Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | ||||
| 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 | $2^{4}\cdot4\cdot6^{4}\cdot12$ | Cusp orbits | $1^{2}\cdot4^{2}$ | ||
| 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: | 12I0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.0.964 |
Level structure
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 8 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{x^{48}(x^{4}-3y^{4})^{3}(x^{12}+15x^{8}y^{4}+75x^{4}y^{8}-3y^{12})^{3}}{y^{12}x^{52}(x^{4}+y^{4})^{6}(x^{4}+9y^{4})^{2}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.24.0.d.1 | $12$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 24.96.3.dk.1 | $24$ | $2$ | $2$ | $3$ |
| 24.96.3.dm.1 | $24$ | $2$ | $2$ | $3$ |
| 24.96.3.dp.1 | $24$ | $2$ | $2$ | $3$ |
| 24.96.3.dr.2 | $24$ | $2$ | $2$ | $3$ |
| 24.96.3.dx.2 | $24$ | $2$ | $2$ | $3$ |
| 24.96.3.dy.1 | $24$ | $2$ | $2$ | $3$ |
| 24.96.3.ea.1 | $24$ | $2$ | $2$ | $3$ |
| 24.96.3.ef.1 | $24$ | $2$ | $2$ | $3$ |
| 24.144.3.b.1 | $24$ | $3$ | $3$ | $3$ |
| 72.144.3.b.2 | $72$ | $3$ | $3$ | $3$ |
| 72.144.8.c.1 | $72$ | $3$ | $3$ | $8$ |
| 72.144.8.d.1 | $72$ | $3$ | $3$ | $8$ |
| 120.96.3.iu.1 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.ix.1 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.ja.1 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.jd.2 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.jp.2 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.js.1 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.jv.1 | $120$ | $2$ | $2$ | $3$ |
| 120.96.3.jy.1 | $120$ | $2$ | $2$ | $3$ |
| 120.240.16.em.1 | $120$ | $5$ | $5$ | $16$ |
| 120.288.15.eaj.2 | $120$ | $6$ | $6$ | $15$ |
| 168.96.3.gu.1 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.gx.1 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.ha.1 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.hd.2 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.hp.2 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.hs.1 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.hv.1 | $168$ | $2$ | $2$ | $3$ |
| 168.96.3.hy.1 | $168$ | $2$ | $2$ | $3$ |
| 168.384.23.ko.2 | $168$ | $8$ | $8$ | $23$ |
| 264.96.3.gu.1 | $264$ | $2$ | $2$ | $3$ |
| 264.96.3.gx.1 | $264$ | $2$ | $2$ | $3$ |
| 264.96.3.ha.1 | $264$ | $2$ | $2$ | $3$ |
| 264.96.3.hd.2 | $264$ | $2$ | $2$ | $3$ |
| 264.96.3.hp.2 | $264$ | $2$ | $2$ | $3$ |
| 264.96.3.hs.1 | $264$ | $2$ | $2$ | $3$ |
| 264.96.3.hv.1 | $264$ | $2$ | $2$ | $3$ |
| 264.96.3.hy.1 | $264$ | $2$ | $2$ | $3$ |
| 312.96.3.iu.1 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.ix.1 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.ja.1 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.jd.2 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.jp.2 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.js.1 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.jv.1 | $312$ | $2$ | $2$ | $3$ |
| 312.96.3.jy.1 | $312$ | $2$ | $2$ | $3$ |