Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $576$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8B1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.1.202 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}11&23\\12&13\end{bmatrix}$, $\begin{bmatrix}13&14\\12&17\end{bmatrix}$, $\begin{bmatrix}15&10\\20&15\end{bmatrix}$, $\begin{bmatrix}19&2\\16&15\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.24.1.n.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $64$ |
| Full 24-torsion field degree: | $1536$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.c |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 9x $ |
Rational points
This modular curve has infinitely many rational points, including 2 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^4}{3^2}\cdot\frac{58563x^{2}y^{4}z^{2}-241805655x^{2}z^{6}-414xy^{6}z+53754273xy^{2}z^{5}+y^{8}-3018060y^{4}z^{4}+531441z^{8}}{zy^{4}(9x^{2}z+xy^{2}+81z^{3})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.24.0-4.d.1.2 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.24.0-4.d.1.2 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.96.1-24.dh.1.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1-24.dk.1.5 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1-24.dy.1.3 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1-24.ef.1.4 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1-24.ex.1.4 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1-24.ey.1.3 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1-24.fl.1.2 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1-24.fm.1.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.144.5-24.bz.1.7 | $24$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 24.192.5-24.bf.1.15 | $24$ | $4$ | $4$ | $5$ | $2$ | $1^{4}$ |
| 120.96.1-120.kk.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.ko.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.le.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.lm.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.mk.1.4 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.ms.1.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.nm.1.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.nq.1.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.240.9-120.z.1.11 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.288.9-120.czd.1.16 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 120.480.17-120.sv.1.15 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
| 168.96.1-168.kk.1.7 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.ko.1.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.le.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.lm.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.mk.1.6 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.ms.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.nm.1.6 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.nq.1.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.384.13-168.bd.1.33 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 264.96.1-264.kk.1.8 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.ko.1.8 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.le.1.8 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.lm.1.8 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.mk.1.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.ms.1.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.nm.1.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.nq.1.3 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.kk.1.5 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.ko.1.5 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.le.1.6 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.lm.1.8 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.mk.1.6 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.ms.1.6 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.nm.1.7 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.nq.1.5 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |