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: | 24.72.1.363 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&7\\4&7\end{bmatrix}$, $\begin{bmatrix}7&19\\10&17\end{bmatrix}$, $\begin{bmatrix}9&1\\14&3\end{bmatrix}$, $\begin{bmatrix}15&5\\2&21\end{bmatrix}$, $\begin{bmatrix}23&13\\8&5\end{bmatrix}$ |
| 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 $ | $=$ | $ 6 x^{2} - z w $ |
| $=$ | $6 y^{2} - 4 z^{2} + w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 6 y^{2} z^{2} - 9 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{(4z^{3}-w^{3})^{3}(4z^{3}+w^{3})^{3}}{w^{6}z^{12}}$ |
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 |
|---|---|---|---|---|---|---|
| 24.36.0.by.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.0.ce.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.1.gp.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.ra.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.rb.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.rc.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.rd.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.so.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.sp.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.sq.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.sr.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.ty.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.tz.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.ua.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.ub.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.uo.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.up.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.uq.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.ur.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.9.fx.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.of.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bak.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bao.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.btq.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bts.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.bug.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bui.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 72.216.13.my.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.5.kru.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.krv.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.krw.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.krx.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ksk.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ksl.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ksm.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ksn.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kug.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kuh.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kui.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kuj.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kuw.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kux.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kuy.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kuz.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.9.bfza.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfzb.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfzq.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfzr.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgbm.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgbn.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgcc.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgcd.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.5.hrt.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hru.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hrv.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hrw.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hsj.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hsk.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hsl.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hsm.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.huf.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hug.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.huh.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hui.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.huv.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.huw.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hux.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.huy.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.9.bbvu.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbvv.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbwk.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbwl.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbyg.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbyh.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbyw.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbyx.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.5.hru.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hrv.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hrw.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hrx.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hsk.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hsl.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hsm.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hsn.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hug.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.huh.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hui.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.huj.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.huw.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hux.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.huy.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.huz.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.9.bcbu.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcbv.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcck.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bccl.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bceg.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bceh.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcew.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcex.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.5.hru.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hrv.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hrw.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hrx.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hsk.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hsl.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hsm.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hsn.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hug.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.huh.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hui.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.huj.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.huw.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hux.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.huy.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.huz.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.9.bbwc.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbwd.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbws.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbwt.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbyo.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbyp.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbze.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbzf.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |