Invariants
| Level: | $114$ | $\SL_2$-level: | $6$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $2^{3}\cdot6^{3}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6I0 |
Level structure
| $\GL_2(\Z/114\Z)$-generators: | $\begin{bmatrix}28&75\\97&44\end{bmatrix}$, $\begin{bmatrix}89&46\\48&79\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 114.24.0.b.1 for the level structure with $-I$) |
| Cyclic 114-isogeny field degree: | $20$ |
| Cyclic 114-torsion field degree: | $360$ |
| Full 114-torsion field degree: | $738720$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_1(3)$ | $3$ | $6$ | $6$ | $0$ | $0$ |
| 38.6.0.b.1 | $38$ | $8$ | $4$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_1(6)$ | $6$ | $2$ | $2$ | $0$ | $0$ |
| 114.16.0-114.a.1.4 | $114$ | $3$ | $3$ | $0$ | $?$ |
| 114.24.0-6.a.1.2 | $114$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 114.144.1-114.d.1.2 | $114$ | $3$ | $3$ | $1$ |
| 228.96.1-228.i.1.2 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.k.1.4 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.u.1.2 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.w.1.2 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.bg.1.2 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.bi.1.2 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.bo.1.2 | $228$ | $2$ | $2$ | $1$ |
| 228.96.1-228.bq.1.4 | $228$ | $2$ | $2$ | $1$ |