Invariants
| Level: | $84$ | $\SL_2$-level: | $14$ | ||||
| Index: | $32$ | $\PSL_2$-index: | $16$ | ||||
| Genus: | $0 = 1 + \frac{ 16 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $2\cdot14$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $4$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 14B0 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}41&9\\27&38\end{bmatrix}$, $\begin{bmatrix}54&53\\43&44\end{bmatrix}$, $\begin{bmatrix}81&26\\50&49\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 28.16.0.b.1 for the level structure with $-I$) |
| Cyclic 84-isogeny field degree: | $24$ |
| Cyclic 84-torsion field degree: | $576$ |
| Full 84-torsion field degree: | $290304$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Maps to other modular curves
$j$-invariant map of degree 16 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^{14}\cdot3^{14}}\cdot\frac{x^{16}(7x^{4}+468x^{2}y^{2}+9072y^{4})(49x^{4}+1260x^{2}y^{2}+1296y^{4})^{3}}{y^{14}x^{18}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 21.16.0-7.a.1.2 | $21$ | $2$ | $2$ | $0$ | $0$ |
| 84.16.0-7.a.1.4 | $84$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 84.96.2-28.f.1.4 | $84$ | $3$ | $3$ | $2$ |
| 84.96.2-28.f.2.2 | $84$ | $3$ | $3$ | $2$ |
| 84.96.2-28.g.1.1 | $84$ | $3$ | $3$ | $2$ |
| 84.96.2-28.j.1.1 | $84$ | $3$ | $3$ | $2$ |
| 84.96.4-84.c.1.6 | $84$ | $3$ | $3$ | $4$ |
| 84.128.3-28.d.1.2 | $84$ | $4$ | $4$ | $3$ |
| 84.128.3-84.e.1.3 | $84$ | $4$ | $4$ | $3$ |
| 84.224.5-28.h.1.2 | $84$ | $7$ | $7$ | $5$ |
| 252.96.2-252.q.1.8 | $252$ | $3$ | $3$ | $2$ |
| 252.96.2-252.q.2.6 | $252$ | $3$ | $3$ | $2$ |
| 252.96.2-252.r.1.7 | $252$ | $3$ | $3$ | $2$ |
| 252.96.2-252.r.2.5 | $252$ | $3$ | $3$ | $2$ |
| 252.96.2-252.u.1.8 | $252$ | $3$ | $3$ | $2$ |
| 252.96.2-252.u.2.6 | $252$ | $3$ | $3$ | $2$ |