Invariants
| Level: | $84$ | $\SL_2$-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
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}11&42\\14&79\end{bmatrix}$, $\begin{bmatrix}30&35\\23&66\end{bmatrix}$, $\begin{bmatrix}44&13\\3&70\end{bmatrix}$, $\begin{bmatrix}63&34\\2&79\end{bmatrix}$, $\begin{bmatrix}68&27\\35&76\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 12.24.0.g.1 for the level structure with $-I$) |
| Cyclic 84-isogeny field degree: | $8$ |
| Cyclic 84-torsion field degree: | $192$ |
| Full 84-torsion field degree: | $193536$ |
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 |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $12$ | $6$ | $0$ | $0$ |
| 28.12.0-4.c.1.2 | $28$ | $4$ | $4$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 28.12.0-4.c.1.2 | $28$ | $4$ | $4$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 84.96.0-12.c.1.2 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-12.c.1.7 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-12.c.2.2 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-12.c.2.7 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-12.c.3.2 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-12.c.3.7 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-12.c.4.2 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-12.c.4.7 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-84.c.1.8 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-84.c.1.9 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-84.c.2.8 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-84.c.2.9 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-84.c.3.3 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-84.c.3.14 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-84.c.4.3 | $84$ | $2$ | $2$ | $0$ |
| 84.96.0-84.c.4.14 | $84$ | $2$ | $2$ | $0$ |
| 84.96.1-12.b.1.2 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-12.h.1.2 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-12.k.1.1 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.k.1.1 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-12.l.1.1 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.l.1.10 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.o.1.7 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.p.1.7 | $84$ | $2$ | $2$ | $1$ |
| 84.144.1-12.f.1.5 | $84$ | $3$ | $3$ | $1$ |
| 84.384.11-84.bm.1.29 | $84$ | $8$ | $8$ | $11$ |
| 168.96.0-24.bs.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bs.1.32 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bs.2.1 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bs.2.32 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bt.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bt.1.32 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bt.2.1 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bt.2.32 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bu.1.1 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bu.1.16 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bu.2.1 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bu.2.16 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bu.3.2 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bu.3.15 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bu.4.2 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-24.bu.4.15 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.do.1.8 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.do.1.57 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.do.2.20 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.do.2.45 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dp.1.12 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dp.1.53 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dp.2.8 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dp.2.57 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dq.1.10 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dq.1.53 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dq.2.18 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dq.2.45 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dq.3.6 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dq.3.57 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dq.4.6 | $168$ | $2$ | $2$ | $0$ |
| 168.96.0-168.dq.4.57 | $168$ | $2$ | $2$ | $0$ |
| 168.96.1-24.cg.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.es.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.ik.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.in.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.iq.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.iq.1.32 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.ir.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.ir.1.47 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.is.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.is.1.31 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.it.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.it.1.29 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.iu.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.iu.1.29 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.iv.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.iv.1.31 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.iw.1.2 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.iw.1.31 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.ix.1.1 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-24.ix.1.32 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.za.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zd.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zm.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zp.1.5 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zs.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zs.1.61 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zt.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zt.1.53 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zu.1.32 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zu.1.33 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zv.1.16 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zv.1.49 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zw.1.16 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zw.1.49 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zx.1.32 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zx.1.33 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zy.1.12 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zy.1.53 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zz.1.4 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.zz.1.61 | $168$ | $2$ | $2$ | $1$ |
| 168.96.2-24.f.1.1 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.f.1.32 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.f.2.1 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.f.2.32 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.f.1.12 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.f.1.53 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.f.2.8 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.f.2.57 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.g.1.1 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.g.1.32 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.g.2.1 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-24.g.2.32 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.g.1.8 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.g.1.57 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.g.2.20 | $168$ | $2$ | $2$ | $2$ |
| 168.96.2-168.g.2.45 | $168$ | $2$ | $2$ | $2$ |
| 252.144.1-36.c.1.8 | $252$ | $3$ | $3$ | $1$ |
| 252.144.4-36.d.1.8 | $252$ | $3$ | $3$ | $4$ |
| 252.144.4-36.f.1.4 | $252$ | $3$ | $3$ | $4$ |