Invariants
| Level: | $24$ | $\SL_2$-level: | $6$ | ||||
| Index: | $6$ | $\PSL_2$-index: | $6$ | ||||
| Genus: | $0 = 1 + \frac{ 6 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 1 }{2}$ | ||||||
| Cusps: | $1$ (which is rational) | Cusp widths | $6$ | Cusp orbits | $1$ | ||
| Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4,-8$) |
Other labels
| Cummins and Pauli (CP) label: | 6B0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.6.0.12 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&5\\23&14\end{bmatrix}$, $\begin{bmatrix}8&21\\9&16\end{bmatrix}$, $\begin{bmatrix}13&10\\10&5\end{bmatrix}$, $\begin{bmatrix}19&15\\0&7\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $48$ |
| Cyclic 24-torsion field degree: | $384$ |
| Full 24-torsion field degree: | $12288$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 8 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 6 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{x^{6}(3x^{2}+2y^{2})^{3}}{x^{12}}$ |
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 |
|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 24.12.1.p.1 | $24$ | $2$ | $2$ | $1$ |
| 24.12.1.q.1 | $24$ | $2$ | $2$ | $1$ |
| 24.12.1.s.1 | $24$ | $2$ | $2$ | $1$ |
| 24.12.1.t.1 | $24$ | $2$ | $2$ | $1$ |
| 24.12.1.bb.1 | $24$ | $2$ | $2$ | $1$ |
| 24.12.1.bc.1 | $24$ | $2$ | $2$ | $1$ |
| 24.12.1.be.1 | $24$ | $2$ | $2$ | $1$ |
| 24.12.1.bf.1 | $24$ | $2$ | $2$ | $1$ |
| 24.18.0.g.1 | $24$ | $3$ | $3$ | $0$ |
| 24.24.0.fe.1 | $24$ | $4$ | $4$ | $0$ |
| 72.18.1.b.1 | $72$ | $3$ | $3$ | $1$ |
| 72.54.1.a.1 | $72$ | $9$ | $9$ | $1$ |
| 120.12.1.ea.1 | $120$ | $2$ | $2$ | $1$ |
| 120.12.1.eb.1 | $120$ | $2$ | $2$ | $1$ |
| 120.12.1.eg.1 | $120$ | $2$ | $2$ | $1$ |
| 120.12.1.eh.1 | $120$ | $2$ | $2$ | $1$ |
| 120.12.1.em.1 | $120$ | $2$ | $2$ | $1$ |
| 120.12.1.en.1 | $120$ | $2$ | $2$ | $1$ |
| 120.12.1.es.1 | $120$ | $2$ | $2$ | $1$ |
| 120.12.1.et.1 | $120$ | $2$ | $2$ | $1$ |
| 120.30.2.bc.1 | $120$ | $5$ | $5$ | $2$ |
| 120.36.1.sm.1 | $120$ | $6$ | $6$ | $1$ |
| 120.60.3.gm.1 | $120$ | $10$ | $10$ | $3$ |
| 168.12.1.dw.1 | $168$ | $2$ | $2$ | $1$ |
| 168.12.1.dx.1 | $168$ | $2$ | $2$ | $1$ |
| 168.12.1.ec.1 | $168$ | $2$ | $2$ | $1$ |
| 168.12.1.ed.1 | $168$ | $2$ | $2$ | $1$ |
| 168.12.1.ei.1 | $168$ | $2$ | $2$ | $1$ |
| 168.12.1.ej.1 | $168$ | $2$ | $2$ | $1$ |
| 168.12.1.eo.1 | $168$ | $2$ | $2$ | $1$ |
| 168.12.1.ep.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.4.m.1 | $168$ | $8$ | $8$ | $4$ |
| 168.126.5.q.1 | $168$ | $21$ | $21$ | $5$ |
| 168.168.9.ea.1 | $168$ | $28$ | $28$ | $9$ |
| 264.12.1.dw.1 | $264$ | $2$ | $2$ | $1$ |
| 264.12.1.dx.1 | $264$ | $2$ | $2$ | $1$ |
| 264.12.1.ec.1 | $264$ | $2$ | $2$ | $1$ |
| 264.12.1.ed.1 | $264$ | $2$ | $2$ | $1$ |
| 264.12.1.ei.1 | $264$ | $2$ | $2$ | $1$ |
| 264.12.1.ej.1 | $264$ | $2$ | $2$ | $1$ |
| 264.12.1.eo.1 | $264$ | $2$ | $2$ | $1$ |
| 264.12.1.ep.1 | $264$ | $2$ | $2$ | $1$ |
| 264.72.6.m.1 | $264$ | $12$ | $12$ | $6$ |
| 264.330.19.e.1 | $264$ | $55$ | $55$ | $19$ |
| 264.330.23.i.1 | $264$ | $55$ | $55$ | $23$ |
| 312.12.1.dw.1 | $312$ | $2$ | $2$ | $1$ |
| 312.12.1.dx.1 | $312$ | $2$ | $2$ | $1$ |
| 312.12.1.ec.1 | $312$ | $2$ | $2$ | $1$ |
| 312.12.1.ed.1 | $312$ | $2$ | $2$ | $1$ |
| 312.12.1.ei.1 | $312$ | $2$ | $2$ | $1$ |
| 312.12.1.ej.1 | $312$ | $2$ | $2$ | $1$ |
| 312.12.1.eo.1 | $312$ | $2$ | $2$ | $1$ |
| 312.12.1.ep.1 | $312$ | $2$ | $2$ | $1$ |
| 312.84.5.bg.1 | $312$ | $14$ | $14$ | $5$ |