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: | 24.72.1.369 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}9&5\\8&3\end{bmatrix}$, $\begin{bmatrix}15&2\\2&21\end{bmatrix}$, $\begin{bmatrix}19&1\\2&1\end{bmatrix}$, $\begin{bmatrix}19&10\\16&23\end{bmatrix}$, $\begin{bmatrix}21&8\\20&21\end{bmatrix}$ |
| 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} + 3 y w $ |
| $=$ | $4 y^{2} + 2 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 3z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{3}{2}w$ |
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.cf.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.1.gs.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.mi.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.mj.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.mk.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.ml.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.og.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.oh.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.oi.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.oj.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.ow.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.ox.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.oy.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.oz.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.pm.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.pn.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.po.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.pp.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.9.hv.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.qz.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bcj.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bcp.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.blx.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.bmc.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.bmf.1 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{8}$ |
| 24.144.9.bmk.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 72.216.13.ml.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.5.kne.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.knf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kng.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.knh.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.knu.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.knv.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.knw.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.knx.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kpq.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kpr.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kps.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kpt.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kqg.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kqh.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kqi.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kqj.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.9.bfug.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfui.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfuw.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfuy.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfws.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfwu.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfxi.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfxk.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.5.hnd.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hne.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hnf.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hng.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hnt.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hnu.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hnv.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hnw.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hpp.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hpq.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hpr.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hps.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hqf.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hqg.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hqh.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hqi.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.9.bbra.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbrc.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbrq.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbrs.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbtm.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbto.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbuc.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbue.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.5.hne.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hnf.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hng.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hnh.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hnu.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hnv.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hnw.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hnx.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hpq.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hpr.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hps.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hpt.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hqg.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hqh.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hqi.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hqj.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.9.bbxa.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bbxc.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bbxq.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bbxs.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bbzm.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bbzo.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcac.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcae.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.5.hne.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hnf.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hng.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hnh.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hnu.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hnv.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hnw.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hnx.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hpq.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hpr.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hps.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hpt.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hqg.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hqh.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hqi.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hqj.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.9.bbri.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbrk.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbry.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbsa.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbtu.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbtw.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbuk.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbum.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |