Invariants
| Level: | $168$ | $\SL_2$-level: | $12$ | Newform level: | $192$ | ||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (all of which are rational) | Cusp widths | $2\cdot4\cdot6\cdot12$ | Cusp orbits | $1^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12F1 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}17&16\\42&127\end{bmatrix}$, $\begin{bmatrix}45&100\\134&91\end{bmatrix}$, $\begin{bmatrix}87&86\\98&57\end{bmatrix}$, $\begin{bmatrix}104&111\\81&92\end{bmatrix}$, $\begin{bmatrix}106&101\\63&128\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.24.1.et.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $32$ |
| Cyclic 168-torsion field degree: | $1536$ |
| Full 168-torsion field degree: | $3096576$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 192.2.a.d |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} + x^{2} - 97x - 385 $ |
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 $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6}\cdot\frac{8x^{2}y^{6}+4848x^{2}y^{4}z^{2}+784384x^{2}y^{2}z^{4}+42025984x^{2}z^{6}+136xy^{6}z+58272xy^{4}z^{3}+9451776xy^{2}z^{5}+511207424xz^{7}+y^{8}+784y^{6}z^{2}+219760y^{4}z^{4}+29704960y^{2}z^{6}+1522164736z^{8}}{z^{4}y^{2}(16x^{2}+192xz+y^{2}+560z^{2})}$ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.6.0.f.1 | $8$ | $8$ | $4$ | $0$ | $0$ | full Jacobian |
| 21.8.0-3.a.1.2 | $21$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 42.24.0-6.a.1.4 | $42$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 168.24.0-6.a.1.9 | $168$ | $2$ | $2$ | $0$ | $?$ | 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 |
|---|---|---|---|---|---|---|
| 168.96.1-24.bw.1.17 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.cf.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.er.1.12 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.es.1.12 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.je.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.jf.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.jh.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-24.ji.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bzm.1.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bzn.1.12 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bzp.1.14 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bzq.1.14 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bzy.1.15 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.bzz.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.cab.1.12 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1-168.cac.1.12 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.144.3-24.ug.1.1 | $168$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 168.384.13-168.pf.1.48 | $168$ | $8$ | $8$ | $13$ | $?$ | not computed |