Invariants
| Level: | $4$ | $\SL_2$-level: | $4$ | ||||
| Index: | $4$ | $\PSL_2$-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 and Zureick-Brown (RZB) label: | X2a |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 4.4.0.2 |
Level structure
| $\GL_2(\Z/4\Z)$-generators: | $\begin{bmatrix}1&2\\0&3\end{bmatrix}$, $\begin{bmatrix}2&3\\1&1\end{bmatrix}$ |
| $\GL_2(\Z/4\Z)$-subgroup: | $C_2\times A_4$ |
| Contains $-I$: | no $\quad$ (see 2.2.0.a.1 for the level structure with $-I$) |
| Cyclic 4-isogeny field degree: | $6$ |
| Cyclic 4-torsion field degree: | $12$ |
| Full 4-torsion field degree: | $24$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 32740 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 2 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{2}(x^{2}+1728y^{2})}{y^{2}x^{2}}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 4.12.0-2.a.1.2 | $4$ | $3$ | $3$ | $0$ |
| 4.16.0-4.a.1.1 | $4$ | $4$ | $4$ | $0$ |
| 12.12.1-6.a.1.3 | $12$ | $3$ | $3$ | $1$ |
| 12.16.0-6.a.1.3 | $12$ | $4$ | $4$ | $0$ |
| 20.20.0-10.a.1.2 | $20$ | $5$ | $5$ | $0$ |
| 20.24.1-10.a.1.2 | $20$ | $6$ | $6$ | $1$ |
| 20.40.1-10.a.1.4 | $20$ | $10$ | $10$ | $1$ |
| 28.12.0-14.a.1.3 | $28$ | $3$ | $3$ | $0$ |
| 28.12.0-14.a.1.4 | $28$ | $3$ | $3$ | $0$ |
| 28.32.0-14.a.1.4 | $28$ | $8$ | $8$ | $0$ |
| 28.84.3-14.a.1.2 | $28$ | $21$ | $21$ | $3$ |
| 28.112.3-14.a.1.1 | $28$ | $28$ | $28$ | $3$ |
| 36.12.0-18.a.1.3 | $36$ | $3$ | $3$ | $0$ |
| 36.12.0-18.a.1.4 | $36$ | $3$ | $3$ | $0$ |
| 36.108.2-18.a.1.2 | $36$ | $27$ | $27$ | $2$ |
| 44.48.2-22.a.1.4 | $44$ | $12$ | $12$ | $2$ |
| 44.220.5-22.a.1.2 | $44$ | $55$ | $55$ | $5$ |
| 44.220.7-22.a.1.1 | $44$ | $55$ | $55$ | $7$ |
| 44.264.9-22.a.1.1 | $44$ | $66$ | $66$ | $9$ |
| 52.12.0-26.a.1.3 | $52$ | $3$ | $3$ | $0$ |
| 52.12.0-26.a.1.4 | $52$ | $3$ | $3$ | $0$ |
| 52.56.1-26.a.1.3 | $52$ | $14$ | $14$ | $1$ |
| 52.312.11-26.a.1.3 | $52$ | $78$ | $78$ | $11$ |
| 52.364.10-26.a.1.2 | $52$ | $91$ | $91$ | $10$ |
| 52.364.12-26.a.1.4 | $52$ | $91$ | $91$ | $12$ |
| 68.72.3-34.a.1.3 | $68$ | $18$ | $18$ | $3$ |
| 68.544.19-34.a.1.3 | $68$ | $136$ | $136$ | $19$ |
| 68.612.22-34.a.1.2 | $68$ | $153$ | $153$ | $22$ |
| 76.12.0-38.a.1.1 | $76$ | $3$ | $3$ | $0$ |
| 76.12.0-38.a.1.4 | $76$ | $3$ | $3$ | $0$ |
| 76.80.2-38.a.1.4 | $76$ | $20$ | $20$ | $2$ |
| 92.96.4-46.a.1.2 | $92$ | $24$ | $24$ | $4$ |
| 116.120.5-58.a.1.3 | $116$ | $30$ | $30$ | $5$ |
| 124.12.0-62.a.1.3 | $124$ | $3$ | $3$ | $0$ |
| 124.12.0-62.a.1.4 | $124$ | $3$ | $3$ | $0$ |
| 124.128.4-62.a.1.2 | $124$ | $32$ | $32$ | $4$ |
| 148.12.0-74.a.1.3 | $148$ | $3$ | $3$ | $0$ |
| 148.12.0-74.a.1.4 | $148$ | $3$ | $3$ | $0$ |
| 148.152.5-74.a.1.4 | $148$ | $38$ | $38$ | $5$ |
| 164.168.7-82.a.1.1 | $164$ | $42$ | $42$ | $7$ |
| 172.12.0-86.a.1.3 | $172$ | $3$ | $3$ | $0$ |
| 172.12.0-86.a.1.4 | $172$ | $3$ | $3$ | $0$ |
| 172.176.6-86.a.1.4 | $172$ | $44$ | $44$ | $6$ |
| 188.192.8-94.a.1.3 | $188$ | $48$ | $48$ | $8$ |
| 212.216.9-106.a.1.2 | $212$ | $54$ | $54$ | $9$ |
| 236.240.10-118.a.1.4 | $236$ | $60$ | $60$ | $10$ |
| 244.12.0-122.a.1.3 | $244$ | $3$ | $3$ | $0$ |
| 244.12.0-122.a.1.4 | $244$ | $3$ | $3$ | $0$ |
| 244.248.9-122.a.1.2 | $244$ | $62$ | $62$ | $9$ |
| 252.12.0-126.a.1.3 | $252$ | $3$ | $3$ | $0$ |
| 252.12.0-126.a.1.6 | $252$ | $3$ | $3$ | $0$ |
| 252.12.0-126.b.1.2 | $252$ | $3$ | $3$ | $0$ |
| 252.12.0-126.b.1.7 | $252$ | $3$ | $3$ | $0$ |
| 268.12.0-134.a.1.2 | $268$ | $3$ | $3$ | $0$ |
| 268.12.0-134.a.1.3 | $268$ | $3$ | $3$ | $0$ |
| 268.272.10-134.a.1.4 | $268$ | $68$ | $68$ | $10$ |
| 284.288.12-142.a.1.4 | $284$ | $72$ | $72$ | $12$ |
| 292.12.0-146.a.1.2 | $292$ | $3$ | $3$ | $0$ |
| 292.12.0-146.a.1.3 | $292$ | $3$ | $3$ | $0$ |
| 292.296.11-146.a.1.4 | $292$ | $74$ | $74$ | $11$ |
| 316.12.0-158.a.1.3 | $316$ | $3$ | $3$ | $0$ |
| 316.12.0-158.a.1.4 | $316$ | $3$ | $3$ | $0$ |
| 316.320.12-158.a.1.4 | $316$ | $80$ | $80$ | $12$ |
| 332.336.14-166.a.1.4 | $332$ | $84$ | $84$ | $14$ |