Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | ||||
| Index: | $24$ | $\PSL_2$-index: | $12$ | ||||
| Genus: | $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $1\cdot2\cdot3\cdot6$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3,-12$) |
Other labels
| Cummins and Pauli (CP) label: | 6F0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.24.0.98 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&15\\18&5\end{bmatrix}$, $\begin{bmatrix}7&10\\12&23\end{bmatrix}$, $\begin{bmatrix}17&19\\0&17\end{bmatrix}$, $\begin{bmatrix}23&0\\18&7\end{bmatrix}$, $\begin{bmatrix}23&15\\0&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 6.12.0.a.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $3072$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 9048 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6}\cdot\frac{x^{12}(x+2y)^{3}(x^{3}+6x^{2}y-84xy^{2}-568y^{3})^{3}}{y^{6}x^{12}(x-10y)(x+6y)^{3}(x+8y)^{2}}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 24.48.0-6.a.1.11 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-6.b.1.3 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-12.d.1.10 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-12.f.1.6 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-12.g.1.24 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-12.h.1.4 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-12.i.1.4 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-12.j.1.8 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.p.1.16 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.y.1.6 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.bw.1.2 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.bx.1.12 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.ca.1.8 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.cb.1.4 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.cc.1.4 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.cd.1.12 | $24$ | $2$ | $2$ | $0$ |
| 24.48.1-12.i.1.1 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1-12.j.1.5 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1-12.k.1.9 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1-12.l.1.27 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1-24.eq.1.4 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1-24.er.1.8 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1-24.es.1.6 | $24$ | $2$ | $2$ | $1$ |
| 24.48.1-24.et.1.16 | $24$ | $2$ | $2$ | $1$ |
| 24.72.0-6.a.1.7 | $24$ | $3$ | $3$ | $0$ |
| 72.72.0-18.a.1.11 | $72$ | $3$ | $3$ | $0$ |
| 72.72.2-18.c.1.5 | $72$ | $3$ | $3$ | $2$ |
| 72.72.2-18.d.1.16 | $72$ | $3$ | $3$ | $2$ |
| 120.48.0-30.a.1.4 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-30.b.1.8 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-60.o.1.9 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-60.p.1.13 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-60.q.1.13 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-60.r.1.13 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-60.s.1.14 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-60.t.1.14 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.fm.1.32 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.fn.1.14 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.fo.1.2 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.fp.1.20 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.fq.1.20 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.fr.1.6 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.fs.1.17 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ft.1.21 | $120$ | $2$ | $2$ | $0$ |
| 120.48.1-60.v.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-60.w.1.1 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-60.x.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-60.y.1.2 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.iu.1.12 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.iv.1.24 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.iw.1.6 | $120$ | $2$ | $2$ | $1$ |
| 120.48.1-120.ix.1.28 | $120$ | $2$ | $2$ | $1$ |
| 120.120.4-30.b.1.2 | $120$ | $5$ | $5$ | $4$ |
| 120.144.3-30.a.1.11 | $120$ | $6$ | $6$ | $3$ |
| 120.240.7-30.h.1.17 | $120$ | $10$ | $10$ | $7$ |
| 168.48.0-42.b.1.12 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-42.c.1.9 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-84.m.1.10 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-84.n.1.10 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-84.o.1.14 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-84.p.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-84.q.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-84.r.1.4 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.fg.1.32 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.fh.1.20 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.fi.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.fj.1.23 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.fk.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.fl.1.12 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.fm.1.29 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.fn.1.21 | $168$ | $2$ | $2$ | $0$ |
| 168.48.1-84.m.1.13 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-84.n.1.9 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-84.o.1.3 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-84.p.1.7 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-168.hk.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-168.hl.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-168.hm.1.18 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1-168.hn.1.30 | $168$ | $2$ | $2$ | $1$ |
| 168.192.5-42.a.1.25 | $168$ | $8$ | $8$ | $5$ |
| 168.504.16-42.a.1.51 | $168$ | $21$ | $21$ | $16$ |
| 264.48.0-66.a.1.12 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-66.b.1.8 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-132.m.1.14 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-132.n.1.16 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-132.o.1.14 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-132.p.1.6 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-132.q.1.4 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-132.r.1.2 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.fg.1.32 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.fh.1.22 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.fi.1.17 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.fj.1.27 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.fk.1.26 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.fl.1.20 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.fm.1.17 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.fn.1.11 | $264$ | $2$ | $2$ | $0$ |
| 264.48.1-132.m.1.15 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-132.n.1.13 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-132.o.1.3 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-132.p.1.1 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-264.hk.1.26 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-264.hl.1.30 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-264.hm.1.6 | $264$ | $2$ | $2$ | $1$ |
| 264.48.1-264.hn.1.24 | $264$ | $2$ | $2$ | $1$ |
| 264.288.9-66.a.1.46 | $264$ | $12$ | $12$ | $9$ |
| 312.48.0-78.b.1.4 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-78.c.1.7 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-156.o.1.6 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-156.p.1.6 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-156.q.1.4 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-156.r.1.4 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-156.s.1.2 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-156.t.1.8 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.fm.1.32 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.fn.1.8 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.fo.1.19 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.fp.1.11 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.fq.1.6 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.fr.1.26 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.fs.1.21 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ft.1.19 | $312$ | $2$ | $2$ | $0$ |
| 312.48.1-156.m.1.10 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-156.n.1.16 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-156.o.1.14 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-156.p.1.12 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-312.hk.1.19 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-312.hl.1.11 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-312.hm.1.3 | $312$ | $2$ | $2$ | $1$ |
| 312.48.1-312.hn.1.29 | $312$ | $2$ | $2$ | $1$ |
| 312.336.11-78.a.1.43 | $312$ | $14$ | $14$ | $11$ |