Invariants
| Level: | $40$ | $\SL_2$-level: | $4$ | ||||
| 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 | $2^{2}\cdot4^{2}$ | 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: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 4E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.24.0.67 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&16\\3&29\end{bmatrix}$, $\begin{bmatrix}7&20\\33&9\end{bmatrix}$, $\begin{bmatrix}13&0\\18&1\end{bmatrix}$, $\begin{bmatrix}31&4\\34&27\end{bmatrix}$, $\begin{bmatrix}39&28\\38&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 4.12.0.d.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $12$ |
| Cyclic 40-torsion field degree: | $192$ |
| 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 746 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^2}\cdot\frac{(4x-y)^{12}(64x^{2}-32xy+y^{2})^{3}(64x^{2}+32xy+y^{2})^{3}}{y^{2}x^{2}(4x-y)^{12}(64x^{2}+y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.12.0-4.c.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.12.0-4.c.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 40.48.0-4.c.1.1 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-20.d.1.1 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-8.p.1.2 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-8.q.1.4 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-8.t.1.2 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.v.1.4 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-8.w.1.2 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.w.1.4 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-8.x.1.2 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.z.1.2 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.be.1.4 | $40$ | $2$ | $2$ | $0$ |
| 40.48.0-40.bf.1.4 | $40$ | $2$ | $2$ | $0$ |
| 40.48.1-8.m.1.2 | $40$ | $2$ | $2$ | $1$ |
| 40.48.1-40.m.1.7 | $40$ | $2$ | $2$ | $1$ |
| 40.48.1-8.n.1.2 | $40$ | $2$ | $2$ | $1$ |
| 40.48.1-40.n.1.8 | $40$ | $2$ | $2$ | $1$ |
| 40.120.4-20.h.1.4 | $40$ | $5$ | $5$ | $4$ |
| 40.144.3-20.l.1.1 | $40$ | $6$ | $6$ | $3$ |
| 40.240.7-20.p.1.6 | $40$ | $10$ | $10$ | $7$ |
| 120.48.0-12.e.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-60.h.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.u.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.v.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.z.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.bc.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-24.bd.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.bh.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.bi.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.bl.1.11 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.bq.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.br.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.48.1-24.m.1.7 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.m.1.13 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-24.n.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.n.1.13 | $120$ | $2$ | $2$ | $1$ |
| 120.72.2-12.p.1.4 | $120$ | $3$ | $3$ | $2$ |
| 120.96.1-12.h.1.1 | $120$ | $4$ | $4$ | $1$ |
| 280.48.0-28.d.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-140.h.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.t.1.4 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.u.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.x.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.ba.1.4 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-56.bb.1.2 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.bh.1.6 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.bi.1.7 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.bl.1.5 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.bq.1.6 | $280$ | $2$ | $2$ | $0$ |
| 280.48.0-280.br.1.6 | $280$ | $2$ | $2$ | $0$ |
| 280.48.1-56.m.1.3 | $280$ | $2$ | $2$ | $1$ |
| 280.48.1-280.m.1.11 | $280$ | $2$ | $2$ | $1$ |
| 280.48.1-56.n.1.5 | $280$ | $2$ | $2$ | $1$ |
| 280.48.1-280.n.1.11 | $280$ | $2$ | $2$ | $1$ |
| 280.192.5-28.h.1.4 | $280$ | $8$ | $8$ | $5$ |
| 280.504.16-28.p.1.6 | $280$ | $21$ | $21$ | $16$ |