Invariants
| Level: | $33$ | $\SL_2$-level: | $3$ | ||||
| Index: | $8$ | $\PSL_2$-index: | $4$ | ||||
| Genus: | $0 = 1 + \frac{ 4 }{12} - \frac{ 0 }{4} - \frac{ 1 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $1\cdot3$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $1$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3,-12,-27$) |
Other labels
| Cummins and Pauli (CP) label: | 3B0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 33.8.0.1 |
Level structure
| $\GL_2(\Z/33\Z)$-generators: | $\begin{bmatrix}17&30\\12&13\end{bmatrix}$, $\begin{bmatrix}26&32\\6&17\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 3.4.0.a.1 for the level structure with $-I$) |
| Cyclic 33-isogeny field degree: | $12$ |
| Cyclic 33-torsion field degree: | $240$ |
| Full 33-torsion field degree: | $79200$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 78278 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 4 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^3}\cdot\frac{x^{4}(x-18y)^{3}(x+30y)}{y^{3}x^{4}(x-24y)}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 33.24.0-3.a.1.1 | $33$ | $3$ | $3$ | $0$ |
| 66.16.0-6.a.1.1 | $66$ | $2$ | $2$ | $0$ |
| 66.16.0-6.b.1.1 | $66$ | $2$ | $2$ | $0$ |
| 66.24.0-6.a.1.1 | $66$ | $3$ | $3$ | $0$ |
| 99.24.0-9.a.1.1 | $99$ | $3$ | $3$ | $0$ |
| 99.24.0-9.b.1.1 | $99$ | $3$ | $3$ | $0$ |
| 99.24.1-9.a.1.1 | $99$ | $3$ | $3$ | $1$ |
| 132.16.0-12.a.1.1 | $132$ | $2$ | $2$ | $0$ |
| 132.16.0-12.b.1.1 | $132$ | $2$ | $2$ | $0$ |
| 132.32.1-12.a.1.8 | $132$ | $4$ | $4$ | $1$ |
| 165.40.1-15.a.1.1 | $165$ | $5$ | $5$ | $1$ |
| 165.48.1-15.a.1.8 | $165$ | $6$ | $6$ | $1$ |
| 165.80.2-15.a.1.5 | $165$ | $10$ | $10$ | $2$ |
| 231.64.1-21.a.1.7 | $231$ | $8$ | $8$ | $1$ |
| 231.168.5-21.a.1.8 | $231$ | $21$ | $21$ | $5$ |
| 231.224.6-21.a.1.4 | $231$ | $28$ | $28$ | $6$ |
| 264.16.0-24.a.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.16.0-24.b.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.16.0-24.c.1.3 | $264$ | $2$ | $2$ | $0$ |
| 264.16.0-24.d.1.5 | $264$ | $2$ | $2$ | $0$ |
| 330.16.0-30.a.1.4 | $330$ | $2$ | $2$ | $0$ |
| 330.16.0-30.b.1.3 | $330$ | $2$ | $2$ | $0$ |
| 33.96.3-33.a.1.7 | $33$ | $12$ | $12$ | $3$ |
| 33.440.13-33.a.1.3 | $33$ | $55$ | $55$ | $13$ |
| 33.440.14-33.a.1.5 | $33$ | $55$ | $55$ | $14$ |
| 33.528.17-33.a.1.8 | $33$ | $66$ | $66$ | $17$ |
| 66.16.0-66.a.1.2 | $66$ | $2$ | $2$ | $0$ |
| 66.16.0-66.b.1.3 | $66$ | $2$ | $2$ | $0$ |
| 132.16.0-132.a.1.4 | $132$ | $2$ | $2$ | $0$ |
| 132.16.0-132.b.1.6 | $132$ | $2$ | $2$ | $0$ |
| 264.16.0-264.a.1.14 | $264$ | $2$ | $2$ | $0$ |
| 264.16.0-264.b.1.15 | $264$ | $2$ | $2$ | $0$ |
| 264.16.0-264.c.1.15 | $264$ | $2$ | $2$ | $0$ |
| 264.16.0-264.d.1.8 | $264$ | $2$ | $2$ | $0$ |
| 330.16.0-330.a.1.7 | $330$ | $2$ | $2$ | $0$ |
| 330.16.0-330.b.1.7 | $330$ | $2$ | $2$ | $0$ |