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: | 24.72.1.368 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&23\\2&19\end{bmatrix}$, $\begin{bmatrix}7&22\\8&7\end{bmatrix}$, $\begin{bmatrix}15&22\\20&15\end{bmatrix}$, $\begin{bmatrix}19&1\\20&1\end{bmatrix}$, $\begin{bmatrix}19&2\\10&13\end{bmatrix}$ |
| 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: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 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} - 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}w$ |
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.cb.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.36.0.cf.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.ry.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.rz.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.sa.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.sb.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.tq.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.tr.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.ts.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.tt.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.vq.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.vr.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.vs.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.vt.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.wg.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.wh.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.wi.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.wj.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.9.fj.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.qu.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.bhj.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bhs.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.btw.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.buc.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bum.1 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{8}$ |
| 24.144.9.bus.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 72.216.13.mx.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.5.kti.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ktj.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ktk.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ktl.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kty.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ktz.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kua.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kub.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kvu.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kvv.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kvw.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kvx.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kwk.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kwl.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kwm.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kwn.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.9.bfzm.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfzo.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgac.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgae.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgby.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgca.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgco.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgcq.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.5.hth.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hti.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.htj.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.htk.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.htx.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hty.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.htz.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hua.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hvt.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hvu.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hvv.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hvw.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hwj.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hwk.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hwl.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hwm.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.9.bbwg.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbwi.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbww.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbwy.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbys.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbyu.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbzi.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbzk.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.5.hti.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.htj.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.htk.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.htl.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hty.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.htz.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hua.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hub.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hvu.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hvv.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hvw.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hvx.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hwk.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hwl.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hwm.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hwn.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.9.bccg.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcci.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bccw.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bccy.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bces.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bceu.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcfi.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcfk.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.5.hti.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.htj.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.htk.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.htl.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hty.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.htz.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hua.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hub.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hvu.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hvv.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hvw.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hvx.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hwk.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hwl.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hwm.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hwn.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.9.bbwo.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbwq.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbxe.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbxg.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbza.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbzc.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbzq.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbzs.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |