Invariants
| Level: | $114$ | $\SL_2$-level: | $6$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $1\cdot2\cdot3\cdot6$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6F0 |
Level structure
| $\GL_2(\Z/114\Z)$-generators: | $\begin{bmatrix}25&74\\70&93\end{bmatrix}$, $\begin{bmatrix}72&43\\17&70\end{bmatrix}$, $\begin{bmatrix}77&18\\20&109\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 6.12.0.a.1 for the level structure with $-I$) |
| Cyclic 114-isogeny field degree: | $20$ |
| Cyclic 114-torsion field degree: | $720$ |
| Full 114-torsion field degree: | $1477440$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 9048 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6}\cdot\frac{x^{12}(x+2y)^{3}(x^{3}+6x^{2}y-84xy^{2}-568y^{3})^{3}}{y^{6}x^{12}(x-10y)(x+6y)^{3}(x+8y)^{2}}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(2)$ | $2$ | $8$ | $4$ | $0$ | $0$ |
| 57.8.0-3.a.1.1 | $57$ | $3$ | $3$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 57.8.0-3.a.1.1 | $57$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 114.48.0-6.a.1.1 | $114$ | $2$ | $2$ | $0$ |
| 114.48.0-6.b.1.1 | $114$ | $2$ | $2$ | $0$ |
| 114.48.0-114.b.1.1 | $114$ | $2$ | $2$ | $0$ |
| 114.48.0-114.c.1.1 | $114$ | $2$ | $2$ | $0$ |
| 114.72.0-6.a.1.1 | $114$ | $3$ | $3$ | $0$ |
| 114.480.17-114.a.1.10 | $114$ | $20$ | $20$ | $17$ |
| 228.48.0-12.d.1.6 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0-12.f.1.8 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0-12.g.1.12 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0-12.h.1.4 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0-12.i.1.8 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0-12.j.1.8 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0-228.m.1.11 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0-228.n.1.8 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0-228.o.1.6 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0-228.p.1.13 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0-228.q.1.12 | $228$ | $2$ | $2$ | $0$ |
| 228.48.0-228.r.1.8 | $228$ | $2$ | $2$ | $0$ |
| 228.48.1-12.i.1.4 | $228$ | $2$ | $2$ | $1$ |
| 228.48.1-12.j.1.4 | $228$ | $2$ | $2$ | $1$ |
| 228.48.1-12.k.1.4 | $228$ | $2$ | $2$ | $1$ |
| 228.48.1-12.l.1.8 | $228$ | $2$ | $2$ | $1$ |
| 228.48.1-228.m.1.11 | $228$ | $2$ | $2$ | $1$ |
| 228.48.1-228.n.1.7 | $228$ | $2$ | $2$ | $1$ |
| 228.48.1-228.o.1.7 | $228$ | $2$ | $2$ | $1$ |
| 228.48.1-228.p.1.5 | $228$ | $2$ | $2$ | $1$ |