Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | ||||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $0 = 1 + \frac{ 36 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $6^{2}\cdot12^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $8$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1 \le \gamma \le 2$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12H0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.0.53 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}9&22\\22&7\end{bmatrix}$, $\begin{bmatrix}11&0\\14&13\end{bmatrix}$, $\begin{bmatrix}23&0\\0&5\end{bmatrix}$, $\begin{bmatrix}23&13\\16&9\end{bmatrix}$ |
| 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: | $2048$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.
Maps to other modular curves
$j$-invariant map of degree 36 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^6}{3^3}\cdot\frac{(2x-y)^{36}(4992192x^{12}-20155392x^{11}y+38677824x^{10}y^{2}-46656000x^{9}y^{3}+39404880x^{8}y^{4}-24509952x^{7}y^{5}+11488608x^{6}y^{6}-4084992x^{5}y^{7}+1094580x^{4}y^{8}-216000x^{3}y^{9}+29844x^{2}y^{10}-2592xy^{11}+107y^{12})^{3}}{(2x-y)^{36}(6x^{2}-y^{2})^{12}(6x^{2}-4xy+y^{2})^{6}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 12.18.0.i.1 | $12$ | $2$ | $2$ | $0$ | $0$ |
| 24.18.0.e.1 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.18.0.l.1 | $24$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 24.72.3.q.1 | $24$ | $2$ | $2$ | $3$ |
| 24.72.3.dg.1 | $24$ | $2$ | $2$ | $3$ |
| 24.72.3.go.1 | $24$ | $2$ | $2$ | $3$ |
| 24.72.3.gt.1 | $24$ | $2$ | $2$ | $3$ |
| 24.72.3.mb.1 | $24$ | $2$ | $2$ | $3$ |
| 24.72.3.mg.1 | $24$ | $2$ | $2$ | $3$ |
| 24.72.3.mp.1 | $24$ | $2$ | $2$ | $3$ |
| 24.72.3.mu.1 | $24$ | $2$ | $2$ | $3$ |
| 72.108.6.z.1 | $72$ | $3$ | $3$ | $6$ |
| 72.324.16.bh.1 | $72$ | $9$ | $9$ | $16$ |
| 120.72.3.ffy.1 | $120$ | $2$ | $2$ | $3$ |
| 120.72.3.ffz.1 | $120$ | $2$ | $2$ | $3$ |
| 120.72.3.fgf.1 | $120$ | $2$ | $2$ | $3$ |
| 120.72.3.fgg.1 | $120$ | $2$ | $2$ | $3$ |
| 120.72.3.fic.1 | $120$ | $2$ | $2$ | $3$ |
| 120.72.3.fid.1 | $120$ | $2$ | $2$ | $3$ |
| 120.72.3.fij.1 | $120$ | $2$ | $2$ | $3$ |
| 120.72.3.fik.1 | $120$ | $2$ | $2$ | $3$ |
| 120.180.12.ir.1 | $120$ | $5$ | $5$ | $12$ |
| 120.216.11.bcv.1 | $120$ | $6$ | $6$ | $11$ |
| 120.360.23.dhd.1 | $120$ | $10$ | $10$ | $23$ |
| 168.72.3.eug.1 | $168$ | $2$ | $2$ | $3$ |
| 168.72.3.euh.1 | $168$ | $2$ | $2$ | $3$ |
| 168.72.3.eun.1 | $168$ | $2$ | $2$ | $3$ |
| 168.72.3.euo.1 | $168$ | $2$ | $2$ | $3$ |
| 168.72.3.ewk.1 | $168$ | $2$ | $2$ | $3$ |
| 168.72.3.ewl.1 | $168$ | $2$ | $2$ | $3$ |
| 168.72.3.ewr.1 | $168$ | $2$ | $2$ | $3$ |
| 168.72.3.ews.1 | $168$ | $2$ | $2$ | $3$ |
| 168.288.21.bld.1 | $168$ | $8$ | $8$ | $21$ |
| 264.72.3.eug.1 | $264$ | $2$ | $2$ | $3$ |
| 264.72.3.euh.1 | $264$ | $2$ | $2$ | $3$ |
| 264.72.3.eun.1 | $264$ | $2$ | $2$ | $3$ |
| 264.72.3.euo.1 | $264$ | $2$ | $2$ | $3$ |
| 264.72.3.ewk.1 | $264$ | $2$ | $2$ | $3$ |
| 264.72.3.ewl.1 | $264$ | $2$ | $2$ | $3$ |
| 264.72.3.ewr.1 | $264$ | $2$ | $2$ | $3$ |
| 264.72.3.ews.1 | $264$ | $2$ | $2$ | $3$ |
| 312.72.3.eug.1 | $312$ | $2$ | $2$ | $3$ |
| 312.72.3.euh.1 | $312$ | $2$ | $2$ | $3$ |
| 312.72.3.eun.1 | $312$ | $2$ | $2$ | $3$ |
| 312.72.3.euo.1 | $312$ | $2$ | $2$ | $3$ |
| 312.72.3.ewk.1 | $312$ | $2$ | $2$ | $3$ |
| 312.72.3.ewl.1 | $312$ | $2$ | $2$ | $3$ |
| 312.72.3.ewr.1 | $312$ | $2$ | $2$ | $3$ |
| 312.72.3.ews.1 | $312$ | $2$ | $2$ | $3$ |