Invariants
| Level: | $24$ | $\SL_2$-level: | $6$ | ||||
| Index: | $12$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $6^{2}$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $4$ 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: | 6E0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.12.0.6 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}9&13\\16&3\end{bmatrix}$, $\begin{bmatrix}12&1\\13&9\end{bmatrix}$, $\begin{bmatrix}21&8\\7&21\end{bmatrix}$, $\begin{bmatrix}21&23\\23&18\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $24$ |
| Cyclic 24-torsion field degree: | $192$ |
| Full 24-torsion field degree: | $6144$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 14 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^3}\cdot\frac{(x+y)^{12}(x^{2}-18y^{2})^{3}(x^{2}+6y^{2})^{3}}{y^{6}x^{6}(x+y)^{12}}$ |
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 |
|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(3)$ | $3$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 24.24.1.bz.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.ca.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.cc.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.cd.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.ci.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.cj.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.co.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.cp.1 | $24$ | $2$ | $2$ | $1$ |
| 24.36.0.e.1 | $24$ | $3$ | $3$ | $0$ |
| 24.48.1.mi.1 | $24$ | $4$ | $4$ | $1$ |
| 72.36.0.e.1 | $72$ | $3$ | $3$ | $0$ |
| 72.36.1.e.1 | $72$ | $3$ | $3$ | $1$ |
| 72.36.1.g.1 | $72$ | $3$ | $3$ | $1$ |
| 120.24.1.oa.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.ob.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.od.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.oe.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.om.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.on.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.op.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.oq.1 | $120$ | $2$ | $2$ | $1$ |
| 120.60.4.hm.1 | $120$ | $5$ | $5$ | $4$ |
| 120.72.3.gqo.1 | $120$ | $6$ | $6$ | $3$ |
| 120.120.7.bny.1 | $120$ | $10$ | $10$ | $7$ |
| 168.24.1.lf.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.lg.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.li.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.lj.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.lr.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.ls.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.lu.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.lv.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.7.ig.1 | $168$ | $8$ | $8$ | $7$ |
| 168.252.14.g.1 | $168$ | $21$ | $21$ | $14$ |
| 168.336.21.bng.1 | $168$ | $28$ | $28$ | $21$ |
| 264.24.1.lg.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.lh.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.lj.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.lk.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.ls.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.lt.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.lv.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.lw.1 | $264$ | $2$ | $2$ | $1$ |
| 264.144.11.hy.1 | $264$ | $12$ | $12$ | $11$ |
| 312.24.1.lg.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.lh.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.lj.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.lk.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.ls.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.lt.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.lv.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.lw.1 | $312$ | $2$ | $2$ | $1$ |
| 312.168.11.wy.1 | $312$ | $14$ | $14$ | $11$ |