Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $784$ | ||
| Index: | $672$ | $\PSL_2$-index: | $336$ | ||||
| Genus: | $21 = 1 + \frac{ 336 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $2$ are rational) | Cusp widths | $14^{8}\cdot28^{8}$ | Cusp orbits | $1^{2}\cdot2\cdot3^{2}\cdot6$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $5$ | ||||||
| $\Q$-gonality: | $6 \le \gamma \le 12$ | ||||||
| $\overline{\Q}$-gonality: | $6 \le \gamma \le 12$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 28D21 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.672.21.52 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}1&8\\24&13\end{bmatrix}$, $\begin{bmatrix}1&12\\52&41\end{bmatrix}$, $\begin{bmatrix}13&4\\28&53\end{bmatrix}$, $\begin{bmatrix}23&40\\28&3\end{bmatrix}$, $\begin{bmatrix}53&14\\54&45\end{bmatrix}$, $\begin{bmatrix}54&21\\49&54\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 28.336.21.p.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $4$ |
| Cyclic 56-torsion field degree: | $96$ |
| Full 56-torsion field degree: | $4608$ |
Jacobian
| Conductor: | $2^{60}\cdot7^{37}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{9}\cdot2^{6}$ |
| Newforms: | 14.2.a.a$^{2}$, 98.2.a.b$^{2}$, 112.2.a.a, 112.2.a.b, 112.2.a.c, 196.2.a.b, 196.2.a.c, 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 2 rational cusps but no known non-cuspidal 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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(7)$ | $7$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 8.24.0-4.d.1.2 | $8$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.24.0-4.d.1.2 | $8$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
| 56.336.9-28.c.1.2 | $56$ | $2$ | $2$ | $9$ | $0$ | $1^{6}\cdot2^{3}$ |
| 56.336.9-28.c.1.24 | $56$ | $2$ | $2$ | $9$ | $0$ | $1^{6}\cdot2^{3}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.1344.41-28.ba.1.20 | $56$ | $2$ | $2$ | $41$ | $10$ | $1^{18}\cdot2$ |
| 56.1344.41-28.bb.1.20 | $56$ | $2$ | $2$ | $41$ | $10$ | $1^{18}\cdot2$ |
| 56.1344.41-28.be.1.11 | $56$ | $2$ | $2$ | $41$ | $12$ | $1^{18}\cdot2$ |
| 56.1344.41-28.bf.1.10 | $56$ | $2$ | $2$ | $41$ | $14$ | $1^{18}\cdot2$ |
| 56.1344.41-56.hc.1.12 | $56$ | $2$ | $2$ | $41$ | $12$ | $1^{18}\cdot2$ |
| 56.1344.41-56.hj.1.15 | $56$ | $2$ | $2$ | $41$ | $14$ | $1^{18}\cdot2$ |
| 56.1344.41-56.ie.1.14 | $56$ | $2$ | $2$ | $41$ | $13$ | $1^{18}\cdot2$ |
| 56.1344.41-56.il.1.9 | $56$ | $2$ | $2$ | $41$ | $15$ | $1^{18}\cdot2$ |
| 56.1344.45-28.t.1.7 | $56$ | $2$ | $2$ | $45$ | $11$ | $1^{12}\cdot2^{6}$ |
| 56.1344.45-28.u.1.7 | $56$ | $2$ | $2$ | $45$ | $12$ | $1^{12}\cdot2^{6}$ |
| 56.1344.45-28.v.1.6 | $56$ | $2$ | $2$ | $45$ | $12$ | $1^{20}\cdot2^{2}$ |
| 56.1344.45-28.w.1.6 | $56$ | $2$ | $2$ | $45$ | $12$ | $1^{20}\cdot2^{2}$ |
| 56.1344.45-56.di.1.1 | $56$ | $2$ | $2$ | $45$ | $7$ | $1^{12}\cdot2^{6}$ |
| 56.1344.45-56.dj.1.2 | $56$ | $2$ | $2$ | $45$ | $7$ | $1^{12}\cdot2^{6}$ |
| 56.1344.45-56.dm.1.7 | $56$ | $2$ | $2$ | $45$ | $17$ | $1^{12}\cdot2^{6}$ |
| 56.1344.45-56.dn.1.7 | $56$ | $2$ | $2$ | $45$ | $11$ | $1^{12}\cdot2^{6}$ |
| 56.1344.45-56.ds.1.4 | $56$ | $2$ | $2$ | $45$ | $7$ | $1^{20}\cdot2^{2}$ |
| 56.1344.45-56.dt.1.6 | $56$ | $2$ | $2$ | $45$ | $13$ | $1^{20}\cdot2^{2}$ |
| 56.1344.45-56.dw.1.7 | $56$ | $2$ | $2$ | $45$ | $13$ | $1^{20}\cdot2^{2}$ |
| 56.1344.45-56.dx.1.7 | $56$ | $2$ | $2$ | $45$ | $15$ | $1^{20}\cdot2^{2}$ |
| 56.1344.45-56.ea.1.15 | $56$ | $2$ | $2$ | $45$ | $16$ | $1^{12}\cdot2^{6}$ |
| 56.1344.45-56.eb.1.9 | $56$ | $2$ | $2$ | $45$ | $17$ | $1^{12}\cdot2^{6}$ |
| 56.1344.45-56.ec.1.11 | $56$ | $2$ | $2$ | $45$ | $16$ | $1^{20}\cdot2^{2}$ |
| 56.1344.45-56.ed.1.11 | $56$ | $2$ | $2$ | $45$ | $16$ | $1^{20}\cdot2^{2}$ |
| 56.1344.45-56.em.1.8 | $56$ | $2$ | $2$ | $45$ | $9$ | $1^{20}\cdot2^{2}$ |
| 56.1344.45-56.en.1.8 | $56$ | $2$ | $2$ | $45$ | $19$ | $1^{20}\cdot2^{2}$ |
| 56.1344.45-56.eq.1.3 | $56$ | $2$ | $2$ | $45$ | $11$ | $1^{20}\cdot2^{2}$ |
| 56.1344.45-56.er.1.2 | $56$ | $2$ | $2$ | $45$ | $25$ | $1^{20}\cdot2^{2}$ |
| 56.1344.45-56.eu.1.7 | $56$ | $2$ | $2$ | $45$ | $13$ | $1^{12}\cdot2^{6}$ |
| 56.1344.45-56.ev.1.7 | $56$ | $2$ | $2$ | $45$ | $17$ | $1^{12}\cdot2^{6}$ |
| 56.1344.45-56.ey.1.5 | $56$ | $2$ | $2$ | $45$ | $15$ | $1^{12}\cdot2^{6}$ |
| 56.1344.45-56.ez.1.2 | $56$ | $2$ | $2$ | $45$ | $25$ | $1^{12}\cdot2^{6}$ |
| 56.1344.49-56.bw.1.15 | $56$ | $2$ | $2$ | $49$ | $14$ | $1^{4}\cdot2^{10}\cdot4$ |
| 56.1344.49-56.bx.1.15 | $56$ | $2$ | $2$ | $49$ | $16$ | $1^{4}\cdot2^{10}\cdot4$ |
| 56.1344.49-56.by.1.14 | $56$ | $2$ | $2$ | $49$ | $17$ | $1^{4}\cdot2^{10}\cdot4$ |
| 56.1344.49-56.bz.1.12 | $56$ | $2$ | $2$ | $49$ | $17$ | $1^{4}\cdot2^{10}\cdot4$ |
| 56.1344.49-56.ca.1.10 | $56$ | $2$ | $2$ | $49$ | $19$ | $1^{12}\cdot2^{8}$ |
| 56.1344.49-56.cb.1.13 | $56$ | $2$ | $2$ | $49$ | $15$ | $1^{12}\cdot2^{8}$ |
| 56.1344.49-56.cc.1.11 | $56$ | $2$ | $2$ | $49$ | $19$ | $1^{12}\cdot2^{8}$ |
| 56.1344.49-56.cd.1.13 | $56$ | $2$ | $2$ | $49$ | $15$ | $1^{12}\cdot2^{8}$ |
| 56.2016.61-28.t.1.9 | $56$ | $3$ | $3$ | $61$ | $16$ | $1^{26}\cdot2^{7}$ |