Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $576$ | ||
| Index: | $768$ | $\PSL_2$-index: | $384$ | ||||
| Genus: | $13 = 1 + \frac{ 384 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 40 }{2}$ | ||||||
| Cusps: | $40$ (none of which are rational) | Cusp widths | $4^{16}\cdot8^{4}\cdot12^{16}\cdot24^{4}$ | Cusp orbits | $2^{6}\cdot4^{5}\cdot8$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 8$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24AB13 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.768.13.418 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&6\\12&19\end{bmatrix}$, $\begin{bmatrix}1&14\\0&13\end{bmatrix}$, $\begin{bmatrix}7&8\\0&5\end{bmatrix}$, $\begin{bmatrix}19&18\\12&5\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $D_4\times D_6$ |
| Contains $-I$: | no $\quad$ (see 24.384.13.cf.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $2$ |
| Cyclic 24-torsion field degree: | $8$ |
| Full 24-torsion field degree: | $96$ |
Jacobian
| Conductor: | $2^{63}\cdot3^{17}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}\cdot2^{5}$ |
| Newforms: | 24.2.a.a, 24.2.d.a, 96.2.f.a$^{2}$, 192.2.c.a, 288.2.d.b, 576.2.a.b, 576.2.a.d |
Models
Canonical model in $\mathbb{P}^{ 12 }$ defined by 55 equations
| $ 0 $ | $=$ | $ y d + v r + r s $ |
| $=$ | $x r - w b - u s$ | |
| $=$ | $x d - u v - u s$ | |
| $=$ | $y z - y v - y s - b c + b d$ | |
| $=$ | $\cdots$ |
Rational points
This modular curve has no $\Q_p$ points for $p=43$, and therefore no rational points.
Maps to other modular curves
Map of degree 4 from the canonical model of this modular curve to the canonical model of the modular curve 24.96.3.bo.2 :
| $\displaystyle X$ | $=$ | $\displaystyle u$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -w+t-2u$ |
| $\displaystyle Z$ | $=$ | $\displaystyle x-w+t-2u+2a$ |
Equation of the image curve:
| $0$ | $=$ | $ 6X^{4}-4X^{3}Y+6X^{2}Y^{2}+4XY^{3}-8X^{3}Z-6X^{2}YZ+2Y^{3}Z-3X^{2}Z^{2}-6XYZ^{2}-3Y^{2}Z^{2}+2XZ^{3}+YZ^{3} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.384.5-24.bv.2.12 | $24$ | $2$ | $2$ | $5$ | $1$ | $2^{4}$ |
| 24.384.5-24.bv.2.19 | $24$ | $2$ | $2$ | $5$ | $1$ | $2^{4}$ |
| 24.384.5-24.ck.4.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2^{3}$ |
| 24.384.5-24.ck.4.30 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2^{3}$ |
| 24.384.5-24.cm.1.5 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2^{3}$ |
| 24.384.5-24.cm.1.24 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2^{3}$ |
| 24.384.7-24.bc.2.4 | $24$ | $2$ | $2$ | $7$ | $1$ | $2^{3}$ |
| 24.384.7-24.bc.2.31 | $24$ | $2$ | $2$ | $7$ | $1$ | $2^{3}$ |
| 24.384.7-24.bf.1.16 | $24$ | $2$ | $2$ | $7$ | $1$ | $2^{3}$ |
| 24.384.7-24.bf.1.22 | $24$ | $2$ | $2$ | $7$ | $1$ | $2^{3}$ |
| 24.384.7-24.cf.3.8 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2^{2}$ |
| 24.384.7-24.cf.3.32 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2^{2}$ |
| 24.384.7-24.ch.2.20 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2^{2}$ |
| 24.384.7-24.ch.2.32 | $24$ | $2$ | $2$ | $7$ | $0$ | $1^{2}\cdot2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.1536.33-24.fe.4.3 | $24$ | $2$ | $2$ | $33$ | $2$ | $1^{6}\cdot2^{5}\cdot4$ |
| 24.1536.33-24.gg.3.7 | $24$ | $2$ | $2$ | $33$ | $2$ | $1^{6}\cdot2^{5}\cdot4$ |
| 24.1536.33-24.ha.3.3 | $24$ | $2$ | $2$ | $33$ | $2$ | $1^{6}\cdot2^{5}\cdot4$ |
| 24.1536.33-24.he.4.1 | $24$ | $2$ | $2$ | $33$ | $2$ | $1^{6}\cdot2^{5}\cdot4$ |
| 24.1536.33-24.jj.2.6 | $24$ | $2$ | $2$ | $33$ | $2$ | $1^{6}\cdot2^{5}\cdot4$ |
| 24.1536.33-24.jr.1.8 | $24$ | $2$ | $2$ | $33$ | $2$ | $1^{6}\cdot2^{5}\cdot4$ |
| 24.1536.33-24.kb.1.4 | $24$ | $2$ | $2$ | $33$ | $3$ | $1^{6}\cdot2^{5}\cdot4$ |
| 24.1536.33-24.kd.1.3 | $24$ | $2$ | $2$ | $33$ | $3$ | $1^{6}\cdot2^{5}\cdot4$ |
| 24.2304.57-24.cl.1.10 | $24$ | $3$ | $3$ | $57$ | $3$ | $1^{10}\cdot2^{13}\cdot4^{2}$ |