Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $64$ | ||
| 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: | 24.72.1.342 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&10\\20&3\end{bmatrix}$, $\begin{bmatrix}5&16\\16&1\end{bmatrix}$, $\begin{bmatrix}5&17\\14&1\end{bmatrix}$, $\begin{bmatrix}5&18\\0&19\end{bmatrix}$, $\begin{bmatrix}17&16\\8&5\end{bmatrix}$ |
| 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: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - y w $ |
| $=$ | $4 y^{2} - 2 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 2 y^{2} z^{2} + 4 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
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.cg.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.1.gn.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.xu.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 24.144.5.xu.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 24.144.5.xv.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 24.144.5.xv.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 24.144.5.xw.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 24.144.5.xw.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 24.144.5.xx.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 24.144.5.xx.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 24.144.9.eu.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.mv.1 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{8}$ |
| 24.144.9.wo.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.ws.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.dgb.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.dgc.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.dgj.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.dgk.1 | $24$ | $2$ | $2$ | $9$ | $4$ | $1^{8}$ |
| 48.144.3.a.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.144.3.a.2 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.144.3.q.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.144.3.q.2 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.144.7.yu.1 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{6}$ |
| 48.144.7.yv.1 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{6}$ |
| 48.144.7.zg.1 | $48$ | $2$ | $2$ | $7$ | $2$ | $1^{6}$ |
| 48.144.7.zh.1 | $48$ | $2$ | $2$ | $7$ | $2$ | $1^{6}$ |
| 48.144.7.zk.1 | $48$ | $2$ | $2$ | $7$ | $3$ | $1^{6}$ |
| 48.144.7.zl.1 | $48$ | $2$ | $2$ | $7$ | $3$ | $1^{6}$ |
| 48.144.7.zo.1 | $48$ | $2$ | $2$ | $7$ | $2$ | $1^{6}$ |
| 48.144.7.zp.1 | $48$ | $2$ | $2$ | $7$ | $4$ | $1^{6}$ |
| 48.144.11.js.1 | $48$ | $2$ | $2$ | $11$ | $0$ | $2\cdot4^{2}$ |
| 48.144.11.js.2 | $48$ | $2$ | $2$ | $11$ | $0$ | $2\cdot4^{2}$ |
| 48.144.11.rq.1 | $48$ | $2$ | $2$ | $11$ | $0$ | $2\cdot4^{2}$ |
| 48.144.11.rq.2 | $48$ | $2$ | $2$ | $11$ | $0$ | $2\cdot4^{2}$ |
| 72.216.13.nh.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.5.kws.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kws.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kwt.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kwt.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kwu.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kwu.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kwv.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kwv.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.9.bgcs.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgcu.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgdi.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgdk.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgfe.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgfg.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgfu.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgfw.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.5.hwr.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hwr.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hws.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hws.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hwt.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hwt.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hwu.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hwu.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.9.bbzm.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbzo.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcac.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcae.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcby.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcca.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcco.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bccq.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.144.3.bg.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.144.3.bg.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.144.3.bw.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.144.3.bw.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.144.7.dwc.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.dwd.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.dwg.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.dwh.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.dws.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.dwt.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.dww.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.dwx.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.11.ctc.1 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.ctc.2 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.cua.1 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.cua.2 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 264.144.5.hws.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hws.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hwt.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hwt.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hwu.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hwu.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hwv.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hwv.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.9.bcfm.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcfo.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcgc.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcge.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bchy.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcia.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcio.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bciq.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.5.hws.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hws.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hwt.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hwt.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hwu.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hwu.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hwv.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hwv.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.9.bbzu.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbzw.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcak.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcam.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bccg.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcci.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bccw.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bccy.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |