Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{3}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8O0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.96.0.1041 |
Level structure
| $\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}5&32\\23&11\end{bmatrix}$, $\begin{bmatrix}13&24\\14&33\end{bmatrix}$, $\begin{bmatrix}17&28\\7&29\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.48.0.bk.2 for the level structure with $-I$) |
| Cyclic 40-isogeny field degree: | $6$ |
| Cyclic 40-torsion field degree: | $96$ |
| Full 40-torsion field degree: | $7680$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 2 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2}{3^2\cdot5}\cdot\frac{(2x+y)^{48}(31887616x^{16}+78540800x^{15}y-691768320x^{14}y^{2}-7351654400x^{13}y^{3}-30025580800x^{12}y^{4}-50074752000x^{11}y^{5}+121312000x^{10}y^{6}+206177600000x^{9}y^{7}+509759100000x^{8}y^{8}+515444000000x^{7}y^{9}+758200000x^{6}y^{10}-782418000000x^{5}y^{11}-1172874250000x^{4}y^{12}-717935000000x^{3}y^{13}-168888750000x^{2}y^{14}+47937500000xy^{15}+48656640625y^{16})^{3}}{(2x+y)^{48}(2x^{2}-5y^{2})^{2}(2x^{2}+2xy+5y^{2})^{2}(2x^{2}+20xy+5y^{2})^{4}(76x^{4}+80x^{3}y+60x^{2}y^{2}+200xy^{3}+475y^{4})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.ba.2.3 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-8.ba.2.8 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.bl.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.bl.1.11 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.cb.2.6 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.48.0-40.cb.2.14 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 40.480.16-40.ca.2.4 | $40$ | $5$ | $5$ | $16$ |
| 40.576.15-40.eq.2.4 | $40$ | $6$ | $6$ | $15$ |
| 40.960.31-40.gm.1.9 | $40$ | $10$ | $10$ | $31$ |
| 80.192.1-80.ci.2.5 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.ck.2.7 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.cq.1.5 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.cs.2.1 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.dq.1.12 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.ds.1.7 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.dy.2.3 | $80$ | $2$ | $2$ | $1$ |
| 80.192.1-80.ea.1.6 | $80$ | $2$ | $2$ | $1$ |
| 120.288.8-120.rk.1.17 | $120$ | $3$ | $3$ | $8$ |
| 120.384.7-120.ll.2.18 | $120$ | $4$ | $4$ | $7$ |
| 240.192.1-240.of.1.13 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.oh.2.13 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ov.2.12 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.ox.1.12 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.wv.2.13 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.wx.2.13 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.xl.1.12 | $240$ | $2$ | $2$ | $1$ |
| 240.192.1-240.xn.1.12 | $240$ | $2$ | $2$ | $1$ |