Invariants
| Level: | $258$ | $\SL_2$-level: | $6$ | ||||
| Index: | $24$ | $\PSL_2$-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
| $\GL_2(\Z/258\Z)$-generators: | $\begin{bmatrix}58&245\\31&174\end{bmatrix}$, $\begin{bmatrix}95&66\\212&175\end{bmatrix}$, $\begin{bmatrix}254&127\\257&210\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 258.48.0-258.b.1.1, 258.48.0-258.b.1.2, 258.48.0-258.b.1.3, 258.48.0-258.b.1.4 |
| Cyclic 258-isogeny field degree: | $44$ |
| Cyclic 258-torsion field degree: | $3696$ |
| Full 258-torsion field degree: | $40049856$ |
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 |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $6$ | $6$ | $0$ | $0$ |
| 86.6.0.b.1 | $86$ | $4$ | $4$ | $0$ | $?$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(6)$ | $6$ | $2$ | $2$ | $0$ | $0$ |
| 86.6.0.b.1 | $86$ | $4$ | $4$ | $0$ | $?$ |
| 258.8.0.a.1 | $258$ | $3$ | $3$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 258.72.1.d.1 | $258$ | $3$ | $3$ | $1$ |