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: | 40.12.0.31 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}3&38\\30&29\end{bmatrix}$, $\begin{bmatrix}21&2\\37&11\end{bmatrix}$, $\begin{bmatrix}21&34\\1&23\end{bmatrix}$, $\begin{bmatrix}35&36\\39&37\end{bmatrix}$, $\begin{bmatrix}37&10\\11&27\end{bmatrix}$ |
| 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 53 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{16}}\cdot\frac{x^{12}(5x^{2}-192y^{2})^{3}(5x^{2}-64y^{2})^{3}}{y^{8}x^{12}(5x^{2}-128y^{2})^{2}}$ |
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.s.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.bf.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.bh.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.cc.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.cf.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.db.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.dc.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.dw.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.dx.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.ea.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.eb.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.ew.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.ex.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.fa.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.fb.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.1.bq.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.br.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.bu.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.bv.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.dm.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.dn.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.dq.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.dr.1 | $40$ | $2$ | $2$ | $1$ |
| 40.60.4.cr.1 | $40$ | $5$ | $5$ | $4$ |
| 40.72.3.fr.1 | $40$ | $6$ | $6$ | $3$ |
| 40.120.7.iz.1 | $40$ | $10$ | $10$ | $7$ |
| 120.24.0.ev.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.ex.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.fj.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.fl.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.hi.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.hl.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.it.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.iu.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.kq.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.kr.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.ku.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.kv.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.ns.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.nt.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.nw.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.nx.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.1.gw.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.gx.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.ha.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.hb.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.ne.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.nf.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.ni.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.nj.1 | $120$ | $2$ | $2$ | $1$ |
| 120.36.1.sh.1 | $120$ | $3$ | $3$ | $1$ |
| 120.48.2.u.1 | $120$ | $4$ | $4$ | $2$ |
| 280.24.0.gb.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.gd.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.gl.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.gn.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.hr.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.ht.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.il.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.in.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.kq.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.kr.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.ku.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.kv.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.me.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.mf.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.mi.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.mj.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.1.ik.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.il.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.io.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.ip.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.ko.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.kp.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.ks.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.kt.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.6.z.1 | $280$ | $8$ | $8$ | $6$ |
| 280.252.15.gv.1 | $280$ | $21$ | $21$ | $15$ |
| 280.336.21.yf.1 | $280$ | $28$ | $28$ | $21$ |