Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $1568$ | ||
| Index: | $2688$ | $\PSL_2$-index: | $1344$ | ||||
| Genus: | $97 = 1 + \frac{ 1344 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 32 }{2}$ | ||||||
| Cusps: | $32$ (of which $2$ are rational) | Cusp widths | $28^{16}\cdot56^{16}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot3^{2}\cdot6^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $10$ | ||||||
| $\Q$-gonality: | $14 \le \gamma \le 48$ | ||||||
| $\overline{\Q}$-gonality: | $14 \le \gamma \le 48$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.2688.97.125 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}1&26\\30&41\end{bmatrix}$, $\begin{bmatrix}3&14\\30&55\end{bmatrix}$, $\begin{bmatrix}31&50\\42&11\end{bmatrix}$, $\begin{bmatrix}43&40\\42&41\end{bmatrix}$, $\begin{bmatrix}45&14\\8&15\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.1344.97.df.2 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: | $2^{371}\cdot7^{167}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{13}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| Newforms: | 14.2.a.a$^{3}$, 32.2.a.a$^{2}$, 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.a.a, 224.2.a.b, 224.2.a.c, 224.2.a.d, 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, 1568.2.a.l, 1568.2.a.m, 1568.2.a.o, 1568.2.a.p, 1568.2.a.q, 1568.2.a.r, 1568.2.a.s, 1568.2.a.w, 1568.2.a.x, 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(7)$ | $7$ | $96$ | $48$ | $0$ | $0$ | full Jacobian |
| 8.96.1-8.i.2.5 | $8$ | $28$ | $28$ | $1$ | $0$ | $1^{12}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.1-8.i.2.5 | $8$ | $28$ | $28$ | $1$ | $0$ | $1^{12}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.1344.45-56.u.2.4 | $56$ | $2$ | $2$ | $45$ | $1$ | $1^{4}\cdot2^{11}\cdot4^{2}\cdot6\cdot12$ |
| 56.1344.45-56.u.2.48 | $56$ | $2$ | $2$ | $45$ | $1$ | $1^{4}\cdot2^{11}\cdot4^{2}\cdot6\cdot12$ |
| 56.1344.45-56.bb.1.1 | $56$ | $2$ | $2$ | $45$ | $1$ | $1^{4}\cdot2^{11}\cdot4^{2}\cdot6\cdot12$ |
| 56.1344.45-56.bb.1.47 | $56$ | $2$ | $2$ | $45$ | $1$ | $1^{4}\cdot2^{11}\cdot4^{2}\cdot6\cdot12$ |
| 56.1344.49-56.c.1.1 | $56$ | $2$ | $2$ | $49$ | $10$ | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.1344.49-56.c.1.43 | $56$ | $2$ | $2$ | $49$ | $10$ | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.5376.193-56.gb.1.8 | $56$ | $2$ | $2$ | $193$ | $24$ | $1^{30}\cdot2^{15}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.go.2.3 | $56$ | $2$ | $2$ | $193$ | $40$ | $1^{30}\cdot2^{15}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.iv.1.8 | $56$ | $2$ | $2$ | $193$ | $22$ | $1^{30}\cdot2^{15}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.ji.2.5 | $56$ | $2$ | $2$ | $193$ | $36$ | $1^{30}\cdot2^{15}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.lj.1.3 | $56$ | $2$ | $2$ | $193$ | $22$ | $1^{24}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.lq.2.3 | $56$ | $2$ | $2$ | $193$ | $32$ | $1^{24}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.lz.1.1 | $56$ | $2$ | $2$ | $193$ | $24$ | $1^{24}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.me.2.3 | $56$ | $2$ | $2$ | $193$ | $34$ | $1^{24}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.mp.2.1 | $56$ | $2$ | $2$ | $193$ | $32$ | $1^{40}\cdot2^{10}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.ms.1.1 | $56$ | $2$ | $2$ | $193$ | $24$ | $1^{40}\cdot2^{10}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.nf.2.1 | $56$ | $2$ | $2$ | $193$ | $32$ | $1^{40}\cdot2^{10}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.ng.1.1 | $56$ | $2$ | $2$ | $193$ | $24$ | $1^{40}\cdot2^{10}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.oo.1.1 | $56$ | $2$ | $2$ | $193$ | $32$ | $1^{40}\cdot2^{10}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.op.2.1 | $56$ | $2$ | $2$ | $193$ | $24$ | $1^{40}\cdot2^{10}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.pc.1.1 | $56$ | $2$ | $2$ | $193$ | $32$ | $1^{40}\cdot2^{10}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.pf.2.1 | $56$ | $2$ | $2$ | $193$ | $24$ | $1^{40}\cdot2^{10}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.pq.2.1 | $56$ | $2$ | $2$ | $193$ | $24$ | $1^{24}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.pv.1.1 | $56$ | $2$ | $2$ | $193$ | $34$ | $1^{24}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.qe.2.1 | $56$ | $2$ | $2$ | $193$ | $22$ | $1^{24}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.ql.1.1 | $56$ | $2$ | $2$ | $193$ | $32$ | $1^{24}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.5376.193-56.vo.2.1 | $56$ | $2$ | $2$ | $193$ | $29$ | $1^{30}\cdot2^{15}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.vz.2.1 | $56$ | $2$ | $2$ | $193$ | $30$ | $1^{30}\cdot2^{15}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.xs.2.1 | $56$ | $2$ | $2$ | $193$ | $25$ | $1^{30}\cdot2^{15}\cdot4^{6}\cdot6^{2}$ |
| 56.5376.193-56.yd.2.1 | $56$ | $2$ | $2$ | $193$ | $30$ | $1^{30}\cdot2^{15}\cdot4^{6}\cdot6^{2}$ |
| 56.8064.289-56.su.2.1 | $56$ | $3$ | $3$ | $289$ | $38$ | $1^{38}\cdot2^{33}\cdot4^{7}\cdot6^{6}\cdot12^{2}$ |