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: | 24.72.1.353 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&7\\4&21\end{bmatrix}$, $\begin{bmatrix}5&12\\6&11\end{bmatrix}$, $\begin{bmatrix}9&10\\20&9\end{bmatrix}$, $\begin{bmatrix}11&23\\10&5\end{bmatrix}$, $\begin{bmatrix}13&14\\22&19\end{bmatrix}$ |
| 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 $ | $=$ | $ 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.bm.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.0.cj.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.1.go.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.ww.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.wx.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.wy.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.wz.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.zi.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.zj.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.zk.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.zl.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.zy.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.zz.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.baa.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.bab.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bao.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bap.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.baq.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bar.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.9.bq.1 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{8}$ |
| 24.144.9.sp.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.bch.1 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{8}$ |
| 24.144.9.bcr.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.dhl.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.dhm.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.dht.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.dhu.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 48.144.3.e.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $2$ |
| 48.144.3.e.2 | $48$ | $2$ | $2$ | $3$ | $2$ | $2$ |
| 48.144.3.u.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.144.3.u.2 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.144.7.baa.1 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{6}$ |
| 48.144.7.baa.2 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{6}$ |
| 48.144.7.bad.1 | $48$ | $2$ | $2$ | $7$ | $4$ | $1^{6}$ |
| 48.144.7.bad.2 | $48$ | $2$ | $2$ | $7$ | $4$ | $1^{6}$ |
| 48.144.7.bae.1 | $48$ | $2$ | $2$ | $7$ | $2$ | $1^{6}$ |
| 48.144.7.bae.2 | $48$ | $2$ | $2$ | $7$ | $2$ | $1^{6}$ |
| 48.144.7.baf.1 | $48$ | $2$ | $2$ | $7$ | $5$ | $1^{6}$ |
| 48.144.7.baf.2 | $48$ | $2$ | $2$ | $7$ | $5$ | $1^{6}$ |
| 48.144.11.jw.1 | $48$ | $2$ | $2$ | $11$ | $2$ | $2^{3}\cdot4$ |
| 48.144.11.jw.2 | $48$ | $2$ | $2$ | $11$ | $2$ | $2^{3}\cdot4$ |
| 48.144.11.ru.1 | $48$ | $2$ | $2$ | $11$ | $2$ | $2^{3}\cdot4$ |
| 48.144.11.ru.2 | $48$ | $2$ | $2$ | $11$ | $2$ | $2^{3}\cdot4$ |
| 72.216.13.nr.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.5.kxq.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kxr.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kxs.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kxt.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kyg.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kyh.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kyi.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kyj.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lac.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lad.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lae.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.laf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.las.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lat.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lau.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lav.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.9.bgec.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgee.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bges.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgeu.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bggo.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bggq.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bghe.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bghg.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.5.hxp.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hxq.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hxr.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hxs.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hyf.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hyg.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hyh.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hyi.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iab.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iac.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iad.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iae.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iar.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ias.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iat.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iau.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.9.bcaw.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcay.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcbm.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcbo.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcdi.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcdk.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcdy.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcea.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.144.3.bk.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.144.3.bk.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.144.3.ca.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.144.3.ca.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.144.7.dxi.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.dxi.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.dxj.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.dxj.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.dxm.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.dxm.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.dxn.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.dxn.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.11.ctg.1 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.ctg.2 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.cue.1 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.cue.2 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 264.144.5.hxq.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hxr.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hxs.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hxt.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hyg.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hyh.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hyi.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hyj.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iac.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iad.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iae.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iaf.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ias.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iat.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iau.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iav.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.9.bcgw.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcgy.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bchm.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcho.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcji.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcjk.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcjy.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcka.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.5.hxq.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hxr.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hxs.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hxt.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hyg.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hyh.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hyi.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hyj.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iac.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iad.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iae.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iaf.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ias.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iat.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iau.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iav.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.9.bcbe.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcbg.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcbu.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcbw.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcdq.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcds.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bceg.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcei.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |