Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $576$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 16 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $12^{2}\cdot24^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $16$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24H1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.72.1.84 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&16\\4&23\end{bmatrix}$, $\begin{bmatrix}5&20\\16&5\end{bmatrix}$, $\begin{bmatrix}17&0\\0&19\end{bmatrix}$, $\begin{bmatrix}19&15\\6&7\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $16$ |
| Cyclic 24-torsion field degree: | $128$ |
| Full 24-torsion field degree: | $1024$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - 2 z w $ |
| $=$ | $3 y^{2} - 4 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 3 y^{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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(16z^{6}+w^{6})^{3}}{w^{6}z^{12}}$ |
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.q.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.0.ch.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.1.gr.1 | $24$ | $2$ | $2$ | $1$ | $1$ | 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 |
|---|---|---|---|---|---|---|
| 24.144.9.fa.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.ui.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.vt.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.wh.1 | $24$ | $2$ | $2$ | $9$ | $6$ | $1^{8}$ |
| 24.144.9.eih.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.eij.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.eix.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.eiz.1 | $24$ | $2$ | $2$ | $9$ | $4$ | $1^{8}$ |
| 72.216.13.ny.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.9.bghv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bghx.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgil.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgin.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgkh.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgkj.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgkx.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgkz.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcep.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcer.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcff.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcfh.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bchb.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bchd.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bchr.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcht.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bckp.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bckr.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bclf.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bclh.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcnb.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcnd.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcnr.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcnt.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcex.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcez.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcfn.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcfp.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bchj.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bchl.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bchz.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcib.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |