Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 2 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (of which $1$ is rational) | Cusp widths | $2^{2}\cdot8$ | Cusp orbits | $1\cdot2$ | ||
| Elliptic points: | $2$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 8D0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.12.0.34 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&10\\12&23\end{bmatrix}$, $\begin{bmatrix}5&16\\38&9\end{bmatrix}$, $\begin{bmatrix}13&8\\13&27\end{bmatrix}$, $\begin{bmatrix}23&8\\14&31\end{bmatrix}$, $\begin{bmatrix}29&30\\24&7\end{bmatrix}$ |
| 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 36 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{16}}\cdot\frac{x^{12}(5x^{2}+64y^{2})^{3}(5x^{2}+192y^{2})^{3}}{y^{8}x^{12}(5x^{2}+128y^{2})^{2}}$ |
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 |
|---|---|---|---|---|---|
| 4.6.0.e.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 |
|---|---|---|---|---|
| 40.24.0.j.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.p.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.v.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.y.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.cg.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.cj.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.cr.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.cs.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.dm.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.dn.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.dq.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.dr.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.em.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.en.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.eq.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.er.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.1.bg.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.bh.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.bk.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.bl.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.dc.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.dd.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.dg.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.dh.1 | $40$ | $2$ | $2$ | $1$ |
| 40.60.4.ck.1 | $40$ | $5$ | $5$ | $4$ |
| 40.72.3.fe.1 | $40$ | $6$ | $6$ | $3$ |
| 40.120.7.ig.1 | $40$ | $10$ | $10$ | $7$ |
| 120.24.0.ek.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.em.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.ey.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.fa.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.he.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.hh.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.hz.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.ia.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.jq.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.jr.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.ju.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.jv.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.ms.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.mt.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.mw.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.mx.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.1.fw.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.fx.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.ga.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.gb.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.me.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.mf.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.mi.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.mj.1 | $120$ | $2$ | $2$ | $1$ |
| 120.36.1.rk.1 | $120$ | $3$ | $3$ | $1$ |
| 120.48.2.j.1 | $120$ | $4$ | $4$ | $2$ |
| 280.24.0.fm.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.fo.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.fq.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.fs.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.gw.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.gy.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.ha.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.hc.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.ja.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.jb.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.je.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.jf.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.le.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.lf.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.li.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.lj.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.1.hk.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.hl.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.ho.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.hp.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.iy.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.iz.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.jc.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.jd.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.6.o.1 | $280$ | $8$ | $8$ | $6$ |
| 280.252.15.fy.1 | $280$ | $21$ | $21$ | $15$ |
| 280.336.21.xi.1 | $280$ | $28$ | $28$ | $21$ |