Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $192$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $6^{4}\cdot12^{4}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $8$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12T1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.72.1.51 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&19\\8&15\end{bmatrix}$, $\begin{bmatrix}7&15\\12&17\end{bmatrix}$, $\begin{bmatrix}19&15\\0&1\end{bmatrix}$, $\begin{bmatrix}23&3\\6&23\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $64$ |
| Full 24-torsion field degree: | $1024$ |
Jacobian
| Conductor: | $2^{6}\cdot3$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 192.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 6 x y + 6 y^{2} - w^{2} $ |
| $=$ | $3 x^{2} + 6 x y - 6 y^{2} - z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} + x^{2} y^{2} - 4 x^{2} z^{2} - 3 z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{2}{3}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{(z^{6}+2w^{6})^{3}}{w^{12}z^{6}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.36.0.d.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.0.c.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.1.fi.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.144.5.g.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.bc.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.dn.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.dr.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.iv.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.ix.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.jf.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.jh.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 72.216.9.bf.1 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.216.9.br.1 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.216.9.cp.1 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.144.5.eww.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ewx.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.exd.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.exe.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.eza.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ezb.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ezh.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ezi.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ciw.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cix.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cjd.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cje.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cla.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.clb.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.clh.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cli.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ciw.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cix.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cjd.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cje.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cla.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.clb.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.clh.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cli.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ciw.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cix.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cjd.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cje.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cla.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.clb.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.clh.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cli.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |