Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $4\cdot8$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $4$ 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: | 8B0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.12.0.17 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}19&16\\6&27\end{bmatrix}$, $\begin{bmatrix}19&16\\22&1\end{bmatrix}$, $\begin{bmatrix}37&16\\16&27\end{bmatrix}$, $\begin{bmatrix}39&18\\11&29\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $384$ |
| 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 6 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^2}{5^4}\cdot\frac{x^{12}(25x^{4}+64y^{4})^{3}}{y^{4}x^{20}}$ |
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 |
|---|---|---|---|---|---|
| 4.6.0.d.1 | $4$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 40.24.1.d.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.g.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.j.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.l.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.q.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.t.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.u.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.x.1 | $40$ | $2$ | $2$ | $1$ |
| 40.60.4.cv.1 | $40$ | $5$ | $5$ | $4$ |
| 40.72.3.fz.1 | $40$ | $6$ | $6$ | $3$ |
| 40.120.7.jl.1 | $40$ | $10$ | $10$ | $7$ |
| 120.24.1.hp.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.hr.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.ht.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.hv.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.if.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.ih.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.ij.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.il.1 | $120$ | $2$ | $2$ | $1$ |
| 120.36.0.gn.1 | $120$ | $3$ | $3$ | $0$ |
| 120.48.3.dd.1 | $120$ | $4$ | $4$ | $3$ |
| 280.24.1.mc.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.md.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.mg.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.mh.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.ms.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.mt.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.mw.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.mx.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.7.z.1 | $280$ | $8$ | $8$ | $7$ |
| 280.252.14.j.1 | $280$ | $21$ | $21$ | $14$ |
| 280.336.21.zf.1 | $280$ | $28$ | $28$ | $21$ |