Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $32$ | ||
| 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: | 24.72.1.85 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&11\\10&17\end{bmatrix}$, $\begin{bmatrix}17&10\\4&19\end{bmatrix}$, $\begin{bmatrix}23&4\\20&11\end{bmatrix}$, $\begin{bmatrix}23&13\\14&23\end{bmatrix}$ |
| 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: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x^{2} - z w $ |
| $=$ | $3 y^{2} - 4 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} - 3 y^{2} z^{2} + 9 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(16z^{6}+w^{6})^{3}}{w^{6}z^{12}}$ |
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.cc.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.1.gm.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.gm.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.pg.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.vu.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.wd.1 | $24$ | $2$ | $2$ | $9$ | $5$ | $1^{8}$ |
| 24.144.9.bsl.1 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{8}$ |
| 24.144.9.bsn.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.btb.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.btd.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 48.144.7.ui.1 | $48$ | $2$ | $2$ | $7$ | $3$ | $1^{6}$ |
| 48.144.7.uj.1 | $48$ | $2$ | $2$ | $7$ | $3$ | $1^{6}$ |
| 48.144.7.uk.1 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{6}$ |
| 48.144.7.ul.1 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{6}$ |
| 72.216.13.mr.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.9.bfxw.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfxx.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfym.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfyn.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgai.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgaj.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgay.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgaz.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbuq.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbur.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbvg.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbvh.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbxc.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbxd.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbxs.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbxt.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.144.7.dsg.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.dsh.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.dsi.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.dsj.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 264.144.9.bcaq.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcar.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcbg.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcbh.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcdc.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcdd.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcds.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcdt.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbuy.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbuz.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbvo.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbvp.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbxk.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbxl.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbya.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbyb.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |