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: | 40.12.0.32 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}9&8\\25&19\end{bmatrix}$, $\begin{bmatrix}9&16\\28&13\end{bmatrix}$, $\begin{bmatrix}15&18\\6&13\end{bmatrix}$, $\begin{bmatrix}23&22\\2&1\end{bmatrix}$, $\begin{bmatrix}23&24\\34&39\end{bmatrix}$ |
| 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 48 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^6}{5^4}\cdot\frac{x^{12}(5x^{2}-8y^{2})^{3}(15x^{2}-8y^{2})^{3}}{x^{20}(5x^{2}-4y^{2})^{2}}$ |
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.k.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.s.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.w.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.y.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.ch.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.ci.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.cq.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.ct.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.do.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.dp.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.ds.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.dt.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.eo.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.ep.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.es.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.et.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.1.bi.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.bj.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.bm.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.bn.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.de.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.df.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.di.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.dj.1 | $40$ | $2$ | $2$ | $1$ |
| 40.60.4.cl.1 | $40$ | $5$ | $5$ | $4$ |
| 40.72.3.ff.1 | $40$ | $6$ | $6$ | $3$ |
| 40.120.7.ih.1 | $40$ | $10$ | $10$ | $7$ |
| 120.24.0.el.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.en.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.ez.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.fb.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.hf.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.hg.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.hy.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.ib.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.js.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.jt.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.jw.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.jx.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.mu.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.mv.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.my.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.mz.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.1.fy.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.fz.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.gc.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.gd.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.mg.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.mh.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.mk.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.ml.1 | $120$ | $2$ | $2$ | $1$ |
| 120.36.1.rl.1 | $120$ | $3$ | $3$ | $1$ |
| 120.48.2.k.1 | $120$ | $4$ | $4$ | $2$ |
| 280.24.0.fn.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.fp.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.fr.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.ft.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.gx.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.gz.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.hb.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.hd.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.jc.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.jd.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.jg.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.jh.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.lg.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.lh.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.lk.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.ll.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.1.hm.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.hn.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.hq.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.hr.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.ja.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.jb.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.je.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.jf.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.6.p.1 | $280$ | $8$ | $8$ | $6$ |
| 280.252.15.fz.1 | $280$ | $21$ | $21$ | $15$ |
| 280.336.21.xj.1 | $280$ | $28$ | $28$ | $21$ |