Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | ||||
| 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 | $1^{2}\cdot2\cdot8$ | 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: | none |
Other labels
| Cummins and Pauli (CP) label: | 8C0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.24.0.145 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&0\\12&23\end{bmatrix}$, $\begin{bmatrix}3&1\\16&1\end{bmatrix}$, $\begin{bmatrix}3&8\\20&5\end{bmatrix}$, $\begin{bmatrix}7&16\\16&15\end{bmatrix}$, $\begin{bmatrix}9&17\\4&13\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 8.12.0.n.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 5199 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 12 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{x^{12}(x^{4}-16x^{2}y^{2}+16y^{4})^{3}}{y^{8}x^{14}(x-4y)(x+4y)}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 24.48.0-8.i.1.2 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-8.k.1.3 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-8.q.1.2 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-8.r.1.3 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-8.ba.1.3 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-8.ba.1.6 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-8.ba.2.3 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-8.ba.2.6 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-8.bb.1.3 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-8.bb.1.6 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-8.bb.2.3 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-8.bb.2.6 | $24$ | $2$ | $2$ | $0$ |
| 48.48.0-16.e.1.8 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-16.e.1.9 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-16.e.2.1 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-16.e.2.16 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-16.f.1.8 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-16.f.1.9 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-16.f.2.1 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-16.f.2.16 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-16.g.1.8 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-16.g.1.9 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-16.h.1.8 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-16.h.1.9 | $48$ | $2$ | $2$ | $0$ |
| 48.48.1-16.a.1.8 | $48$ | $2$ | $2$ | $1$ |
| 48.48.1-16.a.1.9 | $48$ | $2$ | $2$ | $1$ |
| 48.48.1-16.b.1.8 | $48$ | $2$ | $2$ | $1$ |
| 48.48.1-16.b.1.9 | $48$ | $2$ | $2$ | $1$ |
| 24.48.0-24.bh.1.2 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.bj.1.10 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.bl.1.6 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.bn.1.12 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.by.1.3 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.by.1.14 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.by.2.8 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.by.2.9 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.bz.1.1 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.bz.1.16 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.bz.2.6 | $24$ | $2$ | $2$ | $0$ |
| 24.48.0-24.bz.2.11 | $24$ | $2$ | $2$ | $0$ |
| 24.72.2-24.cj.1.18 | $24$ | $3$ | $3$ | $2$ |
| 24.96.1-24.ir.1.22 | $24$ | $4$ | $4$ | $1$ |
| 120.48.0-40.bj.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.bl.1.9 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.bn.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.bp.1.9 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.ca.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.ca.1.10 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.ca.2.3 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.ca.2.14 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.cb.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.cb.1.14 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.cb.2.7 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-40.cb.2.10 | $120$ | $2$ | $2$ | $0$ |
| 120.120.4-40.bl.1.16 | $120$ | $5$ | $5$ | $4$ |
| 120.144.3-40.bx.1.7 | $120$ | $6$ | $6$ | $3$ |
| 120.240.7-40.cj.1.5 | $120$ | $10$ | $10$ | $7$ |
| 48.48.0-48.e.1.5 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-48.e.1.28 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-48.e.2.9 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-48.e.2.24 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-48.f.1.1 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-48.f.1.32 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-48.f.2.13 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-48.f.2.20 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-48.g.1.16 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-48.g.1.17 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-48.h.1.1 | $48$ | $2$ | $2$ | $0$ |
| 48.48.0-48.h.1.32 | $48$ | $2$ | $2$ | $0$ |
| 48.48.1-48.a.1.1 | $48$ | $2$ | $2$ | $1$ |
| 48.48.1-48.a.1.32 | $48$ | $2$ | $2$ | $1$ |
| 48.48.1-48.b.1.16 | $48$ | $2$ | $2$ | $1$ |
| 48.48.1-48.b.1.17 | $48$ | $2$ | $2$ | $1$ |
| 168.48.0-56.bf.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bh.1.9 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bj.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bl.1.11 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bu.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bu.1.10 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bu.2.8 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bu.2.9 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bv.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bv.1.9 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bv.2.6 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-56.bv.2.11 | $168$ | $2$ | $2$ | $0$ |
| 168.192.5-56.bl.1.21 | $168$ | $8$ | $8$ | $5$ |
| 168.504.16-56.cj.1.21 | $168$ | $21$ | $21$ | $16$ |
| 240.48.0-80.m.1.11 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-80.m.1.22 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-80.m.2.12 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-80.m.2.21 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-80.n.1.12 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-80.n.1.21 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-80.n.2.10 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-80.n.2.23 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-80.o.1.13 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-80.o.1.20 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-80.p.1.11 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-80.p.1.22 | $240$ | $2$ | $2$ | $0$ |
| 240.48.1-80.a.1.11 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-80.a.1.22 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-80.b.1.13 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-80.b.1.20 | $240$ | $2$ | $2$ | $1$ |
| 264.48.0-88.bf.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-88.bh.1.10 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-88.bj.1.1 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-88.bl.1.11 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-88.bu.1.7 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-88.bu.1.10 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-88.bu.2.3 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-88.bu.2.14 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-88.bv.1.5 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-88.bv.1.12 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-88.bv.2.4 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-88.bv.2.13 | $264$ | $2$ | $2$ | $0$ |
| 264.288.9-88.bl.1.24 | $264$ | $12$ | $12$ | $9$ |
| 312.48.0-104.bj.1.4 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-104.bl.1.9 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-104.bn.1.4 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-104.bp.1.9 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-104.ca.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-104.ca.1.12 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-104.ca.2.6 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-104.ca.2.11 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-104.cb.1.6 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-104.cb.1.11 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-104.cb.2.5 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-104.cb.2.12 | $312$ | $2$ | $2$ | $0$ |
| 312.336.11-104.bx.1.24 | $312$ | $14$ | $14$ | $11$ |
| 120.48.0-120.dd.1.17 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.df.1.24 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.dh.1.16 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.dj.1.20 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ei.1.7 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ei.1.26 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ei.2.3 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ei.2.30 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ej.1.3 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ej.1.30 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ej.2.11 | $120$ | $2$ | $2$ | $0$ |
| 120.48.0-120.ej.2.22 | $120$ | $2$ | $2$ | $0$ |
| 168.48.0-168.cx.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.cz.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.db.1.9 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.dd.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ec.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ec.1.26 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ec.2.13 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ec.2.20 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ed.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ed.1.28 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ed.2.15 | $168$ | $2$ | $2$ | $0$ |
| 168.48.0-168.ed.2.18 | $168$ | $2$ | $2$ | $0$ |
| 240.48.0-240.m.1.14 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.m.1.51 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.m.2.14 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.m.2.51 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.n.1.6 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.n.1.59 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.n.2.30 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.n.2.35 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.o.1.22 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.o.1.43 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.p.1.10 | $240$ | $2$ | $2$ | $0$ |
| 240.48.0-240.p.1.55 | $240$ | $2$ | $2$ | $0$ |
| 240.48.1-240.a.1.6 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-240.a.1.59 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-240.b.1.22 | $240$ | $2$ | $2$ | $1$ |
| 240.48.1-240.b.1.43 | $240$ | $2$ | $2$ | $1$ |
| 264.48.0-264.cx.1.11 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.cz.1.17 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.db.1.7 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.dd.1.17 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.ec.1.15 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.ec.1.18 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.ec.2.14 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.ec.2.19 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.ed.1.16 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.ed.1.17 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.ed.2.13 | $264$ | $2$ | $2$ | $0$ |
| 264.48.0-264.ed.2.20 | $264$ | $2$ | $2$ | $0$ |
| 312.48.0-312.dd.1.6 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.df.1.6 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.dh.1.6 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.dj.1.4 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ei.1.7 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ei.1.26 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ei.2.13 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ei.2.20 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ej.1.5 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ej.1.28 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ej.2.15 | $312$ | $2$ | $2$ | $0$ |
| 312.48.0-312.ej.2.18 | $312$ | $2$ | $2$ | $0$ |