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: | $\begin{bmatrix}82&105\\129&82\end{bmatrix}$, $\begin{bmatrix}88&65\\39&6\end{bmatrix}$, $\begin{bmatrix}117&38\\180&163\end{bmatrix}$, $\begin{bmatrix}186&127\\25&28\end{bmatrix}$, $\begin{bmatrix}219&84\\22&61\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 224.48.3.d.1 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.1.2 | $224$ | $2$ | $2$ | $1$ | $?$ |
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.h.1.6 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.i.1.19 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.m.1.10 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.be.1.7 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.be.2.14 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.bf.1.6 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.bf.2.15 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.bh.1.3 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.bk.2.5 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.bl.2.9 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.bo.2.9 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.by.1.9 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.by.2.25 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.bz.1.5 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.bz.2.13 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.cv.1.10 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.cv.2.2 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.cy.1.10 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.cy.2.2 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.dh.1.9 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.dh.2.1 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.dk.1.9 | $224$ | $2$ | $2$ | $5$ |
| 224.192.5-224.dk.2.1 | $224$ | $2$ | $2$ | $5$ |