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: | 40.48.0.50 |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^8\cdot5^4}\cdot\frac{(3x+17y)^{48}(2762181x^{8}+86377752x^{7}y+1238842188x^{6}y^{2}+10665466344x^{5}y^{3}+60049436670x^{4}y^{4}+224568650664x^{3}y^{5}+539556665868x^{2}y^{6}+754911043032xy^{7}+467813715301y^{8})^{3}(8273421x^{8}+261687672x^{7}y+3627046188x^{6}y^{2}+28756446984x^{5}y^{3}+142586719470x^{4}y^{4}+452748644424x^{3}y^{5}+899473962348x^{2}y^{6}+1023713071032xy^{7}+512491508461y^{8})^{3}}{(3x+11y)^{8}(3x+17y)^{56}(3x^{2}+14xy+3y^{2})^{4}(63x^{2}+534xy+1183y^{2})^{4}(2349x^{4}+37044x^{3}y+225774x^{2}y^{2}+635124xy^{3}+700109y^{4})^{4}}$ |
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 |
|---|---|---|---|---|---|
| $X_{\mathrm{sp}}(4)$ | $4$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0.h.2 | $40$ | $2$ | $2$ | $0$ | $0$ |
| 40.24.0.i.1 | $40$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 40.96.1.a.2 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.b.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.e.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.f.2 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.l.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.m.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.n.2 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.o.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.p.2 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.q.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.w.1 | $40$ | $2$ | $2$ | $1$ |
| 40.96.1.x.2 | $40$ | $2$ | $2$ | $1$ |
| 40.96.3.r.1 | $40$ | $2$ | $2$ | $3$ |
| 40.96.3.s.1 | $40$ | $2$ | $2$ | $3$ |
| 40.96.3.u.1 | $40$ | $2$ | $2$ | $3$ |
| 40.96.3.x.2 | $40$ | $2$ | $2$ | $3$ |
| 40.240.16.c.1 | $40$ | $5$ | $5$ | $16$ |
| 40.288.15.e.2 | $40$ | $6$ | $6$ | $15$ |
| 40.480.31.g.1 | $40$ | $10$ | $10$ | $31$ |
| 120.96.1.g.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.h.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.w.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.x.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.bl.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.bm.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.bp.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.bq.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.bv.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.bw.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.cs.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.ct.1 | $120$ | $2$ | $2$ | $1$ |
| 120.96.3.bl.1 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.bm.1 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.by.1 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.bz.1 | $120$ | $2$ | $2$ | $3$ |
| 120.144.8.h.1 | $120$ | $3$ | $3$ | $8$ |
| 120.192.7.g.2 | $120$ | $4$ | $4$ | $7$ |
| 280.96.1.o.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.p.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.y.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.z.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.bn.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.bo.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.bp.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.bq.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.bx.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.by.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.ck.1 | $280$ | $2$ | $2$ | $1$ |
| 280.96.1.cl.2 | $280$ | $2$ | $2$ | $1$ |
| 280.96.3.ca.1 | $280$ | $2$ | $2$ | $3$ |
| 280.96.3.cb.1 | $280$ | $2$ | $2$ | $3$ |
| 280.96.3.cc.1 | $280$ | $2$ | $2$ | $3$ |
| 280.96.3.cd.1 | $280$ | $2$ | $2$ | $3$ |
| 280.384.23.g.1 | $280$ | $8$ | $8$ | $23$ |