Invariants
| Level: | $24$ | $\SL_2$-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 | $2^{4}\cdot4^{2}\cdot8^{4}$ | 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: | 8O0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.0.197 |
Level structure
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 6 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^4}\cdot\frac{(x+y)^{48}(x^{16}+16x^{15}y+144x^{14}y^{2}+896x^{13}y^{3}+4112x^{12}y^{4}+14400x^{11}y^{5}+38944x^{10}y^{6}+81088x^{9}y^{7}+147312x^{8}y^{8}+302464x^{7}y^{9}+654208x^{6}y^{10}+1115136x^{5}y^{11}+1317632x^{4}y^{12}+1046528x^{3}y^{13}+562176x^{2}y^{14}+206848xy^{15}+49408y^{16})^{3}}{y^{8}(x+y)^{56}(x^{2}+2xy-2y^{2})^{4}(x^{2}+2xy+4y^{2})^{8}(x^{4}+4x^{3}y+24x^{2}y^{2}+40xy^{3}+28y^{4})^{2}}$ |
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.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 24.24.0.i.2 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.24.0.m.1 | $24$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 24.96.1.s.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.t.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.x.2 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.y.2 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.bm.2 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.bn.2 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.bo.1 | $24$ | $2$ | $2$ | $1$ |
| 24.96.1.bp.1 | $24$ | $2$ | $2$ | $1$ |
| 24.144.8.er.1 | $24$ | $3$ | $3$ | $8$ |
| 24.192.7.cx.1 | $24$ | $4$ | $4$ | $7$ |
| 120.96.1.nq.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.nr.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.ns.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.nt.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.og.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.oh.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.oi.2 | $120$ | $2$ | $2$ | $1$ |
| 120.96.1.oj.2 | $120$ | $2$ | $2$ | $1$ |
| 120.240.16.dr.1 | $120$ | $5$ | $5$ | $16$ |
| 120.288.15.cci.1 | $120$ | $6$ | $6$ | $15$ |
| 168.96.1.nq.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.nr.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.ns.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.nt.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.og.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.oh.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.oi.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1.oj.1 | $168$ | $2$ | $2$ | $1$ |
| 168.384.23.jd.2 | $168$ | $8$ | $8$ | $23$ |
| 264.96.1.nq.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.nr.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.ns.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.nt.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.og.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.oh.2 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.oi.1 | $264$ | $2$ | $2$ | $1$ |
| 264.96.1.oj.1 | $264$ | $2$ | $2$ | $1$ |
| 312.96.1.nq.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.nr.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.ns.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.nt.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.og.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.oh.2 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.oi.1 | $312$ | $2$ | $2$ | $1$ |
| 312.96.1.oj.1 | $312$ | $2$ | $2$ | $1$ |