Invariants
| Level: | $168$ | $\SL_2$-level: | $8$ | Newform level: | $3136$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (of which $2$ are rational) | Cusp widths | $4^{2}\cdot8^{2}$ | 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: | 8B1 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}39&8\\104&59\end{bmatrix}$, $\begin{bmatrix}39&46\\8&107\end{bmatrix}$, $\begin{bmatrix}53&88\\64&49\end{bmatrix}$, $\begin{bmatrix}83&150\\150&157\end{bmatrix}$, $\begin{bmatrix}99&98\\124&51\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.24.1.b.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $128$ |
| Cyclic 168-torsion field degree: | $6144$ |
| Full 168-torsion field degree: | $3096576$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 3136.2.a.m |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + 49x $ |
Rational points
This modular curve is an elliptic curve, but the rank has not been computed
Maps to other modular curves
$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^8}{7^4}\cdot\frac{7203x^{2}y^{4}z^{2}-49xy^{6}z+17294403xy^{2}z^{5}+y^{8}+13841287201z^{8}}{z^{2}y^{4}x^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.24.0-4.a.1.2 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 168.24.0-4.a.1.2 | $168$ | $2$ | $2$ | $0$ | $?$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 168.96.1-56.a.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-56.d.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-56.n.1.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-56.z.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bd.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bf.1.6 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-56.bk.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-56.bn.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-56.bp.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bp.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-56.br.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.br.1.6 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.en.1.3 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.ep.1.7 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.er.1.3 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.et.1.7 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.144.5-168.b.1.4 | $168$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 168.192.5-168.b.1.8 | $168$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 168.384.13-56.f.1.24 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |