Invariants
| Level: | $40$ | $\SL_2$-level: | $5$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $1^{2}\cdot5^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 5D0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.12.0.45 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}25&17\\34&33\end{bmatrix}$, $\begin{bmatrix}25&28\\18&35\end{bmatrix}$, $\begin{bmatrix}26&1\\19&8\end{bmatrix}$, $\begin{bmatrix}26&7\\3&5\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | 80.24.0-40.bp.2.1, 80.24.0-40.bp.2.2, 80.24.0-40.bp.2.3, 80.24.0-40.bp.2.4, 80.24.0-40.bp.2.5, 80.24.0-40.bp.2.6, 80.24.0-40.bp.2.7, 80.24.0-40.bp.2.8, 80.24.0-40.bp.2.9, 80.24.0-40.bp.2.10, 80.24.0-40.bp.2.11, 80.24.0-40.bp.2.12, 80.24.0-40.bp.2.13, 80.24.0-40.bp.2.14, 80.24.0-40.bp.2.15, 80.24.0-40.bp.2.16, 240.24.0-40.bp.2.1, 240.24.0-40.bp.2.2, 240.24.0-40.bp.2.3, 240.24.0-40.bp.2.4, 240.24.0-40.bp.2.5, 240.24.0-40.bp.2.6, 240.24.0-40.bp.2.7, 240.24.0-40.bp.2.8, 240.24.0-40.bp.2.9, 240.24.0-40.bp.2.10, 240.24.0-40.bp.2.11, 240.24.0-40.bp.2.12, 240.24.0-40.bp.2.13, 240.24.0-40.bp.2.14, 240.24.0-40.bp.2.15, 240.24.0-40.bp.2.16 |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $61440$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 3 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^6\,\frac{(x-10y)^{12}(19x^{4}-1024x^{3}y+19664x^{2}y^{2}-161024xy^{3}+478144y^{4})^{3}}{(x-10y)^{12}(3x^{2}-64xy+328y^{2})^{5}(31x^{2}-688xy+3656y^{2})}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(5)$ | $5$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 40.24.1.bz.2 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.cb.2 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.cf.2 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.ch.2 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.cl.2 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.cn.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.cr.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.ct.1 | $40$ | $2$ | $2$ | $1$ |
| 40.36.0.b.1 | $40$ | $3$ | $3$ | $0$ |
| 40.48.3.j.2 | $40$ | $4$ | $4$ | $3$ |
| 40.60.0.b.1 | $40$ | $5$ | $5$ | $0$ |
| 120.24.1.ep.2 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.er.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.ev.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.ex.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.jn.2 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.jp.2 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.jt.2 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.jv.2 | $120$ | $2$ | $2$ | $1$ |
| 120.36.2.rl.1 | $120$ | $3$ | $3$ | $2$ |
| 120.48.1.cah.1 | $120$ | $4$ | $4$ | $1$ |
| 200.60.0.b.2 | $200$ | $5$ | $5$ | $0$ |
| 280.24.1.gt.2 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.gu.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.gw.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.gx.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.hf.2 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.hg.2 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.hi.2 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.hj.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.5.gj.1 | $280$ | $8$ | $8$ | $5$ |
| 280.252.16.kj.1 | $280$ | $21$ | $21$ | $16$ |
| 280.336.21.kj.2 | $280$ | $28$ | $28$ | $21$ |