Invariants
| Level: | $40$ | $\SL_2$-level: | $4$ | ||||
| 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 | $4^{3}$ | 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: | 4F0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.12.0.28 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}11&14\\38&21\end{bmatrix}$, $\begin{bmatrix}15&4\\11&37\end{bmatrix}$, $\begin{bmatrix}15&8\\13&13\end{bmatrix}$, $\begin{bmatrix}21&20\\27&11\end{bmatrix}$, $\begin{bmatrix}35&8\\19&25\end{bmatrix}$ |
| 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\cdot3^3\cdot29^3\,\frac{(15x+y)^{12}(15x^{2}-2y^{2})^{3}(495x^{2}+240xy+182y^{2})^{3}}{(15x+y)^{16}(405x^{2}-120xy-178y^{2})^{4}}$ |
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.a.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.q.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.z.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.bb.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.ck.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.cm.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.cv.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.cy.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.de.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.dk.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.ee.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.eg.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.em.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.es.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.ew.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.0.ey.1 | $40$ | $2$ | $2$ | $0$ |
| 40.24.1.bi.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.bk.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.bo.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.bu.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.cw.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.cy.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.ds.1 | $40$ | $2$ | $2$ | $1$ |
| 40.24.1.dy.1 | $40$ | $2$ | $2$ | $1$ |
| 40.60.4.co.1 | $40$ | $5$ | $5$ | $4$ |
| 40.72.3.fk.1 | $40$ | $6$ | $6$ | $3$ |
| 40.120.7.io.1 | $40$ | $10$ | $10$ | $7$ |
| 120.24.0.eo.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.er.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.fc.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.ff.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.ho.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.hu.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.ij.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.iq.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.ji.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.jo.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.ky.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.la.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.na.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.ng.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.nk.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.0.nm.1 | $120$ | $2$ | $2$ | $0$ |
| 120.24.1.gg.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.gi.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.gm.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.gs.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.ly.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.ma.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.nk.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.nq.1 | $120$ | $2$ | $2$ | $1$ |
| 120.36.1.rs.1 | $120$ | $3$ | $3$ | $1$ |
| 120.48.2.n.1 | $120$ | $4$ | $4$ | $2$ |
| 280.24.0.fu.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.fx.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.ge.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.gh.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.hg.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.hn.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.ia.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.ih.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.ji.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.jo.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.ki.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.kk.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.lm.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.ls.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.lw.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.0.ly.1 | $280$ | $2$ | $2$ | $0$ |
| 280.24.1.hu.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.hw.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.ia.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.ig.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.ji.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.jk.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.ke.1 | $280$ | $2$ | $2$ | $1$ |
| 280.24.1.kk.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.6.s.1 | $280$ | $8$ | $8$ | $6$ |
| 280.252.15.gg.1 | $280$ | $21$ | $21$ | $15$ |
| 280.336.21.xq.1 | $280$ | $28$ | $28$ | $21$ |