Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
| Index: | $1344$ | $\PSL_2$-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 | $28^{32}\cdot56^{8}$ | 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: | 56.1344.93.5 |
Level structure
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(7)$ | $7$ | $48$ | $48$ | $0$ | $0$ | full Jacobian |
| 8.48.0.e.2 | $8$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.0.e.2 | $8$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
| 56.672.45.c.1 | $56$ | $2$ | $2$ | $45$ | $12$ | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.672.45.u.2 | $56$ | $2$ | $2$ | $45$ | $1$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
| 56.672.45.bb.2 | $56$ | $2$ | $2$ | $45$ | $1$ | $1^{12}\cdot2^{7}\cdot4\cdot6\cdot12$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.2688.185.v.1 | $56$ | $2$ | $2$ | $185$ | $27$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.2688.185.bb.1 | $56$ | $2$ | $2$ | $185$ | $35$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.2688.185.bv.1 | $56$ | $2$ | $2$ | $185$ | $25$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.2688.185.cb.2 | $56$ | $2$ | $2$ | $185$ | $31$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.2688.185.cx.2 | $56$ | $2$ | $2$ | $185$ | $33$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.2688.185.dd.1 | $56$ | $2$ | $2$ | $185$ | $24$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.2688.185.dv.1 | $56$ | $2$ | $2$ | $185$ | $33$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.2688.185.eb.1 | $56$ | $2$ | $2$ | $185$ | $20$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.2688.193.lq.1 | $56$ | $2$ | $2$ | $193$ | $32$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.2688.193.ql.1 | $56$ | $2$ | $2$ | $193$ | $32$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.2688.193.qp.1 | $56$ | $2$ | $2$ | $193$ | $36$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.2688.193.qz.1 | $56$ | $2$ | $2$ | $193$ | $36$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.2688.193.rh.1 | $56$ | $2$ | $2$ | $193$ | $33$ | $1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$ |
| 56.2688.193.rn.2 | $56$ | $2$ | $2$ | $193$ | $33$ | $1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$ |
| 56.2688.193.rz.2 | $56$ | $2$ | $2$ | $193$ | $33$ | $1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$ |
| 56.2688.193.sb.2 | $56$ | $2$ | $2$ | $193$ | $33$ | $1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$ |
| 56.2688.193.sp.2 | $56$ | $2$ | $2$ | $193$ | $33$ | $1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$ |
| 56.2688.193.sr.1 | $56$ | $2$ | $2$ | $193$ | $33$ | $1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$ |
| 56.2688.193.td.2 | $56$ | $2$ | $2$ | $193$ | $33$ | $1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$ |
| 56.2688.193.tj.2 | $56$ | $2$ | $2$ | $193$ | $33$ | $1^{32}\cdot2^{16}\cdot4^{6}\cdot6^{2}$ |
| 56.2688.193.tr.1 | $56$ | $2$ | $2$ | $193$ | $31$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.2688.193.ub.1 | $56$ | $2$ | $2$ | $193$ | $31$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.2688.193.ue.1 | $56$ | $2$ | $2$ | $193$ | $29$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.2688.193.ul.1 | $56$ | $2$ | $2$ | $193$ | $29$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.4032.277.by.2 | $56$ | $3$ | $3$ | $277$ | $41$ | $1^{58}\cdot2^{21}\cdot4^{6}\cdot6^{6}\cdot12^{2}$ |