Invariants
| Level: | $84$ | $\SL_2$-level: | $6$ | ||||
| Index: | $24$ | $\PSL_2$-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 | $2^{3}\cdot6^{3}$ | 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: | 6I0 |
Level structure
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 |
|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $6$ | $6$ | $0$ | $0$ |
| 28.6.0.a.1 | $28$ | $4$ | $4$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(6)$ | $6$ | $2$ | $2$ | $0$ | $0$ |
| 28.6.0.a.1 | $28$ | $4$ | $4$ | $0$ | $0$ |
| 84.8.0.a.1 | $84$ | $3$ | $3$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 84.48.1.j.1 | $84$ | $2$ | $2$ | $1$ |
| 84.48.1.l.1 | $84$ | $2$ | $2$ | $1$ |
| 84.48.1.v.1 | $84$ | $2$ | $2$ | $1$ |
| 84.48.1.x.1 | $84$ | $2$ | $2$ | $1$ |
| 84.48.1.bh.1 | $84$ | $2$ | $2$ | $1$ |
| 84.48.1.bj.1 | $84$ | $2$ | $2$ | $1$ |
| 84.48.1.bp.1 | $84$ | $2$ | $2$ | $1$ |
| 84.48.1.br.1 | $84$ | $2$ | $2$ | $1$ |
| 84.72.1.j.1 | $84$ | $3$ | $3$ | $1$ |
| 84.192.11.ci.1 | $84$ | $8$ | $8$ | $11$ |
| 168.48.1.yy.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.ze.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.baq.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.baw.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.byl.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.byr.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.bzj.1 | $168$ | $2$ | $2$ | $1$ |
| 168.48.1.bzp.1 | $168$ | $2$ | $2$ | $1$ |
| 252.72.1.f.1 | $252$ | $3$ | $3$ | $1$ |
| 252.72.4.p.1 | $252$ | $3$ | $3$ | $4$ |
| 252.72.4.ba.1 | $252$ | $3$ | $3$ | $4$ |