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: | $6$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot3^{2}\cdot4\cdot12$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12E0 |
Level structure
| $\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}43&30\\48&37\end{bmatrix}$, $\begin{bmatrix}52&77\\3&38\end{bmatrix}$, $\begin{bmatrix}70&33\\3&34\end{bmatrix}$, $\begin{bmatrix}75&28\\4&3\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 84.24.0.o.1 for the level structure with $-I$) |
| Cyclic 84-isogeny field degree: | $16$ |
| Cyclic 84-torsion field degree: | $384$ |
| 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 but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 3.8.0-3.a.1.1 | $3$ | $6$ | $6$ | $0$ | $0$ |
| 28.6.0.c.1 | $28$ | $8$ | $4$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 6.24.0-6.a.1.2 | $6$ | $2$ | $2$ | $0$ | $0$ |
| 84.24.0-6.a.1.7 | $84$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 84.96.1-84.b.1.8 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.h.1.11 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.r.1.8 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.t.1.8 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.bl.1.8 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.bn.1.8 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.bo.1.8 | $84$ | $2$ | $2$ | $1$ |
| 84.96.1-84.br.1.11 | $84$ | $2$ | $2$ | $1$ |
| 84.144.1-84.n.1.1 | $84$ | $3$ | $3$ | $1$ |
| 84.384.11-84.ck.1.32 | $84$ | $8$ | $8$ | $11$ |
| 168.96.1-168.gg.1.16 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.kg.1.16 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bae.1.16 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bak.1.16 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.byx.1.16 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bzd.1.16 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bzh.1.16 | $168$ | $2$ | $2$ | $1$ |
| 168.96.1-168.bzq.1.16 | $168$ | $2$ | $2$ | $1$ |
| 252.144.1-252.h.1.2 | $252$ | $3$ | $3$ | $1$ |
| 252.144.4-252.r.1.15 | $252$ | $3$ | $3$ | $4$ |
| 252.144.4-252.bc.1.2 | $252$ | $3$ | $3$ | $4$ |