Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | ||||
| 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 | $4\cdot8$ | 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: | 8B0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.12.0.21 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&4\\0&7\end{bmatrix}$, $\begin{bmatrix}3&11\\14&1\end{bmatrix}$, $\begin{bmatrix}19&11\\6&7\end{bmatrix}$, $\begin{bmatrix}21&19\\2&19\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $16$ |
| Cyclic 24-torsion field degree: | $128$ |
| 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 8 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^8}{3^4}\cdot\frac{(3x+y)^{12}(9x^{4}+y^{4})^{3}}{y^{4}x^{8}(3x+y)^{12}}$ |
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 |
|---|---|---|---|---|---|
| 4.6.0.d.1 | $4$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 24.24.1.c.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.e.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.i.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.k.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.r.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.s.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.v.1 | $24$ | $2$ | $2$ | $1$ |
| 24.24.1.w.1 | $24$ | $2$ | $2$ | $1$ |
| 24.36.0.cc.1 | $24$ | $3$ | $3$ | $0$ |
| 24.48.3.ci.1 | $24$ | $4$ | $4$ | $3$ |
| 72.324.22.ja.1 | $72$ | $27$ | $27$ | $22$ |
| 120.24.1.oy.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.oz.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.pc.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.pd.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.po.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.pp.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.ps.1 | $120$ | $2$ | $2$ | $1$ |
| 120.24.1.pt.1 | $120$ | $2$ | $2$ | $1$ |
| 120.60.4.ho.1 | $120$ | $5$ | $5$ | $4$ |
| 120.72.3.gqq.1 | $120$ | $6$ | $6$ | $3$ |
| 120.120.7.boa.1 | $120$ | $10$ | $10$ | $7$ |
| 168.24.1.md.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.me.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.mh.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.mi.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.mt.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.mu.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.mx.1 | $168$ | $2$ | $2$ | $1$ |
| 168.24.1.my.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.7.ii.1 | $168$ | $8$ | $8$ | $7$ |
| 168.252.14.i.1 | $168$ | $21$ | $21$ | $14$ |
| 168.336.21.bni.1 | $168$ | $28$ | $28$ | $21$ |
| 264.24.1.me.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.mf.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.mi.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.mj.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.mu.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.mv.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.my.1 | $264$ | $2$ | $2$ | $1$ |
| 264.24.1.mz.1 | $264$ | $2$ | $2$ | $1$ |
| 264.144.11.ia.1 | $264$ | $12$ | $12$ | $11$ |
| 312.24.1.me.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.mf.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.mi.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.mj.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.mu.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.mv.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.my.1 | $312$ | $2$ | $2$ | $1$ |
| 312.24.1.mz.1 | $312$ | $2$ | $2$ | $1$ |
| 312.168.11.xa.1 | $312$ | $14$ | $14$ | $11$ |