Invariants
| Level: | $168$ | $\SL_2$-level: | $12$ | Newform level: | $48$ | ||
| Index: | $64$ | $\PSL_2$-index: | $32$ | ||||
| Genus: | $1 = 1 + \frac{ 32 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $4^{2}\cdot12^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 32$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12I1 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}65&141\\71&94\end{bmatrix}$, $\begin{bmatrix}70&155\\13&114\end{bmatrix}$, $\begin{bmatrix}134&55\\129&118\end{bmatrix}$, $\begin{bmatrix}150&29\\11&51\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.32.1.a.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $96$ |
| Cyclic 168-torsion field degree: | $4608$ |
| Full 168-torsion field degree: | $2322432$ |
Jacobian
| Conductor: | $?$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 48.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - 2 x z + 2 y w $ |
| $=$ | $3 x^{2} + 2 x z - 9 y^{2} + 2 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 2 x^{3} y + x^{2} y^{2} - 12 x^{2} z^{2} - 4 x y z^{2} - 2 y^{2} z^{2} + 36 z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 32 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2\cdot3^2\,\frac{z(1316412xyz^{4}w+944640xyz^{2}w^{3}+6912xyw^{5}-492764xz^{6}-2558700xz^{4}w^{2}-222720xz^{2}w^{4}+2304xw^{6}+492772yz^{5}w+2229408yz^{3}w^{3}+195840yzw^{5}+z^{7}-237780z^{5}w^{2}+49952z^{3}w^{4}+34560zw^{6})}{14400xyz^{5}w+16896xyz^{3}w^{3}+1233xyzw^{5}-5184xz^{7}-32384xz^{5}w^{2}-9271xz^{3}w^{4}+51xzw^{6}+5184yz^{6}w+28736yz^{4}w^{3}+7719yz^{2}w^{5}+72yw^{7}-2624z^{6}w^{2}-434z^{4}w^{4}+837z^{2}w^{6}+18w^{8}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.32.1.a.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}-2X^{3}Y+X^{2}Y^{2}-12X^{2}Z^{2}-4XYZ^{2}-2Y^{2}Z^{2}+36Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 84.32.1-12.a.1.4 | $84$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.16.0-24.a.1.1 | $168$ | $4$ | $4$ | $0$ | $?$ | full Jacobian |
| 168.32.1-12.a.1.3 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 168.192.3-24.cw.1.8 | $168$ | $3$ | $3$ | $3$ | $?$ | not computed |
| 168.192.5-24.dl.1.3 | $168$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 168.256.7-24.b.1.4 | $168$ | $4$ | $4$ | $7$ | $?$ | not computed |
| 168.512.17-168.a.1.17 | $168$ | $8$ | $8$ | $17$ | $?$ | not computed |