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: | 24.72.1.370 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&23\\2&9\end{bmatrix}$, $\begin{bmatrix}5&11\\20&19\end{bmatrix}$, $\begin{bmatrix}17&3\\6&23\end{bmatrix}$, $\begin{bmatrix}21&19\\8&3\end{bmatrix}$, $\begin{bmatrix}23&9\\12&13\end{bmatrix}$ |
| 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} + 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.bp.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.my.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.mz.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.na.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.nb.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.nq.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.nr.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.ns.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.nt.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.qc.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.qd.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.qe.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.qf.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.qs.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.qt.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.qu.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.qv.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.9.hr.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.qy.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bfn.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bfw.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.blp.1 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{8}$ |
| 24.144.9.blu.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.bmn.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.bms.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 72.216.13.mh.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.5.kok.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kol.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kom.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kon.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kpa.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kpb.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kpc.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kpd.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kqw.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kqx.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kqy.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kqz.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.krm.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.krn.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kro.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.krp.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.9.bfuo.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfuq.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfve.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfvg.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfxa.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfxc.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfxq.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfxs.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.5.hoj.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hok.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hol.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hom.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hoz.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hpa.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hpb.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hpc.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hqv.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hqw.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hqx.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hqy.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hrl.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hrm.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hrn.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hro.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.9.bbri.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbrk.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbry.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbsa.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbtu.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbtw.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbuk.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbum.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.5.hok.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hol.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hom.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hon.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hpa.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hpb.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hpc.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hpd.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hqw.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hqx.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hqy.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hqz.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hrm.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hrn.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hro.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hrp.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.9.bbxi.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bbxk.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bbxy.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bbya.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bbzu.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bbzw.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcak.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcam.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.5.hok.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hol.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hom.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hon.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hpa.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hpb.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hpc.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hpd.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hqw.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hqx.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hqy.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hqz.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hrm.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hrn.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hro.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hrp.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.9.bbrq.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbrs.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbsg.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbsi.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbuc.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbue.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbus.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbuu.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |