Invariants
| Level: | $40$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $4^{8}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| 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: | 8N0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.48.0.305 |
Level structure
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 5 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6\cdot5^2}\cdot\frac{(x-2y)^{48}(390625x^{16}+5000000x^{14}y^{2}+252000000x^{12}y^{4}+1523200000x^{10}y^{6}+4275200000x^{8}y^{8}+3899392000x^{6}y^{10}+1651507200x^{4}y^{12}+83886080x^{2}y^{14}+16777216y^{16})^{3}}{y^{4}x^{4}(x-2y)^{48}(5x^{2}-8y^{2})^{8}(5x^{2}+8y^{2})^{4}(25x^{4}+240x^{2}y^{2}+64y^{4})^{4}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.24.0.e.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0.e.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0.h.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 40.96.1.j.2 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.z.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.bk.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.bo.2 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.bv.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.bz.2 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.cf.2 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.ch.1 | $40$ | $2$ | $2$ | $1$ |
| 40.240.16.u.2 | $40$ | $5$ | $5$ | $16$ |
| 40.288.15.bh.1 | $40$ | $6$ | $6$ | $15$ |
| 40.480.31.bp.2 | $40$ | $10$ | $10$ | $31$ |
| 120.96.1.gn.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.gt.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.hs.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.hy.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.mv.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.nb.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.ob.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.oh.1 | $120$ | $2$ | $2$ | $1$ |
| 120.144.8.cx.2 | $120$ | $3$ | $3$ | $8$ |
| 120.192.7.cs.2 | $120$ | $4$ | $4$ | $7$ |
| 280.96.1.hd.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.hh.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.ht.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.hx.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.jp.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.jt.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.kf.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.kj.1 | $280$ | $2$ | $2$ | $1$ |
| 280.384.23.bo.1 | $280$ | $8$ | $8$ | $23$ |