Invariants
| Level: | $42$ | $\SL_2$-level: | $42$ | Newform level: | $588$ | ||
| Index: | $1008$ | $\PSL_2$-index: | $504$ | ||||
| Genus: | $34 = 1 + \frac{ 504 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 18 }{2}$ | ||||||
| Cusps: | $18$ (none of which are rational) | Cusp widths | $14^{9}\cdot42^{9}$ | Cusp orbits | $3^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $6$ | ||||||
| $\Q$-gonality: | $7 \le \gamma \le 16$ | ||||||
| $\overline{\Q}$-gonality: | $7 \le \gamma \le 16$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 42.1008.34.3 |
Level structure
| $\GL_2(\Z/42\Z)$-generators: | $\begin{bmatrix}5&6\\22&19\end{bmatrix}$, $\begin{bmatrix}7&10\\24&35\end{bmatrix}$, $\begin{bmatrix}15&34\\20&41\end{bmatrix}$ |
| $\GL_2(\Z/42\Z)$-subgroup: | $C_{48}:D_6$ |
| Contains $-I$: | no $\quad$ (see 42.504.34.a.1 for the level structure with $-I$) |
| Cyclic 42-isogeny field degree: | $8$ |
| Cyclic 42-torsion field degree: | $96$ |
| Full 42-torsion field degree: | $576$ |
Jacobian
| Conductor: | $2^{26}\cdot3^{20}\cdot7^{68}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{10}\cdot2^{12}$ |
| Newforms: | 98.2.a.b$^{4}$, 147.2.a.c$^{3}$, 147.2.a.d$^{3}$, 147.2.a.e$^{3}$, 196.2.a.b$^{2}$, 196.2.a.c$^{2}$, 294.2.a.d$^{2}$, 294.2.a.e$^{2}$, 588.2.a.a |
Rational points
This modular curve has no $\Q_p$ points for $p=5,149$, and therefore no 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 | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X(2)$ | $2$ | $168$ | $84$ | $0$ | $0$ | full Jacobian |
| 3.8.0-3.a.1.1 | $3$ | $126$ | $126$ | $0$ | $0$ | full Jacobian |
| $X_{\mathrm{ns}}^+(7)$ | $7$ | $48$ | $24$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 6.48.0-6.a.1.2 | $6$ | $21$ | $21$ | $0$ | $0$ | full Jacobian |
| 42.336.12-42.a.1.8 | $42$ | $3$ | $3$ | $12$ | $2$ | $1^{6}\cdot2^{8}$ |
| 42.504.16-42.a.1.1 | $42$ | $2$ | $2$ | $16$ | $4$ | $1^{6}\cdot2^{6}$ |
| 42.504.16-42.a.1.12 | $42$ | $2$ | $2$ | $16$ | $4$ | $1^{6}\cdot2^{6}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 42.2016.67-42.a.1.4 | $42$ | $2$ | $2$ | $67$ | $12$ | $1^{33}$ |
| 42.2016.67-42.b.1.4 | $42$ | $2$ | $2$ | $67$ | $22$ | $1^{33}$ |
| 42.2016.67-42.c.1.6 | $42$ | $2$ | $2$ | $67$ | $9$ | $1^{33}$ |
| 42.2016.67-42.d.1.4 | $42$ | $2$ | $2$ | $67$ | $18$ | $1^{33}$ |
| 42.3024.109-42.a.1.1 | $42$ | $3$ | $3$ | $109$ | $27$ | $1^{27}\cdot2^{22}\cdot4$ |