Invariants
| Level: | $224$ | $\SL_2$-level: | $32$ | Newform level: | $1$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $3 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $4^{2}\cdot8\cdot32$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 32C3 |
Level structure
| $\GL_2(\Z/224\Z)$-generators: | $\begin{bmatrix}56&99\\187&176\end{bmatrix}$, $\begin{bmatrix}64&9\\155&26\end{bmatrix}$, $\begin{bmatrix}102&103\\121&128\end{bmatrix}$, $\begin{bmatrix}119&162\\102&91\end{bmatrix}$, $\begin{bmatrix}191&152\\174&189\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 224.48.3.c.2 for the level structure with $-I$) |
| Cyclic 224-isogeny field degree: | $32$ |
| Cyclic 224-torsion field degree: | $1536$ |
| Full 224-torsion field degree: | $8257536$ |
Rational points
This modular curve has 4 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 |
|---|---|---|---|---|---|
| 16.48.1-16.b.1.6 | $16$ | $2$ | $2$ | $1$ | $0$ |
| 224.48.1-16.b.1.7 | $224$ | $2$ | $2$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 224.192.5-224.c.1.11 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.f.1.10 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.i.1.19 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.k.1.10 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.z.1.15 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.z.2.16 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.bc.1.16 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.bc.2.14 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.bi.1.10 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.bj.1.3 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.bm.1.10 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.bn.1.9 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.bt.1.13 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.bt.2.15 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.bw.1.15 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.bw.2.11 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.cs.1.16 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.cs.2.8 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.ct.1.6 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.ct.2.8 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.de.1.6 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.de.2.2 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.df.1.5 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.df.2.4 | $224$ | $2$ | $2$ | $5$ |