Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $288$ | ||
| 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: | $0$ | ||||||
| $\Q$-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: | 24.72.1.362 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&13\\22&23\end{bmatrix}$, $\begin{bmatrix}5&17\\10&5\end{bmatrix}$, $\begin{bmatrix}7&3\\18&5\end{bmatrix}$, $\begin{bmatrix}11&6\\0&5\end{bmatrix}$ |
| 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: | $2^{5}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 288.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x^{2} + 3 y w $ |
| $=$ | $4 y^{2} - 2 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{4} - 2 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{3}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{3}{2}w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{(8z^{6}-12z^{4}w^{2}+6z^{2}w^{4}+3w^{6})^{3}}{w^{6}(2z^{2}-w^{2})^{6}}$ |
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 |
|---|---|---|---|---|---|---|
| 24.36.0.bg.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.0.cd.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.1.gq.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 |
|---|---|---|---|---|---|---|
| 24.144.9.it.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.qx.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.wr.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.wx.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.bkr.1 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{8}$ |
| 24.144.9.bkw.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bkz.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.ble.1 | $24$ | $2$ | $2$ | $9$ | $4$ | $1^{8}$ |
| 72.216.13.md.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.9.bfta.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bftc.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bftq.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfts.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfvm.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfvo.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfwc.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfwe.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbpu.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbpw.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbqk.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbqm.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbsg.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbsi.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbsw.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbsy.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bbvu.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bbvw.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bbwk.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bbwm.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bbyg.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bbyi.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bbyw.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bbyy.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbqc.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbqe.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbqs.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbqu.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbso.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbsq.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbte.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbtg.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |