Invariants
| Level: | $56$ | $\SL_2$-level: | $56$ | Newform level: | $3136$ | ||
| 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 | $7^{8}\cdot14^{4}\cdot56^{4}$ | Cusp orbits | $1^{2}\cdot2\cdot3^{2}\cdot6$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $10$ | ||||||
| $\Q$-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: | 56E21 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.672.21.542 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}11&45\\24&45\end{bmatrix}$, $\begin{bmatrix}33&27\\12&37\end{bmatrix}$, $\begin{bmatrix}35&6\\36&35\end{bmatrix}$, $\begin{bmatrix}37&21\\28&23\end{bmatrix}$, $\begin{bmatrix}49&32\\52&35\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.336.21.ct.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: | $2^{84}\cdot7^{37}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{13}\cdot2^{4}$ |
| Newforms: | 14.2.a.a$^{2}$, 98.2.a.b$^{2}$, 196.2.a.b, 196.2.a.c, 448.2.a.a, 448.2.a.e, 448.2.a.h, 3136.2.a.a, 3136.2.a.bq, 3136.2.a.c, 3136.2.a.k, 3136.2.a.o, 3136.2.a.p, 3136.2.a.t, 3136.2.a.z |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 28.336.9-28.c.1.5 | $28$ | $2$ | $2$ | $9$ | $0$ | $1^{10}\cdot2$ |
| 56.336.9-28.c.1.2 | $56$ | $2$ | $2$ | $9$ | $0$ | $1^{10}\cdot2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.1344.41-56.my.1.12 | $56$ | $2$ | $2$ | $41$ | $12$ | $1^{14}\cdot2^{3}$ |
| 56.1344.41-56.nb.1.16 | $56$ | $2$ | $2$ | $41$ | $17$ | $1^{14}\cdot2^{3}$ |
| 56.1344.41-56.ob.1.16 | $56$ | $2$ | $2$ | $41$ | $26$ | $1^{14}\cdot2^{3}$ |
| 56.1344.41-56.oc.1.16 | $56$ | $2$ | $2$ | $41$ | $15$ | $1^{14}\cdot2^{3}$ |
| 56.1344.41-56.oq.1.24 | $56$ | $2$ | $2$ | $41$ | $16$ | $1^{14}\cdot2^{3}$ |
| 56.1344.41-56.ot.1.16 | $56$ | $2$ | $2$ | $41$ | $18$ | $1^{14}\cdot2^{3}$ |
| 56.1344.41-56.pd.1.16 | $56$ | $2$ | $2$ | $41$ | $21$ | $1^{14}\cdot2^{3}$ |
| 56.1344.41-56.pe.1.16 | $56$ | $2$ | $2$ | $41$ | $15$ | $1^{14}\cdot2^{3}$ |
| 56.1344.45-56.n.1.12 | $56$ | $2$ | $2$ | $45$ | $12$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.cp.1.8 | $56$ | $2$ | $2$ | $45$ | $15$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.er.1.2 | $56$ | $2$ | $2$ | $45$ | $25$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.es.1.16 | $56$ | $2$ | $2$ | $45$ | $23$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.fp.1.16 | $56$ | $2$ | $2$ | $45$ | $17$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.fr.1.16 | $56$ | $2$ | $2$ | $45$ | $21$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.gb.1.12 | $56$ | $2$ | $2$ | $45$ | $21$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.gd.1.12 | $56$ | $2$ | $2$ | $45$ | $18$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.hl.1.12 | $56$ | $2$ | $2$ | $45$ | $17$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.hm.1.12 | $56$ | $2$ | $2$ | $45$ | $21$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.hw.1.16 | $56$ | $2$ | $2$ | $45$ | $21$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.hz.1.16 | $56$ | $2$ | $2$ | $45$ | $18$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.if.1.24 | $56$ | $2$ | $2$ | $45$ | $12$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.ig.1.16 | $56$ | $2$ | $2$ | $45$ | $15$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.jg.1.14 | $56$ | $2$ | $2$ | $45$ | $25$ | $1^{16}\cdot2^{4}$ |
| 56.1344.45-56.jj.1.15 | $56$ | $2$ | $2$ | $45$ | $23$ | $1^{16}\cdot2^{4}$ |
| 56.2016.61-56.id.1.28 | $56$ | $3$ | $3$ | $61$ | $21$ | $1^{26}\cdot2^{7}$ |