Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $784$ | ||
| Index: | $2016$ | $\PSL_2$-index: | $1008$ | ||||
| Genus: | $70 = 1 + \frac{ 1008 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 30 }{2}$ | ||||||
| Cusps: | $30$ (none of which are rational) | Cusp widths | $28^{24}\cdot56^{6}$ | Cusp orbits | $3^{4}\cdot6^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $6$ | ||||||
| $\Q$-gonality: | $10 \le \gamma \le 21$ | ||||||
| $\overline{\Q}$-gonality: | $10 \le \gamma \le 21$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.2016.70.5 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}33&32\\4&9\end{bmatrix}$, $\begin{bmatrix}33&40\\12&41\end{bmatrix}$, $\begin{bmatrix}37&16\\32&33\end{bmatrix}$, $\begin{bmatrix}41&28\\18&43\end{bmatrix}$, $\begin{bmatrix}45&44\\28&53\end{bmatrix}$, $\begin{bmatrix}49&20\\8&21\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.1008.70.l.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $192$ |
| Full 56-torsion field degree: | $1536$ |
Jacobian
| Conductor: | $2^{194}\cdot7^{140}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{10}\cdot2^{12}\cdot6^{2}\cdot12^{2}$ |
| Newforms: | 98.2.a.b$^{4}$, 196.2.a.b$^{3}$, 196.2.a.c$^{3}$, 392.2.a.c$^{2}$, 392.2.a.f$^{2}$, 392.2.a.g$^{2}$, 392.2.b.e$^{2}$, 392.2.b.g$^{2}$, 784.2.a.a, 784.2.a.d, 784.2.a.h, 784.2.a.k, 784.2.a.l, 784.2.a.m |
Rational points
This modular curve has no $\Q_p$ points for $p=3,5,11,23,37,67,103,149$, 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(7)$ | $7$ | $96$ | $48$ | $0$ | $0$ | full Jacobian |
| 8.96.0-8.c.1.10 | $8$ | $21$ | $21$ | $0$ | $0$ | 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.c.1.10 | $8$ | $21$ | $21$ | $0$ | $0$ | full Jacobian |
| 28.1008.34-28.b.1.2 | $28$ | $2$ | $2$ | $34$ | $6$ | $6^{2}\cdot12^{2}$ |
| 56.1008.34-28.b.1.32 | $56$ | $2$ | $2$ | $34$ | $6$ | $6^{2}\cdot12^{2}$ |
| 56.1008.34-56.z.1.7 | $56$ | $2$ | $2$ | $34$ | $1$ | $1^{6}\cdot2^{6}\cdot6\cdot12$ |
| 56.1008.34-56.z.1.64 | $56$ | $2$ | $2$ | $34$ | $1$ | $1^{6}\cdot2^{6}\cdot6\cdot12$ |
| 56.1008.34-56.z.2.1 | $56$ | $2$ | $2$ | $34$ | $1$ | $1^{6}\cdot2^{6}\cdot6\cdot12$ |
| 56.1008.34-56.z.2.60 | $56$ | $2$ | $2$ | $34$ | $1$ | $1^{6}\cdot2^{6}\cdot6\cdot12$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.4032.139-56.h.1.4 | $56$ | $2$ | $2$ | $139$ | $16$ | $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.139-56.l.1.3 | $56$ | $2$ | $2$ | $139$ | $30$ | $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.139-56.bi.1.7 | $56$ | $2$ | $2$ | $139$ | $12$ | $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.139-56.bm.1.2 | $56$ | $2$ | $2$ | $139$ | $21$ | $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.139-56.cj.1.6 | $56$ | $2$ | $2$ | $139$ | $19$ | $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.139-56.cn.1.8 | $56$ | $2$ | $2$ | $139$ | $18$ | $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.139-56.dh.1.5 | $56$ | $2$ | $2$ | $139$ | $15$ | $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.139-56.dl.1.5 | $56$ | $2$ | $2$ | $139$ | $21$ | $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.pd.1.7 | $56$ | $2$ | $2$ | $145$ | $23$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.4032.145-56.pd.2.1 | $56$ | $2$ | $2$ | $145$ | $23$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.4032.145-56.pe.1.6 | $56$ | $2$ | $2$ | $145$ | $24$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.4032.145-56.pe.2.1 | $56$ | $2$ | $2$ | $145$ | $24$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.4032.145-56.pf.1.4 | $56$ | $2$ | $2$ | $145$ | $21$ | $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.pf.2.1 | $56$ | $2$ | $2$ | $145$ | $21$ | $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.pg.1.7 | $56$ | $2$ | $2$ | $145$ | $18$ | $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.pg.2.1 | $56$ | $2$ | $2$ | $145$ | $18$ | $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.ph.1.4 | $56$ | $2$ | $2$ | $145$ | $25$ | $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.ph.2.1 | $56$ | $2$ | $2$ | $145$ | $25$ | $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.pi.1.6 | $56$ | $2$ | $2$ | $145$ | $26$ | $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.pi.2.1 | $56$ | $2$ | $2$ | $145$ | $26$ | $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$ |
| 56.4032.145-56.pj.1.7 | $56$ | $2$ | $2$ | $145$ | $20$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.4032.145-56.pj.2.1 | $56$ | $2$ | $2$ | $145$ | $20$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.4032.145-56.pk.1.7 | $56$ | $2$ | $2$ | $145$ | $19$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.4032.145-56.pk.2.1 | $56$ | $2$ | $2$ | $145$ | $19$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.4032.151-56.ex.1.4 | $56$ | $2$ | $2$ | $151$ | $23$ | $1^{7}\cdot2^{15}\cdot4^{4}\cdot12\cdot16$ |
| 56.4032.151-56.ey.1.6 | $56$ | $2$ | $2$ | $151$ | $19$ | $1^{7}\cdot2^{15}\cdot4^{4}\cdot12\cdot16$ |
| 56.4032.151-56.ez.1.7 | $56$ | $2$ | $2$ | $151$ | $24$ | $1^{7}\cdot2^{15}\cdot4^{4}\cdot12\cdot16$ |
| 56.4032.151-56.fa.1.4 | $56$ | $2$ | $2$ | $151$ | $20$ | $1^{7}\cdot2^{15}\cdot4^{4}\cdot12\cdot16$ |
| 56.4032.151-56.fb.1.4 | $56$ | $2$ | $2$ | $151$ | $18$ | $1^{23}\cdot2^{11}\cdot4^{4}\cdot8\cdot12$ |
| 56.4032.151-56.fc.1.7 | $56$ | $2$ | $2$ | $151$ | $25$ | $1^{23}\cdot2^{11}\cdot4^{4}\cdot8\cdot12$ |
| 56.4032.151-56.fd.1.6 | $56$ | $2$ | $2$ | $151$ | $21$ | $1^{23}\cdot2^{11}\cdot4^{4}\cdot8\cdot12$ |
| 56.4032.151-56.fe.1.4 | $56$ | $2$ | $2$ | $151$ | $26$ | $1^{23}\cdot2^{11}\cdot4^{4}\cdot8\cdot12$ |