Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | ||||
| Index: | $24$ | $\PSL_2$-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 | $2^{2}\cdot4^{2}$ | 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: | 4E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.24.0.290 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&2\\3&35\end{bmatrix}$, $\begin{bmatrix}15&22\\24&13\end{bmatrix}$, $\begin{bmatrix}33&22\\39&25\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.12.0.p.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $24$ |
| Cyclic 40-torsion field degree: | $384$ |
| Full 40-torsion field degree: | $30720$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 28 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^5}{3^4\cdot5}\cdot\frac{(8x+y)^{12}(31744x^{4}+20480x^{3}y+18240x^{2}y^{2}+3200xy^{3}+775y^{4})^{3}}{(8x+y)^{12}(32x^{2}-5y^{2})^{4}(32x^{2}+8xy+5y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.12.0-4.b.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 40.12.0-4.b.1.3 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 40.120.4-40.bb.1.4 | $40$ | $5$ | $5$ | $4$ |
| 40.144.3-40.bn.1.6 | $40$ | $6$ | $6$ | $3$ |
| 40.240.7-40.bz.1.7 | $40$ | $10$ | $10$ | $7$ |
| 80.48.0-80.i.1.4 | $80$ | $2$ | $2$ | $0$ |
| 80.48.0-80.i.1.8 | $80$ | $2$ | $2$ | $0$ |
| 80.48.0-80.j.1.4 | $80$ | $2$ | $2$ | $0$ |
| 80.48.0-80.j.1.8 | $80$ | $2$ | $2$ | $0$ |
| 80.48.0-80.k.1.4 | $80$ | $2$ | $2$ | $0$ |
| 80.48.0-80.k.1.8 | $80$ | $2$ | $2$ | $0$ |
| 80.48.0-80.l.1.4 | $80$ | $2$ | $2$ | $0$ |
| 80.48.0-80.l.1.8 | $80$ | $2$ | $2$ | $0$ |
| 120.72.2-120.bz.1.13 | $120$ | $3$ | $3$ | $2$ |
| 120.96.1-120.yz.1.24 | $120$ | $4$ | $4$ | $1$ |
| 240.48.0-240.i.1.8 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.i.1.14 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.j.1.8 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.j.1.15 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.k.1.8 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.k.1.15 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.l.1.8 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.l.1.14 | $240$ | $2$ | $2$ | $0$ |
| 280.192.5-280.bb.1.20 | $280$ | $8$ | $8$ | $5$ |
| 280.504.16-280.bz.1.16 | $280$ | $21$ | $21$ | $16$ |