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: | 24.72.1.78 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&3\\0&11\end{bmatrix}$, $\begin{bmatrix}11&5\\8&13\end{bmatrix}$, $\begin{bmatrix}11&14\\4&23\end{bmatrix}$, $\begin{bmatrix}15&16\\14&9\end{bmatrix}$, $\begin{bmatrix}23&2\\16&23\end{bmatrix}$ |
| 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: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ y^{2} - y z + z^{2} - w^{2} $ |
| $=$ | $3 x^{2} + y w + z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 3 x^{2} y z + 3 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:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
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.cf.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.sc.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.sd.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.se.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.sf.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.tu.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.tv.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.tw.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.tx.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.vu.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.vv.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.vw.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.vx.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.wk.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5.wl.1 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.144.5.wm.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.5.wn.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.144.9.fi.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.ud.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bhn.1 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{8}$ |
| 24.144.9.bhu.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.btx.1 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{8}$ |
| 24.144.9.bud.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 24.144.9.bun.1 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{8}$ |
| 24.144.9.but.1 | $24$ | $2$ | $2$ | $9$ | $3$ | $1^{8}$ |
| 72.216.13.mw.1 | $72$ | $3$ | $3$ | $13$ | $?$ | not computed |
| 120.144.5.ktm.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ktn.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kto.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.ktp.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kuc.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kud.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kue.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kuf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kvy.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kvz.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kwa.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kwb.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kwo.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kwp.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kwq.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.5.kwr.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.144.9.bfzn.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bfzp.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgad.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgaf.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgbz.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgcb.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgcp.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.144.9.bgcr.1 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.5.htl.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.htm.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.htn.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hto.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hub.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.huc.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hud.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hue.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hvx.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hvy.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hvz.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hwa.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hwn.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hwo.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hwp.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.5.hwq.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.144.9.bbwh.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbwj.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbwx.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbwz.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbyt.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbyv.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbzj.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.144.9.bbzl.1 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.5.htm.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.htn.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hto.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.htp.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.huc.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hud.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hue.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.huf.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hvy.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hvz.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hwa.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hwb.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hwo.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hwp.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hwq.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.5.hwr.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.144.9.bcch.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bccj.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bccx.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bccz.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcet.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcev.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcfj.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.144.9.bcfl.1 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.5.htm.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.htn.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hto.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.htp.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.huc.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hud.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hue.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.huf.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hvy.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hvz.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hwa.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hwb.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hwo.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hwp.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hwq.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.5.hwr.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.144.9.bbwp.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbwr.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbxf.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbxh.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbzb.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbzd.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbzr.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.144.9.bbzt.1 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |