Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $4$ are rational) | Cusp widths | $1^{2}\cdot2\cdot4\cdot8^{2}$ | Cusp orbits | $1^{4}\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8I0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.0.744 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&18\\8&5\end{bmatrix}$, $\begin{bmatrix}19&11\\0&17\end{bmatrix}$, $\begin{bmatrix}21&2\\4&11\end{bmatrix}$, $\begin{bmatrix}23&20\\16&23\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.24.0.bb.2 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $1536$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 221 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^2\,\frac{(x-y)^{24}(193x^{8}-2784x^{7}y+11384x^{6}y^{2}+12352x^{5}y^{3}-204840x^{4}y^{4}+529792x^{3}y^{5}-289056x^{2}y^{6}-929024xy^{7}+1262608y^{8})^{3}}{(x-6y)^{2}(x-y)^{28}(x+4y)(3x-8y)(x^{2}+8xy-34y^{2})^{8}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 24.24.0-8.n.1.8 | $24$ | $2$ | $2$ | $0$ | $0$ |
| 24.24.0-8.n.1.9 | $24$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 24.96.0-8.l.1.2 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-8.m.2.4 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-8.n.1.5 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-8.p.1.3 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.bj.1.8 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.bl.2.2 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.bn.1.6 | $24$ | $2$ | $2$ | $0$ |
| 24.96.0-24.bp.2.6 | $24$ | $2$ | $2$ | $0$ |
| 24.144.4-24.gl.2.23 | $24$ | $3$ | $3$ | $4$ |
| 24.192.3-24.gi.2.23 | $24$ | $4$ | $4$ | $3$ |
| 48.96.0-16.v.2.2 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-16.x.2.2 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-16.z.2.2 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-16.bb.2.1 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-48.bf.2.9 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-48.bh.2.13 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-48.bn.1.9 | $48$ | $2$ | $2$ | $0$ |
| 48.96.0-48.bp.2.1 | $48$ | $2$ | $2$ | $0$ |
| 48.96.1-16.r.2.8 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-16.t.2.7 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-16.v.2.11 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-16.x.2.8 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.br.2.16 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.bt.1.8 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.bz.2.4 | $48$ | $2$ | $2$ | $1$ |
| 48.96.1-48.cb.2.8 | $48$ | $2$ | $2$ | $1$ |
| 120.96.0-40.bj.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-40.bl.2.6 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-40.bn.1.1 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-40.bp.2.5 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.eg.1.10 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.ek.2.3 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.eo.1.13 | $120$ | $2$ | $2$ | $0$ |
| 120.96.0-120.es.2.11 | $120$ | $2$ | $2$ | $0$ |
| 120.240.8-40.dd.2.1 | $120$ | $5$ | $5$ | $8$ |
| 120.288.7-40.fs.2.1 | $120$ | $6$ | $6$ | $7$ |
| 120.480.15-40.gx.1.9 | $120$ | $10$ | $10$ | $15$ |
| 168.96.0-56.bh.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.bj.2.5 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.bl.1.2 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-56.bn.2.5 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.ec.1.14 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.eg.1.14 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.ek.1.16 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.eo.1.11 | $168$ | $2$ | $2$ | $0$ |
| 168.384.11-56.fe.1.32 | $168$ | $8$ | $8$ | $11$ |
| 240.96.0-80.bn.2.2 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-80.bp.2.1 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-80.bv.1.2 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-80.bx.2.4 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.cp.2.10 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.cr.2.12 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.df.2.10 | $240$ | $2$ | $2$ | $0$ |
| 240.96.0-240.dh.2.9 | $240$ | $2$ | $2$ | $0$ |
| 240.96.1-80.bt.2.13 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-80.bv.1.15 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-80.cb.2.16 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-80.cd.2.15 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.fp.2.24 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.fr.2.23 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.gf.2.21 | $240$ | $2$ | $2$ | $1$ |
| 240.96.1-240.gh.2.23 | $240$ | $2$ | $2$ | $1$ |
| 264.96.0-88.bh.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.bj.2.6 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.bl.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-88.bn.2.6 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.ec.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.eg.2.6 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.ek.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.96.0-264.eo.2.6 | $264$ | $2$ | $2$ | $0$ |
| 312.96.0-104.bj.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-104.bl.2.3 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-104.bn.1.1 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-104.bp.2.7 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.eg.1.14 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.ek.1.15 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.eo.1.16 | $312$ | $2$ | $2$ | $0$ |
| 312.96.0-312.es.1.11 | $312$ | $2$ | $2$ | $0$ |