Invariants
| Level: | $168$ | $\SL_2$-level: | $12$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (all of which are rational) | Cusp widths | $1^{2}\cdot3^{2}\cdot4\cdot12$ | Cusp orbits | $1^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $6$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12E0 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}3&52\\4&51\end{bmatrix}$, $\begin{bmatrix}19&38\\150&107\end{bmatrix}$, $\begin{bmatrix}50&163\\119&18\end{bmatrix}$, $\begin{bmatrix}65&144\\146&163\end{bmatrix}$, $\begin{bmatrix}105&136\\22&123\end{bmatrix}$, $\begin{bmatrix}138&37\\143&100\end{bmatrix}$, $\begin{bmatrix}159&88\\2&25\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.24.0.g.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $16$ |
| Cyclic 168-torsion field degree: | $768$ |
| Full 168-torsion field degree: | $3096576$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 330 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^4}\cdot\frac{x^{24}(3x^{2}-4y^{2})^{3}(3x^{6}-12x^{4}y^{2}+144x^{2}y^{4}-64y^{6})^{3}}{y^{4}x^{36}(x-2y)^{3}(x+2y)^{3}(3x-2y)(3x+2y)}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 168.12.0-4.c.1.2 | $168$ | $4$ | $4$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.96.0-12.c.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-12.c.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-12.c.2.1 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-12.c.2.8 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-12.c.3.1 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-12.c.3.8 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-12.c.4.1 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-12.c.4.8 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-84.c.1.7 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-84.c.1.38 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-84.c.2.7 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-84.c.2.38 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-84.c.3.13 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-84.c.3.32 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-84.c.4.20 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-84.c.4.25 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bs.1.14 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bs.1.19 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bs.2.15 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bs.2.18 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bt.1.14 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bt.1.19 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bt.2.15 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bt.2.18 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bu.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bu.1.9 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bu.2.8 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bu.2.9 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bu.3.7 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bu.3.10 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bu.4.7 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bu.4.10 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.do.1.12 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.do.1.53 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.do.2.8 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.do.2.57 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dp.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dp.1.57 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dp.2.20 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dp.2.45 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dq.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dq.1.57 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dq.2.6 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dq.2.57 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dq.3.18 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dq.3.45 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dq.4.29 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dq.4.34 | $168$ | $2$ | $2$ | $0$ |
| 168.96.1-12.b.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-12.h.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-12.k.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-84.k.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-12.l.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-84.l.1.10 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-84.o.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-84.p.1.7 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.cg.1.6 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.es.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.ik.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.in.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.iq.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.iq.1.25 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.ir.1.7 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.ir.1.42 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.is.1.7 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.is.1.26 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.it.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.it.1.28 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.iu.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.iu.1.28 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.iv.1.7 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.iv.1.26 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.iw.1.7 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.iw.1.26 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.ix.1.8 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.ix.1.25 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.za.1.13 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zd.1.3 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zm.1.15 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zp.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zs.1.16 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zs.1.49 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zt.1.32 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zt.1.33 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zu.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zu.1.53 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zv.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zv.1.61 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zw.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zw.1.61 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zx.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zx.1.53 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zy.1.32 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zy.1.33 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zz.1.16 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zz.1.49 | $168$ | $2$ | $2$ | $1$ |
| 168.96.2-24.f.1.12 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.f.1.21 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.f.2.14 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.f.2.19 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.f.1.8 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.f.1.57 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.f.2.20 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.f.2.45 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.g.1.12 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.g.1.21 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.g.2.14 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.g.2.19 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.g.1.12 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.g.1.53 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.g.2.8 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.g.2.57 | $168$ | $2$ | $2$ | $2$ |
| 168.144.1-12.f.1.13 | $168$ | $3$ | $3$ | $1$ |
| 168.384.11-84.bm.1.14 | $168$ | $8$ | $8$ | $11$ |