Invariants
| Level: | $52$ | $\SL_2$-level: | $4$ | ||||
| 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 $4$ are rational) | Cusp widths | $4^{6}$ | Cusp orbits | $1^{4}\cdot2$ | ||
| 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: | 4G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 52.48.0.13 |
Level structure
| $\GL_2(\Z/52\Z)$-generators: | $\begin{bmatrix}3&24\\20&51\end{bmatrix}$, $\begin{bmatrix}25&26\\44&35\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 4.24.0.b.1 for the level structure with $-I$) |
| Cyclic 52-isogeny field degree: | $14$ |
| Cyclic 52-torsion field degree: | $336$ |
| Full 52-torsion field degree: | $52416$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 61 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{24}(x^{4}-4x^{3}y+8x^{2}y^{2}+16xy^{3}+16y^{4})^{3}(x^{4}+4x^{3}y+8x^{2}y^{2}-16xy^{3}+16y^{4})^{3}}{y^{4}x^{28}(x-2y)^{4}(x+2y)^{4}(x^{2}+4y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 52.24.0-4.a.1.1 | $52$ | $2$ | $2$ | $0$ | $0$ |
| 52.24.0-4.a.1.2 | $52$ | $2$ | $2$ | $0$ | $0$ |
| 52.24.0-4.b.1.1 | $52$ | $2$ | $2$ | $0$ | $0$ |
| 52.24.0-4.b.1.2 | $52$ | $2$ | $2$ | $0$ | $0$ |
| 52.24.0-4.b.1.3 | $52$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 52.672.23-52.b.1.2 | $52$ | $14$ | $14$ | $23$ |
| 52.3744.139-52.b.1.1 | $52$ | $78$ | $78$ | $139$ |
| 52.4368.162-52.c.1.2 | $52$ | $91$ | $91$ | $162$ |
| 52.4368.162-52.d.1.2 | $52$ | $91$ | $91$ | $162$ |
| 104.96.0-8.a.1.2 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-8.a.1.9 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.a.1.2 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.a.1.12 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-8.b.1.1 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-8.b.1.11 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-8.b.2.2 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-8.b.2.9 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.b.1.2 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.b.1.12 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.b.2.6 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.b.2.11 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-8.c.1.1 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-8.c.1.10 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.c.1.8 | $104$ | $2$ | $2$ | $0$ |
| 104.96.0-104.c.1.11 | $104$ | $2$ | $2$ | $0$ |
| 104.96.1-8.g.1.2 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-8.g.1.12 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-8.g.2.2 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-8.h.1.2 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-8.h.1.12 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-8.h.2.2 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.n.1.4 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.n.2.5 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.n.2.16 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.o.1.3 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.o.2.9 | $104$ | $2$ | $2$ | $1$ |
| 104.96.1-104.o.2.16 | $104$ | $2$ | $2$ | $1$ |
| 104.96.2-8.a.1.2 | $104$ | $2$ | $2$ | $2$ |
| 104.96.2-8.a.1.12 | $104$ | $2$ | $2$ | $2$ |
| 104.96.2-104.a.1.6 | $104$ | $2$ | $2$ | $2$ |
| 104.96.2-104.a.1.24 | $104$ | $2$ | $2$ | $2$ |
| 156.144.4-12.b.1.1 | $156$ | $3$ | $3$ | $4$ |
| 156.192.3-12.b.1.1 | $156$ | $4$ | $4$ | $3$ |
| 260.240.8-20.b.1.1 | $260$ | $5$ | $5$ | $8$ |
| 260.288.7-20.b.1.1 | $260$ | $6$ | $6$ | $7$ |
| 260.480.15-20.b.1.3 | $260$ | $10$ | $10$ | $15$ |
| 312.96.0-24.a.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.a.1.11 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.a.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.a.1.23 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.b.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.b.1.11 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.b.2.2 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.b.2.9 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.b.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.b.1.23 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.b.2.2 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.b.2.19 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.c.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-24.c.1.9 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.c.1.3 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.c.1.18 | $312$ | $2$ | $2$ | $0$ |
| 312.96.1-24.n.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.n.2.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.n.2.14 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.n.1.9 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.n.2.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.n.2.21 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.o.1.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.o.2.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-24.o.2.14 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.o.1.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.o.2.3 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1-312.o.2.21 | $312$ | $2$ | $2$ | $1$ |
| 312.96.2-24.a.1.1 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-24.a.1.21 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.a.1.5 | $312$ | $2$ | $2$ | $2$ |
| 312.96.2-312.a.1.41 | $312$ | $2$ | $2$ | $2$ |