Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $4$ are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $1^{4}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.0.580 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}21&8\\6&1\end{bmatrix}$, $\begin{bmatrix}27&24\\22&35\end{bmatrix}$, $\begin{bmatrix}31&16\\38&27\end{bmatrix}$, $\begin{bmatrix}33&8\\5&3\end{bmatrix}$, $\begin{bmatrix}33&32\\12&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.q.1 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $6$ |
| Cyclic 40-torsion field degree: | $96$ |
| Full 40-torsion field degree: | $15360$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 61 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{x^{24}(x^{4}-16x^{3}y+8x^{2}y^{2}+64xy^{3}+16y^{4})^{3}(x^{4}+16x^{3}y+8x^{2}y^{2}-64xy^{3}+16y^{4})^{3}}{y^{2}x^{26}(x-2y)^{2}(x+2y)^{2}(x^{2}+4y^{2})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 40.24.0-4.d.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-4.d.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-8.n.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-8.n.1.6 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-8.n.1.9 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0-8.n.1.11 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 40.96.0-8.n.1.3 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-8.n.1.4 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-8.n.2.1 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-8.n.2.4 | $40$ | $2$ | $2$ | $0$ |
| 40.96.1-8.bb.1.4 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-8.bc.1.3 | $40$ | $2$ | $2$ | $1$ |
| 80.96.0-16.j.1.4 | $80$ | $2$ | $2$ | $0$ |
| 80.96.0-16.j.1.6 | $80$ | $2$ | $2$ | $0$ |
| 80.96.0-16.k.1.3 | $80$ | $2$ | $2$ | $0$ |
| 80.96.0-16.k.1.6 | $80$ | $2$ | $2$ | $0$ |
| 80.96.1-16.g.1.2 | $80$ | $2$ | $2$ | $1$ |
| 80.96.1-16.g.1.6 | $80$ | $2$ | $2$ | $1$ |
| 80.96.1-16.h.1.5 | $80$ | $2$ | $2$ | $1$ |
| 80.96.1-16.h.1.7 | $80$ | $2$ | $2$ | $1$ |
| 80.96.2-16.k.1.5 | $80$ | $2$ | $2$ | $2$ |
| 80.96.2-16.k.1.11 | $80$ | $2$ | $2$ | $2$ |
| 120.96.0-24.be.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.be.1.11 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.be.2.2 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-24.be.2.9 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-24.dj.1.3 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-24.dk.1.5 | $120$ | $2$ | $2$ | $1$ |
| 120.144.4-24.dh.1.17 | $120$ | $3$ | $3$ | $4$ |
| 120.192.3-24.df.1.18 | $120$ | $4$ | $4$ | $3$ |
| 40.96.0-40.bf.1.3 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.bf.1.5 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.bf.2.1 | $40$ | $2$ | $2$ | $0$ |
| 40.96.0-40.bf.2.6 | $40$ | $2$ | $2$ | $0$ |
| 40.96.1-40.cx.1.5 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1-40.cy.1.6 | $40$ | $2$ | $2$ | $1$ |
| 40.240.8-40.bj.1.10 | $40$ | $5$ | $5$ | $8$ |
| 40.288.7-40.cl.1.4 | $40$ | $6$ | $6$ | $7$ |
| 40.480.15-40.dh.1.4 | $40$ | $10$ | $10$ | $15$ |
| 240.96.0-48.j.1.6 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-48.j.1.19 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-48.k.1.4 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-48.k.1.18 | $240$ | $2$ | $2$ | $0$ |
| 240.96.1-48.g.1.3 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-48.g.1.20 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-48.h.1.7 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-48.h.1.24 | $240$ | $2$ | $2$ | $1$ |
| 240.96.2-48.h.1.13 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2-48.h.1.23 | $240$ | $2$ | $2$ | $2$ |
| 280.96.0-56.bd.1.3 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-56.bd.1.8 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-56.bd.2.3 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-56.bd.2.8 | $280$ | $2$ | $2$ | $0$ |
| 280.96.1-56.cx.1.3 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-56.cy.1.3 | $280$ | $2$ | $2$ | $1$ |
| 280.384.11-56.cf.1.8 | $280$ | $8$ | $8$ | $11$ |
| 80.96.0-80.l.1.9 | $80$ | $2$ | $2$ | $0$ |
| 80.96.0-80.l.1.20 | $80$ | $2$ | $2$ | $0$ |
| 80.96.0-80.m.1.3 | $80$ | $2$ | $2$ | $0$ |
| 80.96.0-80.m.1.20 | $80$ | $2$ | $2$ | $0$ |
| 80.96.1-80.g.1.3 | $80$ | $2$ | $2$ | $1$ |
| 80.96.1-80.g.1.17 | $80$ | $2$ | $2$ | $1$ |
| 80.96.1-80.h.1.11 | $80$ | $2$ | $2$ | $1$ |
| 80.96.1-80.h.1.21 | $80$ | $2$ | $2$ | $1$ |
| 80.96.2-80.j.1.13 | $80$ | $2$ | $2$ | $2$ |
| 80.96.2-80.j.1.21 | $80$ | $2$ | $2$ | $2$ |
| 120.96.0-120.dc.1.5 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.dc.1.12 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.dc.2.7 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.dc.2.10 | $120$ | $2$ | $2$ | $0$ |
| 120.96.1-120.ix.1.11 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1-120.iy.1.7 | $120$ | $2$ | $2$ | $1$ |
| 240.96.0-240.l.1.13 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.l.1.22 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.m.1.11 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.m.1.20 | $240$ | $2$ | $2$ | $0$ |
| 240.96.1-240.g.1.4 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.g.1.22 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.h.1.12 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.h.1.30 | $240$ | $2$ | $2$ | $1$ |
| 240.96.2-240.j.1.21 | $240$ | $2$ | $2$ | $2$ |
| 240.96.2-240.j.1.45 | $240$ | $2$ | $2$ | $2$ |
| 280.96.0-280.db.1.6 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.db.1.11 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.db.2.3 | $280$ | $2$ | $2$ | $0$ |
| 280.96.0-280.db.2.14 | $280$ | $2$ | $2$ | $0$ |
| 280.96.1-280.il.1.10 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1-280.im.1.11 | $280$ | $2$ | $2$ | $1$ |