Invariants
| Level: | $52$ | $\SL_2$-level: | $2$ | ||||
| Index: | $2$ | $\PSL_2$-index: | $2$ | ||||
| Genus: | $0 = 1 + \frac{ 2 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 1 }{2}$ | ||||||
| Cusps: | $1$ (which is rational) | Cusp widths | $2$ | Cusp orbits | $1$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 2A0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 52.2.0.1 |
Level structure
| $\GL_2(\Z/52\Z)$-generators: | $\begin{bmatrix}31&3\\16&11\end{bmatrix}$, $\begin{bmatrix}41&50\\21&33\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 52-isogeny field degree: | $84$ |
| Cyclic 52-torsion field degree: | $2016$ |
| Full 52-torsion field degree: | $1257984$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 1778 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 2 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4\cdot3^2}{13}\cdot\frac{(13x+12y)^{2}(156x^{2}-y^{2})}{x^{2}(13x+12y)^{2}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X(1)$ | $1$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 52.6.0.a.1 | $52$ | $3$ | $3$ | $0$ |
| 52.8.0.b.1 | $52$ | $4$ | $4$ | $0$ |
| 52.28.1.b.1 | $52$ | $14$ | $14$ | $1$ |
| 52.156.11.b.1 | $52$ | $78$ | $78$ | $11$ |
| 52.182.10.b.1 | $52$ | $91$ | $91$ | $10$ |
| 52.182.12.b.1 | $52$ | $91$ | $91$ | $12$ |
| 156.6.1.a.1 | $156$ | $3$ | $3$ | $1$ |
| 156.8.0.a.1 | $156$ | $4$ | $4$ | $0$ |
| 260.10.0.a.1 | $260$ | $5$ | $5$ | $0$ |
| 260.12.1.a.1 | $260$ | $6$ | $6$ | $1$ |
| 260.20.1.a.1 | $260$ | $10$ | $10$ | $1$ |