Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $3136$ | ||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-7$) |
Other labels
| Cummins and Pauli (CP) label: | 8F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.48.1.182 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}5&32\\0&31\end{bmatrix}$, $\begin{bmatrix}24&43\\21&8\end{bmatrix}$, $\begin{bmatrix}29&24\\0&43\end{bmatrix}$, $\begin{bmatrix}47&20\\36&3\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 56-isogeny field degree: | $32$ |
| Cyclic 56-torsion field degree: | $768$ |
| Full 56-torsion field degree: | $64512$ |
Jacobian
| Conductor: | $2^{6}\cdot7^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 3136.2.a.m |
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.24.0.bt.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 28.24.0.g.1 | $28$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.0.cl.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.0.el.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.24.1.cf.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.24.1.cx.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.24.1.dm.1 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.96.3.bh.1 | $56$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 56.96.3.bi.1 | $56$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 56.384.25.qy.1 | $56$ | $8$ | $8$ | $25$ | $13$ | $1^{20}\cdot2^{2}$ |
| 56.1008.73.ecn.1 | $56$ | $21$ | $21$ | $73$ | $34$ | $1^{16}\cdot2^{26}\cdot4$ |
| 56.1344.97.dwt.1 | $56$ | $28$ | $28$ | $97$ | $46$ | $1^{36}\cdot2^{28}\cdot4$ |
| 112.96.3.pw.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.py.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.tk.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.3.tm.1 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.96.5.js.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.96.5.ka.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.96.5.to.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 112.96.5.tw.1 | $112$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.96.3.xk.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.96.3.xl.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.144.9.bbar.1 | $168$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 168.192.9.dkt.1 | $168$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 280.96.3.fq.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.96.3.fr.1 | $280$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 280.240.17.ehx.1 | $280$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 280.288.17.qsd.1 | $280$ | $6$ | $6$ | $17$ | $?$ | not computed |