Invariants
| Level: | $30$ | $\SL_2$-level: | $6$ | ||||
| Index: | $16$ | $\PSL_2$-index: | $8$ | ||||
| Genus: | $0 = 1 + \frac{ 8 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $2\cdot6$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6C0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 30.16.0.10 |
Level structure
| $\GL_2(\Z/30\Z)$-generators: | $\begin{bmatrix}29&5\\6&17\end{bmatrix}$, $\begin{bmatrix}29&13\\3&4\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 30.8.0.b.1 for the level structure with $-I$) |
| Cyclic 30-isogeny field degree: | $18$ |
| Cyclic 30-torsion field degree: | $144$ |
| Full 30-torsion field degree: | $8640$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 162 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 8 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^6\cdot3^3\cdot5^3}\cdot\frac{x^{8}(45x^{2}-64y^{2})^{3}(405x^{2}-64y^{2})}{y^{2}x^{14}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 6.8.0-3.a.1.2 | $6$ | $2$ | $2$ | $0$ | $0$ |
| 15.8.0-3.a.1.2 | $15$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 30.48.0-30.b.1.1 | $30$ | $3$ | $3$ | $0$ |
| 30.48.1-30.e.1.2 | $30$ | $3$ | $3$ | $1$ |
| 30.80.2-30.d.1.1 | $30$ | $5$ | $5$ | $2$ |
| 30.96.3-30.d.1.7 | $30$ | $6$ | $6$ | $3$ |
| 30.160.5-30.d.1.7 | $30$ | $10$ | $10$ | $5$ |
| 60.64.1-60.c.1.2 | $60$ | $4$ | $4$ | $1$ |
| 90.48.0-90.b.1.2 | $90$ | $3$ | $3$ | $0$ |
| 90.48.1-90.b.1.4 | $90$ | $3$ | $3$ | $1$ |
| 90.48.2-90.b.1.2 | $90$ | $3$ | $3$ | $2$ |
| 210.128.3-210.b.1.10 | $210$ | $8$ | $8$ | $3$ |
| 210.336.12-210.b.1.12 | $210$ | $21$ | $21$ | $12$ |
| 210.448.15-210.b.1.15 | $210$ | $28$ | $28$ | $15$ |
| 330.192.7-330.b.1.9 | $330$ | $12$ | $12$ | $7$ |