Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
| Index: | $5376$ | $\PSL_2$-index: | $2688$ | ||||
| Genus: | $193 = 1 + \frac{ 2688 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 64 }{2}$ | ||||||
| Cusps: | $64$ (none of which are rational) | Cusp widths | $28^{32}\cdot56^{32}$ | Cusp orbits | $2^{4}\cdot4^{2}\cdot6^{4}\cdot12^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $29$ | ||||||
| $\Q$-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: | 56.5376.193.84 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}9&24\\52&33\end{bmatrix}$, $\begin{bmatrix}17&8\\28&25\end{bmatrix}$, $\begin{bmatrix}19&4\\42&9\end{bmatrix}$, $\begin{bmatrix}45&0\\22&43\end{bmatrix}$, $\begin{bmatrix}53&0\\42&11\end{bmatrix}$ |
| $\GL_2(\Z/56\Z)$-subgroup: | $C_6^2:C_2^4$ |
| Contains $-I$: | no $\quad$ (see 56.2688.193.lc.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: | $2^{858}\cdot7^{335}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{37}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$ |
| Newforms: | 14.2.a.a$^{4}$, 56.2.a.a$^{2}$, 56.2.a.b$^{2}$, 56.2.b.a, 56.2.b.b, 64.2.a.a$^{2}$, 98.2.a.b$^{4}$, 112.2.a.a, 112.2.a.b, 112.2.a.c, 196.2.a.b$^{3}$, 196.2.a.c$^{3}$, 224.2.b.a$^{3}$, 224.2.b.b$^{3}$, 392.2.a.c$^{2}$, 392.2.a.f$^{2}$, 392.2.a.g$^{2}$, 392.2.b.e, 392.2.b.g, 448.2.a.a, 448.2.a.b, 448.2.a.c, 448.2.a.d, 448.2.a.e, 448.2.a.f, 448.2.a.g, 448.2.a.h, 448.2.a.i, 448.2.a.j, 784.2.a.a, 784.2.a.d, 784.2.a.h, 784.2.a.k, 784.2.a.l, 784.2.a.m, 1568.2.b.f$^{3}$, 1568.2.b.g$^{3}$, 3136.2.a.b, 3136.2.a.bc, 3136.2.a.bd, 3136.2.a.bh, 3136.2.a.bi, 3136.2.a.bj, 3136.2.a.bk, 3136.2.a.bl, 3136.2.a.bm, 3136.2.a.bn, 3136.2.a.bo, 3136.2.a.bp, 3136.2.a.br, 3136.2.a.bs, 3136.2.a.bv, 3136.2.a.bw, 3136.2.a.bz, 3136.2.a.h, 3136.2.a.j, 3136.2.a.s, 3136.2.a.u |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=31,79,223$, 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(7)$ | $7$ | $192$ | $96$ | $0$ | $0$ | full Jacobian |
| 8.192.1-8.a.1.6 | $8$ | $28$ | $28$ | $1$ | $0$ | $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.192.1-8.a.1.6 | $8$ | $28$ | $28$ | $1$ | $0$ | $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$ |
| 56.2688.93-56.a.1.6 | $56$ | $2$ | $2$ | $93$ | $7$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.2688.93-56.a.1.21 | $56$ | $2$ | $2$ | $93$ | $7$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.2688.93-56.h.2.6 | $56$ | $2$ | $2$ | $93$ | $7$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.2688.93-56.h.2.27 | $56$ | $2$ | $2$ | $93$ | $7$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.2688.93-56.es.1.2 | $56$ | $2$ | $2$ | $93$ | $12$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.2688.93-56.es.1.31 | $56$ | $2$ | $2$ | $93$ | $12$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.2688.93-56.ez.1.3 | $56$ | $2$ | $2$ | $93$ | $12$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.2688.93-56.ez.1.30 | $56$ | $2$ | $2$ | $93$ | $12$ | $1^{16}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$ |
| 56.2688.97-56.cr.2.3 | $56$ | $2$ | $2$ | $97$ | $29$ | $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ |
| 56.2688.97-56.cr.2.24 | $56$ | $2$ | $2$ | $97$ | $29$ | $2^{4}\cdot4^{4}\cdot6^{4}\cdot12^{4}$ |
| 56.2688.97-56.dx.1.10 | $56$ | $2$ | $2$ | $97$ | $12$ | $1^{24}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.2688.97-56.dx.1.23 | $56$ | $2$ | $2$ | $97$ | $12$ | $1^{24}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.2688.97-56.ee.1.11 | $56$ | $2$ | $2$ | $97$ | $12$ | $1^{24}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
| 56.2688.97-56.ee.1.22 | $56$ | $2$ | $2$ | $97$ | $12$ | $1^{24}\cdot2^{14}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.10752.385-56.gi.2.10 | $56$ | $2$ | $2$ | $385$ | $65$ | $1^{70}\cdot2^{25}\cdot4^{12}\cdot6^{4}$ |
| 56.10752.385-56.go.1.4 | $56$ | $2$ | $2$ | $385$ | $73$ | $1^{70}\cdot2^{25}\cdot4^{12}\cdot6^{4}$ |
| 56.10752.385-56.nk.2.10 | $56$ | $2$ | $2$ | $385$ | $63$ | $1^{70}\cdot2^{25}\cdot4^{12}\cdot6^{4}$ |
| 56.10752.385-56.nq.2.4 | $56$ | $2$ | $2$ | $385$ | $69$ | $1^{70}\cdot2^{25}\cdot4^{12}\cdot6^{4}$ |
| 56.10752.385-56.ye.2.8 | $56$ | $2$ | $2$ | $385$ | $62$ | $1^{70}\cdot2^{25}\cdot4^{12}\cdot6^{4}$ |
| 56.10752.385-56.yi.2.8 | $56$ | $2$ | $2$ | $385$ | $71$ | $1^{70}\cdot2^{25}\cdot4^{12}\cdot6^{4}$ |
| 56.10752.385-56.bci.2.8 | $56$ | $2$ | $2$ | $385$ | $58$ | $1^{70}\cdot2^{25}\cdot4^{12}\cdot6^{4}$ |
| 56.10752.385-56.bcm.2.8 | $56$ | $2$ | $2$ | $385$ | $71$ | $1^{70}\cdot2^{25}\cdot4^{12}\cdot6^{4}$ |
| 56.10752.401-56.ka.1.6 | $56$ | $2$ | $2$ | $401$ | $73$ | $1^{32}\cdot2^{38}\cdot4^{9}\cdot6^{2}\cdot12^{3}\cdot16$ |
| 56.10752.401-56.kh.2.6 | $56$ | $2$ | $2$ | $401$ | $59$ | $1^{32}\cdot2^{38}\cdot4^{9}\cdot6^{2}\cdot12^{3}\cdot16$ |
| 56.10752.401-56.kl.1.6 | $56$ | $2$ | $2$ | $401$ | $77$ | $1^{32}\cdot2^{38}\cdot4^{9}\cdot6^{2}\cdot12^{3}\cdot16$ |
| 56.10752.401-56.ko.1.6 | $56$ | $2$ | $2$ | $401$ | $67$ | $1^{32}\cdot2^{38}\cdot4^{9}\cdot6^{2}\cdot12^{3}\cdot16$ |
| 56.10752.401-56.ks.1.6 | $56$ | $2$ | $2$ | $401$ | $63$ | $1^{64}\cdot2^{32}\cdot4^{12}\cdot6^{2}\cdot8\cdot12$ |
| 56.10752.401-56.ku.1.6 | $56$ | $2$ | $2$ | $401$ | $79$ | $1^{64}\cdot2^{32}\cdot4^{12}\cdot6^{2}\cdot8\cdot12$ |
| 56.10752.401-56.kz.1.6 | $56$ | $2$ | $2$ | $401$ | $63$ | $1^{64}\cdot2^{32}\cdot4^{12}\cdot6^{2}\cdot8\cdot12$ |
| 56.10752.401-56.lb.1.6 | $56$ | $2$ | $2$ | $401$ | $79$ | $1^{64}\cdot2^{32}\cdot4^{12}\cdot6^{2}\cdot8\cdot12$ |
| 56.16128.577-56.oc.2.6 | $56$ | $3$ | $3$ | $577$ | $99$ | $1^{102}\cdot2^{55}\cdot4^{13}\cdot6^{12}\cdot12^{4}$ |