Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $784$ | ||
| Index: | $1008$ | $\PSL_2$-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 | $14^{12}\cdot28^{6}\cdot56^{12}$ | 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: | 56.1008.70.10 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}21&26\\46&17\end{bmatrix}$, $\begin{bmatrix}27&54\\46&51\end{bmatrix}$, $\begin{bmatrix}31&26\\16&25\end{bmatrix}$, $\begin{bmatrix}39&32\\34&5\end{bmatrix}$, $\begin{bmatrix}45&36\\26&47\end{bmatrix}$, $\begin{bmatrix}47&10\\2&23\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 56.2016.70-56.fp.1.1, 56.2016.70-56.fp.1.2, 56.2016.70-56.fp.1.3, 56.2016.70-56.fp.1.4, 56.2016.70-56.fp.1.5, 56.2016.70-56.fp.1.6, 56.2016.70-56.fp.1.7, 56.2016.70-56.fp.1.8, 56.2016.70-56.fp.1.9, 56.2016.70-56.fp.1.10, 56.2016.70-56.fp.1.11, 56.2016.70-56.fp.1.12, 56.2016.70-56.fp.1.13, 56.2016.70-56.fp.1.14, 56.2016.70-56.fp.1.15, 56.2016.70-56.fp.1.16, 56.2016.70-56.fp.1.17, 56.2016.70-56.fp.1.18, 56.2016.70-56.fp.1.19, 56.2016.70-56.fp.1.20, 56.2016.70-56.fp.1.21, 56.2016.70-56.fp.1.22, 56.2016.70-56.fp.1.23, 56.2016.70-56.fp.1.24 |
| Cyclic 56-isogeny field degree: | $8$ |
| Cyclic 56-torsion field degree: | $96$ |
| Full 56-torsion field degree: | $3072$ |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(7)$ | $7$ | $48$ | $48$ | $0$ | $0$ | full Jacobian |
| $X_{\pm1}(2,8)$ | $8$ | $21$ | $21$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\pm1}(2,8)$ | $8$ | $21$ | $21$ | $0$ | $0$ | full Jacobian |
| 56.504.34.z.1 | $56$ | $2$ | $2$ | $34$ | $1$ | $1^{6}\cdot2^{6}\cdot6\cdot12$ |
| 56.504.34.ch.1 | $56$ | $2$ | $2$ | $34$ | $6$ | $6^{2}\cdot12^{2}$ |
| 56.504.34.gl.2 | $56$ | $2$ | $2$ | $34$ | $1$ | $1^{6}\cdot2^{6}\cdot6\cdot12$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.2016.139.lr.1 | $56$ | $2$ | $2$ | $139$ | $16$ | $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$ |
| 56.2016.139.lx.1 | $56$ | $2$ | $2$ | $139$ | $30$ | $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$ |
| 56.2016.139.mp.2 | $56$ | $2$ | $2$ | $139$ | $21$ | $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$ |
| 56.2016.139.mv.2 | $56$ | $2$ | $2$ | $139$ | $15$ | $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$ |
| 56.2016.139.nt.2 | $56$ | $2$ | $2$ | $139$ | $12$ | $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$ |
| 56.2016.139.nz.2 | $56$ | $2$ | $2$ | $139$ | $21$ | $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$ |
| 56.2016.139.or.2 | $56$ | $2$ | $2$ | $139$ | $18$ | $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$ |
| 56.2016.139.ox.2 | $56$ | $2$ | $2$ | $139$ | $19$ | $1^{27}\cdot2^{7}\cdot4^{4}\cdot6^{2}$ |
| 56.2016.145.pk.1 | $56$ | $2$ | $2$ | $145$ | $19$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.2016.145.tr.1 | $56$ | $2$ | $2$ | $145$ | $23$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.2016.145.blb.1 | $56$ | $2$ | $2$ | $145$ | $20$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.2016.145.blj.1 | $56$ | $2$ | $2$ | $145$ | $24$ | $1^{7}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$ |
| 56.2016.145.bsh.1 | $56$ | $2$ | $2$ | $145$ | $25$ | $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$ |
| 56.2016.145.bsp.1 | $56$ | $2$ | $2$ | $145$ | $18$ | $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$ |
| 56.2016.145.btn.1 | $56$ | $2$ | $2$ | $145$ | $26$ | $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$ |
| 56.2016.145.btv.1 | $56$ | $2$ | $2$ | $145$ | $21$ | $1^{23}\cdot2^{12}\cdot4^{4}\cdot6^{2}$ |