Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| 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: | 8G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.0.396 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}11&16\\36&5\end{bmatrix}$, $\begin{bmatrix}13&12\\6&1\end{bmatrix}$, $\begin{bmatrix}21&8\\0&37\end{bmatrix}$, $\begin{bmatrix}37&24\\18&35\end{bmatrix}$, $\begin{bmatrix}39&20\\34&11\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.h.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $192$ |
| Full 40-torsion field degree: | $15360$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 49 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{(x+y)^{24}(x^{4}-2x^{3}y+2x^{2}y^{2}-4xy^{3}+4y^{4})^{3}(x^{4}+2x^{3}y+2x^{2}y^{2}+4xy^{3}+4y^{4})^{3}}{y^{8}x^{8}(x+y)^{24}(x^{2}-2y^{2})^{2}(x^{2}+2y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.24.0-4.b.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-4.b.1.8 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 40.96.0-8.g.1.1 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-8.g.1.4 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-8.h.1.4 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-8.h.1.7 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-8.h.2.6 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-8.h.2.7 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-8.i.1.1 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-8.i.1.4 | $40$ | $2$ | $2$ | $0$ |
| 40.96.1-8.g.1.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-8.g.1.9 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-8.j.1.3 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-8.j.1.5 | $40$ | $2$ | $2$ | $1$ |
| 120.96.0-24.x.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.x.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.y.1.11 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.y.1.16 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.y.2.13 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.y.2.16 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.z.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.z.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-24.bs.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.bs.1.4 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.bt.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.bt.1.8 | $120$ | $2$ | $2$ | $1$ |
| 120.144.4-24.cg.1.40 | $120$ | $3$ | $3$ | $4$ |
| 120.192.3-24.ck.1.34 | $120$ | $4$ | $4$ | $3$ |
| 40.96.0-40.y.1.3 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.y.1.7 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.z.1.14 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.z.1.16 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.z.2.14 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.z.2.16 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.ba.1.3 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.ba.1.7 | $40$ | $2$ | $2$ | $0$ |
| 40.96.1-40.bs.1.8 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.bs.1.15 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.bt.1.8 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.bt.1.15 | $40$ | $2$ | $2$ | $1$ |
| 40.240.8-40.u.1.10 | $40$ | $5$ | $5$ | $8$ |
| 40.288.7-40.bq.1.31 | $40$ | $6$ | $6$ | $7$ |
| 40.480.15-40.cg.1.24 | $40$ | $10$ | $10$ | $15$ |
| 280.96.0-56.w.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-56.w.1.8 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-56.x.1.9 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-56.x.1.14 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-56.x.2.10 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-56.x.2.11 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-56.y.1.4 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-56.y.1.7 | $280$ | $2$ | $2$ | $0$ |
| 280.96.1-56.bs.1.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-56.bs.1.9 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-56.bt.1.4 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-56.bt.1.11 | $280$ | $2$ | $2$ | $1$ |
| 280.384.11-56.bm.1.25 | $280$ | $8$ | $8$ | $11$ |
| 120.96.0-120.cv.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.cv.1.9 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.cw.1.18 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.cw.1.23 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.cw.2.19 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.cw.2.22 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.cx.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.cx.1.9 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-120.fo.1.14 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.fo.1.20 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.fp.1.11 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.fp.1.18 | $120$ | $2$ | $2$ | $1$ |
| 280.96.0-280.cu.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.cu.1.13 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.cv.1.18 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.cv.1.23 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.cv.2.18 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.cv.2.23 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.cw.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.cw.1.11 | $280$ | $2$ | $2$ | $0$ |
| 280.96.1-280.fk.1.6 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.fk.1.28 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.fl.1.3 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.fl.1.22 | $280$ | $2$ | $2$ | $1$ |