Invariants
| Level: | $168$ | $\SL_2$-level: | $24$ | Newform level: | $576$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $4 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $6^{4}\cdot24^{2}$ | Cusp orbits | $2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | not computed | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24D4 |
Level structure
| $\GL_2(\Z/168\Z)$-generators: | $\begin{bmatrix}49&114\\124&167\end{bmatrix}$, $\begin{bmatrix}65&85\\100&129\end{bmatrix}$, $\begin{bmatrix}103&49\\100&125\end{bmatrix}$, $\begin{bmatrix}121&160\\120&107\end{bmatrix}$, $\begin{bmatrix}151&66\\120&79\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 24.72.4.fc.1 for the level structure with $-I$) |
| Cyclic 168-isogeny field degree: | $64$ |
| Cyclic 168-torsion field degree: | $3072$ |
| Full 168-torsion field degree: | $1032192$ |
Models
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 16 y^{2} - z^{2} + 2 z w + 2 w^{2} $ |
| $=$ | $6 x^{3} + y z^{2} + 2 y z w$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} + 4 x^{5} z - 5 x^{4} z^{2} - 3 x^{3} y^{3} - 40 x^{3} z^{3} - 18 x^{2} y^{3} z - 85 x^{2} z^{4} + \cdots - 15 z^{6} $ |
Rational points
This modular curve has no $\Q_p$ points for $p=7$, and therefore no rational points.
Maps to other modular curves
$j$-invariant map of degree 72 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^6}{3}\cdot\frac{(13z^{4}+44z^{3}w+48z^{2}w^{2}+8zw^{3}+4w^{4})^{3}}{z^{2}(z+2w)^{2}(z^{2}-2zw-2w^{2})^{4}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.72.4.fc.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y-\frac{1}{4}w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{4}z+\frac{1}{4}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{6}-3X^{3}Y^{3}+4X^{5}Z-18X^{2}Y^{3}Z-5X^{4}Z^{2}-36XY^{3}Z^{2}-40X^{3}Z^{3}-24Y^{3}Z^{3}-85X^{2}Z^{4}-76XZ^{5}-15Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 168.48.0-24.bk.1.7 | $168$ | $3$ | $3$ | $0$ | $?$ |
| 168.72.2-12.t.1.3 | $168$ | $2$ | $2$ | $2$ | $?$ |
| 168.72.2-12.t.1.6 | $168$ | $2$ | $2$ | $2$ | $?$ |
| 168.72.2-24.ci.1.12 | $168$ | $2$ | $2$ | $2$ | $?$ |
| 168.72.2-24.ci.1.25 | $168$ | $2$ | $2$ | $2$ | $?$ |
| 168.72.2-24.ck.1.16 | $168$ | $2$ | $2$ | $2$ | $?$ |
| 168.72.2-24.ck.1.29 | $168$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 168.288.7-24.vg.1.6 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-24.vi.1.7 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-24.vo.1.4 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-24.vq.1.3 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-24.yi.1.2 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-24.yk.1.3 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-24.yq.1.6 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-24.ys.1.7 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.cvc.1.10 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.cve.1.10 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.cvk.1.5 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.cvm.1.9 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.dae.1.7 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.dag.1.5 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.dam.1.8 | $168$ | $2$ | $2$ | $7$ |
| 168.288.7-168.dao.1.6 | $168$ | $2$ | $2$ | $7$ |