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: | 24.72.1.83 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&16\\4&21\end{bmatrix}$, $\begin{bmatrix}7&19\\2&7\end{bmatrix}$, $\begin{bmatrix}17&11\\10&17\end{bmatrix}$, $\begin{bmatrix}21&5\\22&21\end{bmatrix}$, $\begin{bmatrix}23&14\\8&13\end{bmatrix}$ |
| 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 $ | $=$ | $ 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} + 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
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.cg.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 |
|---|---|---|---|---|---|---|
| 24.144.5.bdq.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 24.144.5.bdq.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 24.144.5.bdr.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 24.144.5.bdr.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 24.144.5.bds.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 24.144.5.bds.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 24.144.5.bdt.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 24.144.5.bdt.2 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 24.144.9.ez.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.nw.1 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{8}$ |
| 24.144.9.vs.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.wb.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.eid.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.eif.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.eit.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.eiv.1 | $24$ | $2$ | $2$ | $9$ | $4$ | $1^{8}$ |
| 48.144.3.j.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.144.3.j.2 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.144.3.z.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.144.3.z.2 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.144.7.beu.1 | $48$ | $2$ | $2$ | $7$ | $2$ | $1^{6}$ |
| 48.144.7.bev.1 | $48$ | $2$ | $2$ | $7$ | $2$ | $1^{6}$ |
| 48.144.7.bfc.1 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{6}$ |
| 48.144.7.bfd.1 | $48$ | $2$ | $2$ | $7$ | $1$ | $1^{6}$ |
| 48.144.7.bfk.1 | $48$ | $2$ | $2$ | $7$ | $3$ | $1^{6}$ |
| 48.144.7.bfl.1 | $48$ | $2$ | $2$ | $7$ | $3$ | $1^{6}$ |
| 48.144.7.bfo.1 | $48$ | $2$ | $2$ | $7$ | $2$ | $1^{6}$ |
| 48.144.7.bfp.1 | $48$ | $2$ | $2$ | $7$ | $4$ | $1^{6}$ |
| 48.144.11.kb.1 | $48$ | $2$ | $2$ | $11$ | $0$ | $2\cdot4^{2}$ |
| 48.144.11.kb.2 | $48$ | $2$ | $2$ | $11$ | $0$ | $2\cdot4^{2}$ |
| 48.144.11.yj.1 | $48$ | $2$ | $2$ | $11$ | $0$ | $2\cdot4^{2}$ |
| 48.144.11.yj.2 | $48$ | $2$ | $2$ | $11$ | $0$ | $2\cdot4^{2}$ |
| 72.216.13.nw.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.5.lck.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lck.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lcl.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lcl.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lcm.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lcm.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lcn.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lcn.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.9.bghr.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bght.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgih.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgij.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgkd.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgkf.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgkt.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgkv.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.5.icj.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.icj.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ick.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ick.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.icl.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.icl.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.icm.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.icm.2 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.9.bcel.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcen.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcfb.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcfd.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcgx.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcgz.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bchn.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bchp.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.144.3.cn.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.144.3.cn.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.144.3.dd.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.144.3.dd.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.144.7.ebs.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.ebt.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.ebw.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.ebx.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.eci.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.ecj.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.ecm.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.ecn.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.11.cuz.1 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.cuz.2 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.cvx.1 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.cvx.2 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 264.144.5.ick.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ick.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.icl.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.icl.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.icm.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.icm.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.icn.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.icn.2 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.9.bckl.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bckn.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bclb.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcld.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcmx.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcmz.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcnn.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcnp.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.5.ick.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ick.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.icl.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.icl.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.icm.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.icm.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.icn.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.icn.2 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.9.bcet.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcev.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcfj.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcfl.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bchf.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bchh.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bchv.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bchx.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |