Invariants
Level: | $84$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $4^{6}\cdot12^{6}$ | Cusp orbits | $1^{2}\cdot2^{5}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $2 \le \gamma \le 3$ | ||||||
$\overline{\Q}$-gonality: | $2 \le \gamma \le 3$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12L3 |
Level structure
$\GL_2(\Z/84\Z)$-generators: | $\begin{bmatrix}15&74\\70&47\end{bmatrix}$, $\begin{bmatrix}29&76\\50&15\end{bmatrix}$, $\begin{bmatrix}81&4\\46&33\end{bmatrix}$, $\begin{bmatrix}81&10\\26&67\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 84.96.3.q.1 for the level structure with $-I$) |
Cyclic 84-isogeny field degree: | $16$ |
Cyclic 84-torsion field degree: | $192$ |
Full 84-torsion field degree: | $48384$ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
12.96.0-12.a.2.9 | $12$ | $2$ | $2$ | $0$ | $0$ |
84.96.0-12.a.2.2 | $84$ | $2$ | $2$ | $0$ | $?$ |
84.96.1-84.c.1.5 | $84$ | $2$ | $2$ | $1$ | $?$ |
84.96.1-84.c.1.18 | $84$ | $2$ | $2$ | $1$ | $?$ |
84.96.2-84.a.1.5 | $84$ | $2$ | $2$ | $2$ | $?$ |
84.96.2-84.a.1.9 | $84$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
84.384.5-84.i.2.3 | $84$ | $2$ | $2$ | $5$ |
84.384.5-84.i.2.8 | $84$ | $2$ | $2$ | $5$ |
84.384.5-84.k.2.6 | $84$ | $2$ | $2$ | $5$ |
84.384.5-84.k.3.8 | $84$ | $2$ | $2$ | $5$ |
84.384.5-84.l.3.6 | $84$ | $2$ | $2$ | $5$ |
84.384.5-84.l.4.8 | $84$ | $2$ | $2$ | $5$ |
84.384.5-84.o.2.7 | $84$ | $2$ | $2$ | $5$ |
84.384.5-84.o.3.4 | $84$ | $2$ | $2$ | $5$ |
168.384.5-168.ji.1.3 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.ji.2.11 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.jt.1.3 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.jt.2.11 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.kb.1.3 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.kb.2.11 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.kw.1.3 | $168$ | $2$ | $2$ | $5$ |
168.384.5-168.kw.2.11 | $168$ | $2$ | $2$ | $5$ |