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 $2$ are rational) | Cusp widths | $2^{2}\cdot4^{3}\cdot8$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $1$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8J0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.0.242 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}9&36\\30&23\end{bmatrix}$, $\begin{bmatrix}11&0\\32&37\end{bmatrix}$, $\begin{bmatrix}23&16\\2&5\end{bmatrix}$, $\begin{bmatrix}23&36\\30&19\end{bmatrix}$, $\begin{bmatrix}39&20\\2&9\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 8.24.0.d.1 for the level structure with $-I$) |
Cyclic 40-isogeny field degree: | $12$ |
Cyclic 40-torsion field degree: | $192$ |
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 136 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^2\,\frac{x^{24}(256x^{8}+256x^{6}y^{2}+80x^{4}y^{4}+8x^{2}y^{6}+y^{8})^{3}}{y^{8}x^{28}(2x^{2}+y^{2})^{2}(4x^{2}+y^{2})^{4}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
40.24.0-4.b.1.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
40.24.0-4.b.1.6 | $40$ | $2$ | $2$ | $0$ | $0$ |
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.a.1.1 | $40$ | $2$ | $2$ | $0$ |
40.96.0-8.b.2.10 | $40$ | $2$ | $2$ | $0$ |
40.96.0-8.d.1.1 | $40$ | $2$ | $2$ | $0$ |
40.96.0-8.e.1.1 | $40$ | $2$ | $2$ | $0$ |
40.96.0-8.g.1.6 | $40$ | $2$ | $2$ | $0$ |
40.96.0-8.h.1.4 | $40$ | $2$ | $2$ | $0$ |
40.96.0-40.i.1.5 | $40$ | $2$ | $2$ | $0$ |
40.96.0-8.j.1.4 | $40$ | $2$ | $2$ | $0$ |
40.96.0-40.j.2.10 | $40$ | $2$ | $2$ | $0$ |
40.96.0-8.k.2.6 | $40$ | $2$ | $2$ | $0$ |
40.96.0-40.m.2.13 | $40$ | $2$ | $2$ | $0$ |
40.96.0-40.n.2.5 | $40$ | $2$ | $2$ | $0$ |
40.96.0-40.q.2.16 | $40$ | $2$ | $2$ | $0$ |
40.96.0-40.r.2.12 | $40$ | $2$ | $2$ | $0$ |
40.96.0-40.u.2.9 | $40$ | $2$ | $2$ | $0$ |
40.96.0-40.v.1.10 | $40$ | $2$ | $2$ | $0$ |
40.96.1-8.e.2.5 | $40$ | $2$ | $2$ | $1$ |
40.96.1-8.i.1.3 | $40$ | $2$ | $2$ | $1$ |
40.96.1-8.l.1.3 | $40$ | $2$ | $2$ | $1$ |
40.96.1-8.m.2.3 | $40$ | $2$ | $2$ | $1$ |
40.96.1-40.bc.2.3 | $40$ | $2$ | $2$ | $1$ |
40.96.1-40.bd.2.7 | $40$ | $2$ | $2$ | $1$ |
40.96.1-40.bg.2.7 | $40$ | $2$ | $2$ | $1$ |
40.96.1-40.bh.2.5 | $40$ | $2$ | $2$ | $1$ |
40.240.8-40.k.2.30 | $40$ | $5$ | $5$ | $8$ |
40.288.7-40.q.2.60 | $40$ | $6$ | $6$ | $7$ |
40.480.15-40.s.2.56 | $40$ | $10$ | $10$ | $15$ |
120.96.0-24.g.2.3 | $120$ | $2$ | $2$ | $0$ |
120.96.0-24.h.2.5 | $120$ | $2$ | $2$ | $0$ |
120.96.0-24.k.1.3 | $120$ | $2$ | $2$ | $0$ |
120.96.0-24.l.1.4 | $120$ | $2$ | $2$ | $0$ |
120.96.0-24.p.1.16 | $120$ | $2$ | $2$ | $0$ |
120.96.0-24.q.2.12 | $120$ | $2$ | $2$ | $0$ |
120.96.0-24.t.2.12 | $120$ | $2$ | $2$ | $0$ |
120.96.0-24.u.2.16 | $120$ | $2$ | $2$ | $0$ |
120.96.0-120.z.2.21 | $120$ | $2$ | $2$ | $0$ |
120.96.0-120.bb.1.11 | $120$ | $2$ | $2$ | $0$ |
120.96.0-120.bh.2.10 | $120$ | $2$ | $2$ | $0$ |
120.96.0-120.bj.2.17 | $120$ | $2$ | $2$ | $0$ |
120.96.0-120.bp.1.26 | $120$ | $2$ | $2$ | $0$ |
120.96.0-120.br.2.23 | $120$ | $2$ | $2$ | $0$ |
120.96.0-120.bx.1.18 | $120$ | $2$ | $2$ | $0$ |
120.96.0-120.bz.2.26 | $120$ | $2$ | $2$ | $0$ |
120.96.1-24.bc.2.10 | $120$ | $2$ | $2$ | $1$ |
120.96.1-24.bd.2.16 | $120$ | $2$ | $2$ | $1$ |
120.96.1-24.bg.2.8 | $120$ | $2$ | $2$ | $1$ |
120.96.1-24.bh.1.7 | $120$ | $2$ | $2$ | $1$ |
120.96.1-120.ds.2.14 | $120$ | $2$ | $2$ | $1$ |
120.96.1-120.du.2.15 | $120$ | $2$ | $2$ | $1$ |
120.96.1-120.ea.2.14 | $120$ | $2$ | $2$ | $1$ |
120.96.1-120.ec.2.14 | $120$ | $2$ | $2$ | $1$ |
120.144.4-24.s.2.63 | $120$ | $3$ | $3$ | $4$ |
120.192.3-24.bn.2.45 | $120$ | $4$ | $4$ | $3$ |
280.96.0-56.g.1.4 | $280$ | $2$ | $2$ | $0$ |
280.96.0-56.h.2.6 | $280$ | $2$ | $2$ | $0$ |
280.96.0-56.k.1.4 | $280$ | $2$ | $2$ | $0$ |
280.96.0-56.l.1.3 | $280$ | $2$ | $2$ | $0$ |
280.96.0-56.o.1.9 | $280$ | $2$ | $2$ | $0$ |
280.96.0-56.p.2.9 | $280$ | $2$ | $2$ | $0$ |
280.96.0-56.s.2.9 | $280$ | $2$ | $2$ | $0$ |
280.96.0-56.t.1.9 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.y.2.5 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.ba.1.6 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.bg.2.4 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.bi.2.2 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.bo.1.17 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.bq.2.22 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.bw.1.25 | $280$ | $2$ | $2$ | $0$ |
280.96.0-280.by.2.18 | $280$ | $2$ | $2$ | $0$ |
280.96.1-56.bc.2.6 | $280$ | $2$ | $2$ | $1$ |
280.96.1-56.bd.2.13 | $280$ | $2$ | $2$ | $1$ |
280.96.1-56.bg.2.7 | $280$ | $2$ | $2$ | $1$ |
280.96.1-56.bh.1.8 | $280$ | $2$ | $2$ | $1$ |
280.96.1-280.do.2.16 | $280$ | $2$ | $2$ | $1$ |
280.96.1-280.dq.2.14 | $280$ | $2$ | $2$ | $1$ |
280.96.1-280.dw.2.12 | $280$ | $2$ | $2$ | $1$ |
280.96.1-280.dy.2.24 | $280$ | $2$ | $2$ | $1$ |
280.384.11-56.p.2.56 | $280$ | $8$ | $8$ | $11$ |