Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
| Index: | $5376$ | $\PSL_2$-index: | $2688$ | ||||
| Genus: | $185 = 1 + \frac{ 2688 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 80 }{2}$ | ||||||
| Cusps: | $80$ (none of which are rational) | Cusp widths | $14^{32}\cdot28^{16}\cdot56^{32}$ | Cusp orbits | $4^{5}\cdot6^{10}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $25$ | ||||||
| $\Q$-gonality: | $27 \le \gamma \le 56$ | ||||||
| $\overline{\Q}$-gonality: | $27 \le \gamma \le 56$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.5376.185.17581 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}19&4\\14&9\end{bmatrix}$, $\begin{bmatrix}25&48\\0&17\end{bmatrix}$, $\begin{bmatrix}39&48\\0&17\end{bmatrix}$, $\begin{bmatrix}41&44\\12&43\end{bmatrix}$, $\begin{bmatrix}55&0\\28&41\end{bmatrix}$ |
| $\GL_2(\Z/56\Z)$-subgroup: | $C_6^2:C_2^4$ |
| Contains $-I$: | no $\quad$ (see 56.2688.185.pe.2 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $4$ |
| Cyclic 56-torsion field degree: | $48$ |
| Full 56-torsion field degree: | $576$ |
Jacobian
| Conductor: | $2^{745}\cdot7^{324}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{59}\cdot2^{23}\cdot4^{8}\cdot6^{4}\cdot12^{2}$ |
| Newforms: | 14.2.a.a$^{6}$, 49.2.a.a$^{4}$, 56.2.a.a$^{2}$, 56.2.a.b$^{2}$, 56.2.b.a$^{2}$, 56.2.b.b$^{2}$, 98.2.a.a$^{3}$, 98.2.a.b$^{3}$, 196.2.a.b$^{4}$, 196.2.a.c$^{2}$, 224.2.b.a$^{2}$, 224.2.b.b$^{2}$, 392.2.a.b, 392.2.a.c$^{2}$, 392.2.a.d, 392.2.a.f$^{2}$, 392.2.a.g, 392.2.a.h, 392.2.b.a, 392.2.b.b, 392.2.b.c, 392.2.b.d, 392.2.b.e$^{2}$, 392.2.b.g, 448.2.a.a$^{2}$, 448.2.a.c$^{2}$, 448.2.a.d$^{2}$, 448.2.a.e$^{2}$, 448.2.a.g$^{2}$, 448.2.a.h$^{2}$, 1568.2.b.a, 1568.2.b.b, 1568.2.b.c, 1568.2.b.d, 1568.2.b.f$^{2}$, 1568.2.b.g, 3136.2.a.b$^{2}$, 3136.2.a.bc$^{2}$, 3136.2.a.bk, 3136.2.a.bm, 3136.2.a.bn, 3136.2.a.bp, 3136.2.a.bq, 3136.2.a.br, 3136.2.a.bs, 3136.2.a.bt, 3136.2.a.c, 3136.2.a.e, 3136.2.a.h$^{2}$, 3136.2.a.j$^{2}$, 3136.2.a.n, 3136.2.a.o, 3136.2.a.p, 3136.2.a.q, 3136.2.a.s$^{2}$, 3136.2.a.u$^{2}$, 3136.2.a.w, 3136.2.a.z |
Rational points
This modular curve has no $\Q_p$ points for $p=3,5,17,\ldots,409$, and therefore no 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 |
|---|---|---|---|---|---|---|
| 7.56.1.b.1 | $7$ | $96$ | $48$ | $1$ | $0$ | $1^{58}\cdot2^{23}\cdot4^{8}\cdot6^{4}\cdot12^{2}$ |
| 8.96.0-8.h.1.8 | $8$ | $56$ | $56$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.2688.89-56.ex.1.22 | $56$ | $2$ | $2$ | $89$ | $3$ | $1^{32}\cdot2^{12}\cdot4^{4}\cdot6^{2}\cdot12$ |
| 56.2688.89-56.ex.1.39 | $56$ | $2$ | $2$ | $89$ | $3$ | $1^{32}\cdot2^{12}\cdot4^{4}\cdot6^{2}\cdot12$ |
| 56.2688.89-56.fa.2.32 | $56$ | $2$ | $2$ | $89$ | $3$ | $1^{32}\cdot2^{12}\cdot4^{4}\cdot6^{2}\cdot12$ |
| 56.2688.89-56.fa.2.41 | $56$ | $2$ | $2$ | $89$ | $3$ | $1^{32}\cdot2^{12}\cdot4^{4}\cdot6^{2}\cdot12$ |
| 56.2688.89-56.im.1.15 | $56$ | $2$ | $2$ | $89$ | $25$ | $2^{8}\cdot4^{8}\cdot6^{4}\cdot12^{2}$ |
| 56.2688.89-56.im.1.21 | $56$ | $2$ | $2$ | $89$ | $25$ | $2^{8}\cdot4^{8}\cdot6^{4}\cdot12^{2}$ |
| 56.2688.93-56.cm.1.20 | $56$ | $2$ | $2$ | $93$ | $12$ | $1^{30}\cdot2^{11}\cdot4^{4}\cdot6^{2}\cdot12$ |
| 56.2688.93-56.cm.1.30 | $56$ | $2$ | $2$ | $93$ | $12$ | $1^{30}\cdot2^{11}\cdot4^{4}\cdot6^{2}\cdot12$ |
| 56.2688.93-56.ct.1.20 | $56$ | $2$ | $2$ | $93$ | $12$ | $1^{30}\cdot2^{11}\cdot4^{4}\cdot6^{2}\cdot12$ |
| 56.2688.93-56.ct.1.30 | $56$ | $2$ | $2$ | $93$ | $12$ | $1^{30}\cdot2^{11}\cdot4^{4}\cdot6^{2}\cdot12$ |
| 56.2688.93-56.ex.2.24 | $56$ | $2$ | $2$ | $93$ | $12$ | $1^{38}\cdot2^{5}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.2688.93-56.ex.2.27 | $56$ | $2$ | $2$ | $93$ | $12$ | $1^{38}\cdot2^{5}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.2688.93-56.ez.1.19 | $56$ | $2$ | $2$ | $93$ | $12$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
| 56.2688.93-56.ez.1.30 | $56$ | $2$ | $2$ | $93$ | $12$ | $1^{38}\cdot2^{9}\cdot4^{6}\cdot6^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.10752.369-56.hx.2.16 | $56$ | $2$ | $2$ | $369$ | $27$ | $2^{16}\cdot4^{16}\cdot8^{8}\cdot12^{2}$ |
| 56.10752.369-56.hz.2.16 | $56$ | $2$ | $2$ | $369$ | $29$ | $2^{16}\cdot4^{16}\cdot8^{8}\cdot12^{2}$ |
| 56.10752.369-56.iv.2.16 | $56$ | $2$ | $2$ | $369$ | $27$ | $2^{16}\cdot4^{16}\cdot8^{8}\cdot12^{2}$ |
| 56.10752.369-56.ix.2.16 | $56$ | $2$ | $2$ | $369$ | $29$ | $2^{16}\cdot4^{16}\cdot8^{8}\cdot12^{2}$ |
| 56.10752.385-56.nk.2.10 | $56$ | $2$ | $2$ | $385$ | $63$ | $1^{48}\cdot2^{34}\cdot4^{9}\cdot6^{4}\cdot12^{2}$ |
| 56.10752.385-56.oa.2.8 | $56$ | $2$ | $2$ | $385$ | $66$ | $1^{48}\cdot2^{34}\cdot4^{9}\cdot6^{4}\cdot12^{2}$ |
| 56.10752.385-56.qn.2.1 | $56$ | $2$ | $2$ | $385$ | $63$ | $1^{48}\cdot2^{34}\cdot4^{9}\cdot6^{4}\cdot12^{2}$ |
| 56.10752.385-56.qx.2.8 | $56$ | $2$ | $2$ | $385$ | $66$ | $1^{48}\cdot2^{34}\cdot4^{9}\cdot6^{4}\cdot12^{2}$ |
| 56.10752.385-56.bqw.2.8 | $56$ | $2$ | $2$ | $385$ | $65$ | $1^{48}\cdot2^{34}\cdot4^{9}\cdot6^{4}\cdot12^{2}$ |
| 56.10752.385-56.bqx.2.8 | $56$ | $2$ | $2$ | $385$ | $70$ | $1^{48}\cdot2^{34}\cdot4^{9}\cdot6^{4}\cdot12^{2}$ |
| 56.10752.385-56.brc.2.8 | $56$ | $2$ | $2$ | $385$ | $65$ | $1^{48}\cdot2^{34}\cdot4^{9}\cdot6^{4}\cdot12^{2}$ |
| 56.10752.385-56.brd.2.8 | $56$ | $2$ | $2$ | $385$ | $70$ | $1^{48}\cdot2^{34}\cdot4^{9}\cdot6^{4}\cdot12^{2}$ |
| 56.10752.385-56.bxh.3.15 | $56$ | $2$ | $2$ | $385$ | $27$ | $2^{12}\cdot4^{12}\cdot8^{9}\cdot12^{2}\cdot16^{2}$ |
| 56.10752.385-56.bxh.4.16 | $56$ | $2$ | $2$ | $385$ | $27$ | $2^{12}\cdot4^{12}\cdot8^{9}\cdot12^{2}\cdot16^{2}$ |
| 56.10752.385-56.bxj.3.15 | $56$ | $2$ | $2$ | $385$ | $29$ | $2^{12}\cdot4^{12}\cdot8^{9}\cdot12^{2}\cdot16^{2}$ |
| 56.10752.385-56.bxj.4.16 | $56$ | $2$ | $2$ | $385$ | $29$ | $2^{12}\cdot4^{12}\cdot8^{9}\cdot12^{2}\cdot16^{2}$ |
| 56.16128.553-56.if.1.24 | $56$ | $3$ | $3$ | $553$ | $85$ | $1^{116}\cdot2^{42}\cdot4^{12}\cdot6^{12}\cdot12^{4}$ |