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: | $4$ (all of which are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8B1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.1.109 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&12\\8&7\end{bmatrix}$, $\begin{bmatrix}9&16\\20&13\end{bmatrix}$, $\begin{bmatrix}13&22\\16&17\end{bmatrix}$, $\begin{bmatrix}21&8\\16&13\end{bmatrix}$, $\begin{bmatrix}23&8\\4&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.24.1.d.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} - 36x $ |
Rational points
This modular curve has infinitely many rational points, including 4 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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{3^4}\cdot\frac{3888x^{2}y^{4}z^{2}+36xy^{6}z+5038848xy^{2}z^{5}+y^{8}+2176782336z^{8}}{z^{2}y^{4}x^{2}}$ |
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.b.1.7 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.24.0-4.b.1.1 | $12$ | $2$ | $2$ | $0$ | $0$ | 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.n.2.15 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1-24.bb.1.5 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1-24.bg.1.2 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1-24.bg.2.6 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1-24.bh.1.6 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1-24.bh.2.4 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1-24.bi.1.5 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1-24.bi.2.2 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1-24.bj.1.3 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1-24.bj.2.4 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1-24.bs.1.15 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1-24.bv.1.7 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.144.5-24.h.1.17 | $24$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 24.192.5-24.h.1.33 | $24$ | $4$ | $4$ | $5$ | $2$ | $1^{4}$ |
| 120.96.1-120.bw.1.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.by.1.12 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.ce.1.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.ce.2.10 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.cf.1.18 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.cf.2.10 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.cg.1.18 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.cg.2.10 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.ch.1.10 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.ch.2.10 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.dc.1.7 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1-120.de.1.8 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.240.9-120.d.1.24 | $120$ | $5$ | $5$ | $9$ | $?$ | not computed |
| 120.288.9-120.it.1.1 | $120$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 120.480.17-120.dv.1.80 | $120$ | $10$ | $10$ | $17$ | $?$ | not computed |
| 168.96.1-168.bw.1.32 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.by.1.26 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.ce.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.ce.2.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.cf.1.20 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.cf.2.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.cg.1.10 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.cg.2.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.ch.1.10 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.ch.2.18 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.dc.1.32 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.de.1.30 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.384.13-168.d.1.36 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |
| 264.96.1-264.bw.1.32 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.by.1.26 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.ce.1.6 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.ce.2.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.cf.1.10 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.cf.2.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.cg.1.10 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.cg.2.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.ch.1.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.ch.2.10 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.dc.1.32 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1-264.de.1.28 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.bw.1.31 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.by.1.25 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.ce.1.4 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.ce.2.4 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.cf.1.20 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.cf.2.8 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.cg.1.18 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.cg.2.4 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.ch.1.10 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.ch.2.10 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.dc.1.31 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1-312.de.1.29 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |