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: | 24.72.1.352 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&5\\20&19\end{bmatrix}$, $\begin{bmatrix}7&9\\12&13\end{bmatrix}$, $\begin{bmatrix}15&23\\22&9\end{bmatrix}$, $\begin{bmatrix}17&5\\10&23\end{bmatrix}$, $\begin{bmatrix}17&9\\0&23\end{bmatrix}$ |
| 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 $ |
| $=$ | $6 y^{2} + 4 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 6 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{(4z^{3}-w^{3})^{3}(4z^{3}+w^{3})^{3}}{w^{6}z^{12}}$ |
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.cb.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.0.cj.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.bda.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bdb.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bdc.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bdd.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bfi.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bfj.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bfk.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bfl.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bhi.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bhj.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.bhk.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bhl.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bhy.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.bhz.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bia.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.bib.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.9.bb.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.rs.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bhh.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bht.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.eju.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.ejw.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.ekk.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.ekm.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 48.144.3.o.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.144.3.o.2 | $48$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 48.144.3.be.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $2$ |
| 48.144.3.be.2 | $48$ | $2$ | $2$ | $3$ | $2$ | $2$ |
| 48.144.7.bgg.1 | $48$ | $2$ | $2$ | $7$ | $3$ | $1^{6}$ |
| 48.144.7.bgg.2 | $48$ | $2$ | $2$ | $7$ | $3$ | $1^{6}$ |
| 48.144.7.bgi.1 | $48$ | $2$ | $2$ | $7$ | $2$ | $1^{6}$ |
| 48.144.7.bgi.2 | $48$ | $2$ | $2$ | $7$ | $2$ | $1^{6}$ |
| 48.144.7.bgk.1 | $48$ | $2$ | $2$ | $7$ | $4$ | $1^{6}$ |
| 48.144.7.bgk.2 | $48$ | $2$ | $2$ | $7$ | $4$ | $1^{6}$ |
| 48.144.7.bgl.1 | $48$ | $2$ | $2$ | $7$ | $3$ | $1^{6}$ |
| 48.144.7.bgl.2 | $48$ | $2$ | $2$ | $7$ | $3$ | $1^{6}$ |
| 48.144.11.kg.1 | $48$ | $2$ | $2$ | $11$ | $0$ | $2^{3}\cdot4$ |
| 48.144.11.kg.2 | $48$ | $2$ | $2$ | $11$ | $0$ | $2^{3}\cdot4$ |
| 48.144.11.yo.1 | $48$ | $2$ | $2$ | $11$ | $4$ | $2^{3}\cdot4$ |
| 48.144.11.yo.2 | $48$ | $2$ | $2$ | $11$ | $4$ | $2^{3}\cdot4$ |
| 72.216.13.od.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.5.lek.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lel.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lem.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.len.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lfa.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lfb.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lfc.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lfd.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lgw.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lgx.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lgy.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lgz.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lhm.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lhn.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lho.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.lhp.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.9.bgji.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgjk.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgjy.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgka.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bglu.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bglw.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgmk.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgmm.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.5.iej.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iek.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iel.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iem.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iez.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ifa.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ifb.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ifc.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.igv.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.igw.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.igx.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.igy.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ihl.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ihm.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.ihn.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.iho.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.9.bcgc.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcge.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcgs.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcgu.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcio.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bciq.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcje.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bcjg.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 240.144.3.cs.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.144.3.cs.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.144.3.di.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.144.3.di.2 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.144.7.ede.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.ede.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.edf.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.edf.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.edi.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.edi.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.edj.1 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.7.edj.2 | $240$ | $2$ | $2$ | $7$ | $?$ | not computed |
| 240.144.11.cve.1 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.cve.2 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.cwc.1 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 240.144.11.cwc.2 | $240$ | $2$ | $2$ | $11$ | $?$ | not computed |
| 264.144.5.iek.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iel.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iem.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ien.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ifa.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ifb.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ifc.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ifd.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.igw.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.igx.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.igy.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.igz.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ihm.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ihn.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.iho.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.ihp.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.9.bcmc.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcme.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcms.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcmu.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcoo.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcoq.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcpe.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcpg.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.5.iek.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iel.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iem.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ien.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ifa.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ifb.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ifc.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ifd.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.igw.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.igx.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.igy.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.igz.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ihm.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ihn.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.iho.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.ihp.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.9.bcgk.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcgm.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcha.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bchc.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bciw.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bciy.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcjm.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bcjo.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |