Invariants
Level: | $24$ | $\SL_2$-level: | $8$ | ||||
Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
Cusps: | $10$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot8^{4}$ | Cusp orbits | $2^{3}\cdot4$ | ||
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: | 8O0 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.96.0.978 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&14\\4&15\end{bmatrix}$, $\begin{bmatrix}17&10\\16&21\end{bmatrix}$, $\begin{bmatrix}19&5\\4&3\end{bmatrix}$ |
$\GL_2(\Z/24\Z)$-subgroup: | Group 768.1086681 |
Contains $-I$: | no $\quad$ (see 24.48.0.bk.2 for the level structure with $-I$) |
Cyclic 24-isogeny field degree: | $4$ |
Cyclic 24-torsion field degree: | $16$ |
Full 24-torsion field degree: | $768$ |
Models
Smooth plane model Smooth plane model
$ 0 $ | $=$ | $ 2 x^{2} + 12 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 |
---|---|---|---|---|---|
8.48.0-8.ba.2.2 | $8$ | $2$ | $2$ | $0$ | $0$ |
24.48.0-8.ba.2.7 | $24$ | $2$ | $2$ | $0$ | $0$ |
24.48.0-24.bj.1.5 | $24$ | $2$ | $2$ | $0$ | $0$ |
24.48.0-24.bj.1.11 | $24$ | $2$ | $2$ | $0$ | $0$ |
24.48.0-24.bz.1.3 | $24$ | $2$ | $2$ | $0$ | $0$ |
24.48.0-24.bz.1.12 | $24$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
24.288.8-24.gk.1.2 | $24$ | $3$ | $3$ | $8$ |
24.384.7-24.el.1.4 | $24$ | $4$ | $4$ | $7$ |
48.192.1-48.cj.2.2 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.cl.2.2 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.cr.1.8 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.ct.1.8 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.dp.1.1 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.dr.1.1 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.dx.2.7 | $48$ | $2$ | $2$ | $1$ |
48.192.1-48.dz.2.7 | $48$ | $2$ | $2$ | $1$ |
120.480.16-120.es.1.8 | $120$ | $5$ | $5$ | $16$ |
240.192.1-240.mb.1.2 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.md.2.2 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.mj.2.15 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.ml.1.15 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.qz.1.2 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.rb.1.2 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.rh.2.15 | $240$ | $2$ | $2$ | $1$ |
240.192.1-240.rj.2.15 | $240$ | $2$ | $2$ | $1$ |