Invariants
Level: | $168$ | $\SL_2$-level: | $12$ | Newform level: | $1$ | ||
Index: | $72$ | $\PSL_2$-index: | $36$ | ||||
Genus: | $2 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $6^{2}\cdot12^{2}$ | Cusp orbits | $1^{2}\cdot2$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 12B2 |
Level structure
$\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}1&95\\104&155\end{bmatrix}$, $\begin{bmatrix}19&75\\96&13\end{bmatrix}$, $\begin{bmatrix}133&25\\156&35\end{bmatrix}$, $\begin{bmatrix}139&41\\20&147\end{bmatrix}$, $\begin{bmatrix}143&138\\48&155\end{bmatrix}$, $\begin{bmatrix}153&115\\20&165\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 84.36.2.s.1 for the level structure with $-I$) |
Cyclic 168-isogeny field degree: | $64$ |
Cyclic 168-torsion field degree: | $3072$ |
Full 168-torsion field degree: | $2064384$ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points.
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
Factor curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
$X_{\mathrm{ns}}^+(3)$ | $3$ | $24$ | $12$ | $0$ | $0$ |
56.24.0-28.g.1.2 | $56$ | $3$ | $3$ | $0$ | $0$ |
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
24.36.1-12.c.1.17 | $24$ | $2$ | $2$ | $1$ | $0$ |
56.24.0-28.g.1.2 | $56$ | $3$ | $3$ | $0$ | $0$ |
168.36.1-12.c.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.