Invariants
Level: | $24$ | $\SL_2$-level: | $4$ | ||||
Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}$ | Cusp orbits | $2^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $1$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 4E0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.24.0.385 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}13&7\\2&13\end{bmatrix}$, $\begin{bmatrix}13&9\\14&17\end{bmatrix}$, $\begin{bmatrix}21&19\\20&7\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 12.12.0.d.1 for the level structure with $-I$) |
Cyclic 24-isogeny field degree: | $16$ |
Cyclic 24-torsion field degree: | $128$ |
Full 24-torsion field degree: | $3072$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 64 x^{2} + 3 y^{2} + 3 z^{2} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
8.12.0-4.a.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
24.12.0-4.a.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
24.72.2-12.o.1.4 | $24$ | $3$ | $3$ | $2$ |
24.96.1-12.g.1.2 | $24$ | $4$ | $4$ | $1$ |
120.120.4-60.h.1.4 | $120$ | $5$ | $5$ | $4$ |
120.144.3-60.di.1.6 | $120$ | $6$ | $6$ | $3$ |
120.240.7-60.p.1.2 | $120$ | $10$ | $10$ | $7$ |
168.192.5-84.h.1.9 | $168$ | $8$ | $8$ | $5$ |
168.504.16-84.p.1.3 | $168$ | $21$ | $21$ | $16$ |
264.288.9-132.h.1.10 | $264$ | $12$ | $12$ | $9$ |
312.336.11-156.l.1.5 | $312$ | $14$ | $14$ | $11$ |