Invariants
| Level: | $15$ | $\SL_2$-level: | $5$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot5^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 5D0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 15.24.0.2 |
Level structure
| $\GL_2(\Z/15\Z)$-generators: | $\begin{bmatrix}10&1\\13&3\end{bmatrix}$, $\begin{bmatrix}13&1\\11&3\end{bmatrix}$ |
| $\GL_2(\Z/15\Z)$-subgroup: | $F_5\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 5.12.0.a.1 for the level structure with $-I$) |
| Cyclic 15-isogeny field degree: | $4$ |
| Cyclic 15-torsion field degree: | $16$ |
| Full 15-torsion field degree: | $960$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 1545 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^{12}\,\frac{x^{12}(29x^{4}+62x^{3}y+16x^{2}y^{2}-32xy^{3}-16y^{4})^{3}}{x^{17}(5x+4y)^{5}(31x^{2}+4xy-16y^{2})}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 15.120.0-5.a.1.2 | $15$ | $5$ | $5$ | $0$ |
| 30.48.1-10.a.1.1 | $30$ | $2$ | $2$ | $1$ |
| 30.48.1-10.b.1.1 | $30$ | $2$ | $2$ | $1$ |
| 30.72.0-10.a.2.3 | $30$ | $3$ | $3$ | $0$ |
| 15.72.2-15.a.1.6 | $15$ | $3$ | $3$ | $2$ |
| 15.96.1-15.a.1.7 | $15$ | $4$ | $4$ | $1$ |
| 60.48.1-20.b.1.2 | $60$ | $2$ | $2$ | $1$ |
| 60.48.1-20.e.1.3 | $60$ | $2$ | $2$ | $1$ |
| 60.96.3-20.i.1.5 | $60$ | $4$ | $4$ | $3$ |
| 75.120.0-25.a.1.1 | $75$ | $5$ | $5$ | $0$ |
| 30.48.1-30.d.1.1 | $30$ | $2$ | $2$ | $1$ |
| 30.48.1-30.i.1.3 | $30$ | $2$ | $2$ | $1$ |
| 105.192.5-35.a.2.8 | $105$ | $8$ | $8$ | $5$ |
| 105.504.16-35.a.2.5 | $105$ | $21$ | $21$ | $16$ |
| 120.48.1-40.bx.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-40.cd.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-40.cj.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-40.cp.1.1 | $120$ | $2$ | $2$ | $1$ |
| 45.648.22-45.a.1.3 | $45$ | $27$ | $27$ | $22$ |
| 165.288.9-55.a.2.3 | $165$ | $12$ | $12$ | $9$ |
| 60.48.1-60.k.1.5 | $60$ | $2$ | $2$ | $1$ |
| 60.48.1-60.bd.1.5 | $60$ | $2$ | $2$ | $1$ |
| 195.336.11-65.a.1.8 | $195$ | $14$ | $14$ | $11$ |
| 210.48.1-70.c.1.1 | $210$ | $2$ | $2$ | $1$ |
| 210.48.1-70.d.1.1 | $210$ | $2$ | $2$ | $1$ |
| 255.432.15-85.a.2.1 | $255$ | $18$ | $18$ | $15$ |
| 285.480.17-95.a.1.6 | $285$ | $20$ | $20$ | $17$ |
| 330.48.1-110.c.1.4 | $330$ | $2$ | $2$ | $1$ |
| 330.48.1-110.d.1.2 | $330$ | $2$ | $2$ | $1$ |
| 120.48.1-120.en.1.11 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.et.1.11 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.jl.1.7 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.jr.1.7 | $120$ | $2$ | $2$ | $1$ |
| 210.48.1-210.s.1.7 | $210$ | $2$ | $2$ | $1$ |
| 210.48.1-210.v.1.5 | $210$ | $2$ | $2$ | $1$ |
| 330.48.1-330.s.1.2 | $330$ | $2$ | $2$ | $1$ |
| 330.48.1-330.v.1.6 | $330$ | $2$ | $2$ | $1$ |