Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $32$ | ||
| 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: | 24.72.1.77 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&21\\12&7\end{bmatrix}$, $\begin{bmatrix}7&2\\10&1\end{bmatrix}$, $\begin{bmatrix}13&16\\2&19\end{bmatrix}$, $\begin{bmatrix}17&7\\4&7\end{bmatrix}$, $\begin{bmatrix}19&23\\2&17\end{bmatrix}$ |
| 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: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ y^{2} - y z + z^{2} - w^{2} $ |
| $=$ | $3 x^{2} - 2 y w - 2 z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 3 x^{2} y z + 3 y^{2} z^{2} - 36 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map of degree 72 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 3^3\,\frac{4374yz^{15}w^{2}-8019yz^{13}w^{4}-11664yz^{11}w^{6}+17739yz^{9}w^{8}+6156yz^{7}w^{10}-5022yz^{5}w^{12}+432yz^{3}w^{14}+225yzw^{16}-729z^{18}+10935z^{14}w^{4}-12879z^{12}w^{6}-10692z^{10}w^{8}+15309z^{8}w^{10}-54z^{6}w^{12}-2430z^{4}w^{14}+540z^{2}w^{16}-125w^{18}}{w^{12}(54yz^{3}w^{2}+9yzw^{4}-27z^{6}+27z^{2}w^{4}+w^{6})}$ |
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 |
|---|---|---|---|---|---|---|
| 12.36.0.t.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.0.ce.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.1.go.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.ru.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.rv.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.rw.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.rx.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.tm.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.tn.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.to.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.tp.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.vm.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.vn.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.vo.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.vp.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.wc.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.wd.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.we.1 | $24$ | $2$ | $2$ | $5$ | $3$ | $1^{4}$ |
| 24.144.5.wf.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.9.fg.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.pj.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.bhm.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.bhr.1 | $24$ | $2$ | $2$ | $9$ | $4$ | $1^{8}$ |
| 24.144.9.btz.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.bub.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bup.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.bur.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 72.216.13.mv.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.5.kte.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ktf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ktg.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kth.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ktu.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ktv.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ktw.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ktx.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kvq.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kvr.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kvs.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kvt.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kwg.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kwh.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kwi.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kwj.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.9.bfzk.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfzl.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgaa.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgab.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgbw.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgbx.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgcm.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgcn.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.5.htd.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hte.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.htf.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.htg.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.htt.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.htu.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.htv.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.htw.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hvp.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hvq.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hvr.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hvs.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hwf.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hwg.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hwh.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hwi.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.9.bbwe.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbwf.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbwu.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbwv.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbyq.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbyr.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbzg.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbzh.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.5.hte.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.htf.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.htg.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hth.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.htu.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.htv.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.htw.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.htx.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hvq.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hvr.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hvs.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hvt.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hwg.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hwh.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hwi.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hwj.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.9.bcce.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bccf.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bccu.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bccv.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bceq.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcer.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcfg.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcfh.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.5.hte.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.htf.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.htg.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hth.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.htu.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.htv.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.htw.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.htx.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hvq.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hvr.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hvs.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hvt.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hwg.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hwh.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hwi.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hwj.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.9.bbwm.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbwn.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbxc.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbxd.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbyy.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbyz.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbzo.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbzp.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |