Invariants
| Level: | $110$ | $\SL_2$-level: | $22$ | Newform level: | $1$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $2^{3}\cdot22^{3}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 22A4 |
Level structure
| $\GL_2(\Z/110\Z)$-generators: | $\begin{bmatrix}31&78\\22&107\end{bmatrix}$, $\begin{bmatrix}68&45\\93&94\end{bmatrix}$, $\begin{bmatrix}93&52\\68&65\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 110.144.4-110.a.1.1, 110.144.4-110.a.1.2, 110.144.4-110.a.1.3, 110.144.4-110.a.1.4, 220.144.4-110.a.1.1, 220.144.4-110.a.1.2, 220.144.4-110.a.1.3, 220.144.4-110.a.1.4, 220.144.4-110.a.1.5, 220.144.4-110.a.1.6, 220.144.4-110.a.1.7, 220.144.4-110.a.1.8, 220.144.4-110.a.1.9, 220.144.4-110.a.1.10, 220.144.4-110.a.1.11, 220.144.4-110.a.1.12, 330.144.4-110.a.1.1, 330.144.4-110.a.1.2, 330.144.4-110.a.1.3, 330.144.4-110.a.1.4 |
| Cyclic 110-isogeny field degree: | $6$ |
| Cyclic 110-torsion field degree: | $240$ |
| Full 110-torsion field degree: | $528000$ |
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 |
|---|---|---|---|---|---|
| 10.6.0.a.1 | $10$ | $12$ | $12$ | $0$ | $0$ |
| $X_0(11)$ | $11$ | $6$ | $6$ | $1$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 10.6.0.a.1 | $10$ | $12$ | $12$ | $0$ | $0$ |
| $X_0(22)$ | $22$ | $2$ | $2$ | $2$ | $0$ |
| 110.24.2.a.1 | $110$ | $3$ | $3$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 110.360.16.a.1 | $110$ | $5$ | $5$ | $16$ |
| 110.360.16.a.2 | $110$ | $5$ | $5$ | $16$ |
| 110.360.16.b.1 | $110$ | $5$ | $5$ | $16$ |
| 110.360.16.b.2 | $110$ | $5$ | $5$ | $16$ |
| 110.360.16.c.1 | $110$ | $5$ | $5$ | $16$ |
| 220.144.9.i.1 | $220$ | $2$ | $2$ | $9$ |
| 220.144.9.k.1 | $220$ | $2$ | $2$ | $9$ |
| 220.144.9.q.1 | $220$ | $2$ | $2$ | $9$ |
| 220.144.9.r.1 | $220$ | $2$ | $2$ | $9$ |
| 220.144.9.y.1 | $220$ | $2$ | $2$ | $9$ |
| 220.144.9.z.1 | $220$ | $2$ | $2$ | $9$ |
| 220.144.9.bg.1 | $220$ | $2$ | $2$ | $9$ |
| 220.144.9.bi.1 | $220$ | $2$ | $2$ | $9$ |
| 330.216.16.a.1 | $330$ | $3$ | $3$ | $16$ |
| 330.288.19.a.1 | $330$ | $4$ | $4$ | $19$ |