56.gw.1.16 | $56$ | $2$ | $2$ | $45$ | $1$ | $2\cdot4\cdot6\cdot12$ |
| 56.1344.45-56.gw.2.28 | $56$ | $2$ | $2$ | $45$ | $1$ | $2\cdot4\cdot6\cdot12$ |
| 56.1344.45-56.gx.1.16 | $56$ | $2$ | $2$ | $45$ | $1$ | $2\cdot4\cdot6\cdot12$ |
| 56.1344.45-56.gx.2.28 | $56$ | $2$ | $2$ | $45$ | $1$ | $2\cdot4\cdot6\cdot12$ |
| 56.1344.45-56.gz.1.22 | $56$ | $2$ | $2$ | $45$ | $12$ | $1^{20}\cdot2^{2}$ |
| 56.1344.45-56.hb.1.22 | $56$ | $2$ | $2$ | $45$ | $8$ | $1^{20}\cdot2^{2}$ |
| 56.1344.45-56.hd.1.22 | $56$ | $2$ | $2$ | $45$ | $12$ | $1^{20}\cdot2^{2}$ |
| 56.1344.45-56.hf.1.22 | $56$ | $2$ | $2$ | $45$ | $8$ | $1^{20}\cdot2^{2}$ |
| 56.1344.45-56.if.1.20 | $56$ | $2$ | $2$ | $45$ | $12$ | $1^{20}\cdot2^{2}$ |
| 56.1344.45-56.ih.1.20 | $56$ | $2$ | $2$ | $45$ | $8$ | $1^{20}\cdot2^{2}$ |
| 56.1344.45-56.ij.1.20 | $56$ | $2$ | $2$ | $45$ | $12$ | $1^{20}\cdot2^{2}$ |
| 56.1344.45-56.il.1.20 | $56$ | $2$ | $2$ | $45$ | $8$ | $1^{20}\cdot2^{2}$ |
| 56.2016.61-56.hf.1.34 | $56$ | $3$ | $3$ | $61$ | $7$ | $1^{26}\cdot2^{7}$ |