Invariants
| Level: | $24$ | $\SL_2$-level: | $6$ | Newform level: | $576$ | ||
| Index: | $72$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $6^{12}$ | Cusp orbits | $2^{2}\cdot4^{2}$ | ||
| Elliptic points: | $0$ 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: | 6F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.72.1.55 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&3\\18&13\end{bmatrix}$, $\begin{bmatrix}9&10\\16&21\end{bmatrix}$, $\begin{bmatrix}17&9\\12&7\end{bmatrix}$, $\begin{bmatrix}17&18\\12&7\end{bmatrix}$ |
| 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: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.f |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 y^{2} + 2 y w - z^{2} $ |
| $=$ | $3 x^{2} - 2 y^{2} + 2 y w - z^{2} + 3 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} - 3 x^{2} y^{2} + 12 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:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^3\,\frac{98304yz^{14}w^{3}+86016yz^{12}w^{5}-147456yz^{10}w^{7}-192512yz^{8}w^{9}+36608yz^{6}w^{11}+167424yz^{4}w^{13}+98432yz^{2}w^{15}+19684yw^{17}-32768z^{18}+122880z^{14}w^{4}+33792z^{12}w^{6}-239616z^{10}w^{8}-199936z^{8}w^{10}+100000z^{6}w^{12}+211584z^{4}w^{14}+108256z^{2}w^{16}+19683w^{18}}{w^{6}z^{6}(1536yz^{4}w+2176yz^{2}w^{3}+728yw^{5}+512z^{6}+2496z^{4}w^{2}+2552z^{2}w^{4}+729w^{6})}$ |
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.a.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.1.co.1 | $24$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 24.36.0.e.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.1.cy.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.fw.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.gc.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.gy.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.he.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.ia.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.ii.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.jc.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.jk.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 72.216.7.dm.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.216.7.eh.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 72.216.13.do.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.5.csz.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ctd.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.cub.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.cuf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dld.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dlh.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dmf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.dmj.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.biz.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.bjd.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.bkb.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.bkf.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.btt.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.btx.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.buv.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.buz.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.biz.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.bjd.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.bkb.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.bkf.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.btt.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.btx.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.buv.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.buz.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.biz.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.bjd.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.bkb.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.bkf.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.btt.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.btx.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.buv.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.buz.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |