Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
| 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 | $2^{5}\cdot6^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $12$ | ||||||
| $\Q$-gonality: | $14 \le \gamma \le 28$ | ||||||
| $\overline{\Q}$-gonality: | $14 \le \gamma \le 28$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.2688.93.113 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}13&28\\34&1\end{bmatrix}$, $\begin{bmatrix}15&12\\36&27\end{bmatrix}$, $\begin{bmatrix}31&8\\4&25\end{bmatrix}$, $\begin{bmatrix}31&28\\52&11\end{bmatrix}$, $\begin{bmatrix}33&32\\32&9\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.1344.93.ez.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $4$ |
| Cyclic 56-torsion field degree: | $48$ |
| Full 56-torsion field degree: | $1152$ |
Jacobian
| Conductor: | $2^{375}\cdot7^{163}$ |
| 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, 56.2.b.a, 56.2.b.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, 392.2.b.e, 392.2.b.g, 448.2.a.a, 448.2.a.c, 448.2.a.d, 448.2.a.e, 448.2.a.g, 448.2.a.h, 1568.2.b.f, 1568.2.b.g, 3136.2.a.b, 3136.2.a.bc, 3136.2.a.bk, 3136.2.a.bm, 3136.2.a.bn, 3136.2.a.bp, 3136.2.a.br, 3136.2.a.bs, 3136.2.a.h, 3136.2.a.j, 3136.2.a.s, 3136.2.a.u |
Rational points
This modular curve has real points and $\Q_p$ points for good $p < 8192$, but no known 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(7)$ | $7$ | $96$ | $48$ | $0$ | $0$ | full Jacobian |
| 8.96.0-8.h.1.8 | $8$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.0-8.h.1.8 | $8$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
| 56.1344.45-56.u.1.46 | $56$ | $2$ | $2$ | $45$ | $1$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.1344.45-56.u.1.62 | $56$ | $2$ | $2$ | $45$ | $1$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.1344.45-56.bb.1.48 | $56$ | $2$ | $2$ | $45$ | $1$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.1344.45-56.bb.1.63 | $56$ | $2$ | $2$ | $45$ | $1$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.1344.45-56.ci.1.39 | $56$ | $2$ | $2$ | $45$ | $12$ | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.1344.45-56.ci.1.40 | $56$ | $2$ | $2$ | $45$ | $12$ | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
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.nc.2.23 | $56$ | $2$ | $2$ | $185$ | $27$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.185-56.ng.1.16 | $56$ | $2$ | $2$ | $185$ | $35$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.185-56.od.1.14 | $56$ | $2$ | $2$ | $185$ | $20$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.185-56.oh.1.14 | $56$ | $2$ | $2$ | $185$ | $33$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.185-56.pe.2.21 | $56$ | $2$ | $2$ | $185$ | $25$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.185-56.pi.1.16 | $56$ | $2$ | $2$ | $185$ | $31$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.185-56.qf.1.12 | $56$ | $2$ | $2$ | $185$ | $24$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.185-56.qj.1.12 | $56$ | $2$ | $2$ | $185$ | $33$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.lc.2.8 | $56$ | $2$ | $2$ | $193$ | $29$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.lq.2.2 | $56$ | $2$ | $2$ | $193$ | $32$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.px.2.5 | $56$ | $2$ | $2$ | $193$ | $29$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.ql.2.6 | $56$ | $2$ | $2$ | $193$ | $32$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.bli.1.12 | $56$ | $2$ | $2$ | $193$ | $31$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.blj.1.9 | $56$ | $2$ | $2$ | $193$ | $36$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.bls.1.11 | $56$ | $2$ | $2$ | $193$ | $31$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.blt.1.11 | $56$ | $2$ | $2$ | $193$ | $36$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.btm.1.13 | $56$ | $2$ | $2$ | $193$ | $33$ | $1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.btn.1.16 | $56$ | $2$ | $2$ | $193$ | $33$ | $1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.bto.1.15 | $56$ | $2$ | $2$ | $193$ | $33$ | $1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.btp.1.15 | $56$ | $2$ | $2$ | $193$ | $33$ | $1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.buq.1.10 | $56$ | $2$ | $2$ | $193$ | $33$ | $1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.bur.1.16 | $56$ | $2$ | $2$ | $193$ | $33$ | $1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.buw.1.15 | $56$ | $2$ | $2$ | $193$ | $33$ | $1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.bux.1.14 | $56$ | $2$ | $2$ | $193$ | $33$ | $1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$ |
| 56.8064.277-56.hv.1.23 | $56$ | $3$ | $3$ | $277$ | $41$ | $1^{58}\cdot2^{21}\cdot4^{6}\cdot6^{6}\cdot12^{2}$ |