Invariants
| Level: | $24$ | $\SL_2$-level: | $6$ | Newform level: | $576$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $6^{12}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 6F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.144.1.30 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&0\\18&7\end{bmatrix}$, $\begin{bmatrix}7&15\\12&11\end{bmatrix}$, $\begin{bmatrix}13&21\\0&1\end{bmatrix}$, $\begin{bmatrix}23&12\\6&5\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.72.1.bn.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $512$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.f |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 216 $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(-6:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^3\cdot3^3}\cdot\frac{(y^{2}+648z^{2})^{3}(y^{6}+48600y^{4}z^{2}-18895680y^{2}z^{4}+2448880128z^{6})^{3}}{z^{2}y^{6}(y^{2}-1944z^{2})^{6}(y^{2}-216z^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.72.0-6.a.1.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.ca.1.5 | $24$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.ca.1.13 | $24$ | $3$ | $3$ | $0$ | $0$ | full Jacobian |
| 24.48.1-24.cl.1.3 | $24$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 24.72.0-6.a.1.5 | $24$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.288.5-24.fv.1.5 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.288.5-24.gb.1.3 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.288.5-24.gx.1.6 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.288.5-24.hd.1.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.288.5-24.hz.1.3 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.288.5-24.ih.1.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.288.5-24.jb.1.3 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.288.5-24.jj.1.2 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 72.432.7-72.cp.1.5 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.432.7-72.cp.1.9 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.432.7-72.dj.1.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.432.7-72.ek.1.5 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.432.10-72.bl.1.3 | $72$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 72.432.10-72.bl.1.5 | $72$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 72.432.13-72.dn.1.5 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.288.5-120.csy.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ctc.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.cua.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.cue.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dlc.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dlg.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dme.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dmi.1.3 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.biy.1.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.bjc.1.8 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.bka.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.bke.1.6 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.bts.1.5 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.btw.1.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.buu.1.5 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.5-168.buy.1.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.288.5-264.biy.1.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.288.5-264.bjc.1.8 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.288.5-264.bka.1.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.288.5-264.bke.1.6 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.288.5-264.bts.1.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.288.5-264.btw.1.8 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.288.5-264.buu.1.5 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.288.5-264.buy.1.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.biy.1.5 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.bjc.1.7 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.bka.1.7 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.bke.1.8 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.bts.1.5 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.btw.1.7 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.buu.1.5 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.288.5-312.buy.1.5 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |