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: | 24.72.1.53 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}9&4\\10&21\end{bmatrix}$, $\begin{bmatrix}9&22\\20&3\end{bmatrix}$, $\begin{bmatrix}15&5\\20&21\end{bmatrix}$, $\begin{bmatrix}17&3\\18&7\end{bmatrix}$ |
| 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.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x y + 2 y^{2} - w^{2} $ |
| $=$ | $x^{2} + 2 x y - 2 y^{2} + z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 12 x^{4} - x^{2} y^{2} - 4 x^{2} z^{2} - 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:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
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.d.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.1.fc.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.j.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.be.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.cf.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.ci.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.ht.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.hv.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.id.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.if.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 72.216.9.z.1 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.216.9.bl.1 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.216.9.cj.1 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.144.5.evu.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.evv.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ewb.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ewc.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.exy.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.exz.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.eyf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.eyg.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.chu.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.chv.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cib.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cic.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cjy.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.cjz.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ckf.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ckg.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.chu.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.chv.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cib.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cic.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cjy.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.cjz.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ckf.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ckg.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.chu.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.chv.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cib.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cic.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cjy.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.cjz.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ckf.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ckg.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |