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: | $\begin{bmatrix}59&28\\156&103\end{bmatrix}$, $\begin{bmatrix}69&124\\34&15\end{bmatrix}$, $\begin{bmatrix}102&25\\151&48\end{bmatrix}$, $\begin{bmatrix}135&62\\74&15\end{bmatrix}$, $\begin{bmatrix}150&79\\85&72\end{bmatrix}$, $\begin{bmatrix}154&5\\93&98\end{bmatrix}$, $\begin{bmatrix}161&166\\38&117\end{bmatrix}$ |
| 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
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $12$ | $6$ | $0$ | $0$ |
| 56.12.0-4.c.1.2 | $56$ | $4$ | $4$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 56.12.0-4.c.1.2 | $56$ | $4$ | $4$ | $0$ | $0$ |
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.1.3 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-12.c.1.6 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-12.c.2.2 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-12.c.2.7 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-12.c.3.3 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-12.c.3.6 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-12.c.4.2 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-12.c.4.7 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-84.c.1.20 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-84.c.1.25 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-84.c.2.20 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-84.c.2.25 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-84.c.3.7 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-84.c.3.38 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-84.c.4.7 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-84.c.4.38 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bs.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bs.1.28 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bs.2.3 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bs.2.30 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bt.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bt.1.28 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bt.2.3 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bt.2.30 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bu.1.5 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bu.1.12 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bu.2.3 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bu.2.14 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bu.3.6 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bu.3.11 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bu.4.4 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bu.4.13 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.do.1.25 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.do.1.40 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.do.2.28 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.do.2.37 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dp.1.21 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dp.1.44 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dp.2.16 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dp.2.49 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dq.1.21 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dq.1.42 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dq.2.13 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dq.2.50 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dq.3.14 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dq.3.49 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dq.4.22 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dq.4.41 | $168$ | $2$ | $2$ | $0$ |
| 168.96.1-12.b.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-12.h.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-12.k.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-84.k.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-12.l.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-84.l.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-84.o.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-84.p.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.cg.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.es.1.6 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.ik.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.in.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.iq.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.iq.1.28 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.ir.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.ir.1.48 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.is.1.6 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.is.1.27 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.it.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.it.1.31 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.iu.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.iu.1.31 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.iv.1.6 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.iv.1.27 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.iw.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.iw.1.32 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.ix.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.ix.1.28 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.za.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zd.1.15 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zm.1.3 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zp.1.9 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zs.1.29 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zs.1.36 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zt.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zt.1.61 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zu.1.24 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zu.1.41 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zv.1.17 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zv.1.48 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zw.1.17 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zw.1.48 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zx.1.24 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zx.1.41 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zy.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zy.1.61 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zz.1.29 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zz.1.36 | $168$ | $2$ | $2$ | $1$ |
| 168.96.2-24.f.1.9 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.f.1.24 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.f.2.5 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.f.2.28 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.f.1.21 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.f.1.44 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.f.2.16 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.f.2.49 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.g.1.9 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.g.1.24 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.g.2.5 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.g.2.28 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.g.1.25 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.g.1.40 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.g.2.28 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.g.2.37 | $168$ | $2$ | $2$ | $2$ |
| 168.144.1-12.f.1.8 | $168$ | $3$ | $3$ | $1$ |
| 168.384.11-84.bm.1.73 | $168$ | $8$ | $8$ | $11$ |