Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $2^{4}\cdot8^{2}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8G0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.24.0.314 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&22\\16&23\end{bmatrix}$, $\begin{bmatrix}7&7\\0&17\end{bmatrix}$, $\begin{bmatrix}9&7\\16&23\end{bmatrix}$, $\begin{bmatrix}19&13\\2&21\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $64$ |
| Full 24-torsion field degree: | $3072$ |
Models
Smooth plane model Smooth plane model
| $ 0 $ | $=$ | $ 3 x^{2} - 24 x y - 48 y^{2} - 2 z^{2} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
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 |
|---|---|---|---|---|---|
| 8.12.0.r.1 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 24.12.0.bp.1 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.12.0.bq.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 |
|---|---|---|---|---|
| 24.48.1.go.1 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1.gp.1 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1.gw.1 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1.gx.1 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1.kg.1 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1.kh.1 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1.ko.1 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1.kp.1 | $24$ | $2$ | $2$ | $1$ |
| 24.72.4.jl.1 | $24$ | $3$ | $3$ | $4$ |
| 24.96.3.hh.1 | $24$ | $4$ | $4$ | $3$ |
| 120.48.1.bca.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.bcb.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.bce.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.bcf.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.bmu.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.bmv.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.bmy.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1.bmz.1 | $120$ | $2$ | $2$ | $1$ |
| 120.120.8.gx.1 | $120$ | $5$ | $5$ | $8$ |
| 120.144.7.gtg.1 | $120$ | $6$ | $6$ | $7$ |
| 120.240.15.bbb.1 | $120$ | $10$ | $10$ | $15$ |
| 168.48.1.bby.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.bbz.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.bcc.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.bcd.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.bms.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.bmt.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.bmw.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.bmx.1 | $168$ | $2$ | $2$ | $1$ |
| 168.192.11.pv.1 | $168$ | $8$ | $8$ | $11$ |
| 264.48.1.bby.1 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1.bbz.1 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1.bcc.1 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1.bcd.1 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1.bms.1 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1.bmt.1 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1.bmw.1 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1.bmx.1 | $264$ | $2$ | $2$ | $1$ |
| 264.288.19.bfu.1 | $264$ | $12$ | $12$ | $19$ |
| 312.48.1.bca.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.bcb.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.bce.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.bcf.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.bmu.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.bmv.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.bmy.1 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1.bmz.1 | $312$ | $2$ | $2$ | $1$ |
| 312.336.23.qa.1 | $312$ | $14$ | $14$ | $23$ |