Invariants
| Level: | $42$ | $\SL_2$-level: | $42$ | Newform level: | $294$ | ||
| Index: | $504$ | $\PSL_2$-index: | $252$ | ||||
| Genus: | $16 = 1 + \frac{ 252 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $7^{3}\cdot14^{3}\cdot21^{3}\cdot42^{3}$ | Cusp orbits | $3^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $4$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 8$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 8$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 42A16 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 42.504.16.5 |
Level structure
| $\GL_2(\Z/42\Z)$-generators: | $\begin{bmatrix}23&15\\18&5\end{bmatrix}$, $\begin{bmatrix}25&26\\12&13\end{bmatrix}$, $\begin{bmatrix}31&38\\36&25\end{bmatrix}$, $\begin{bmatrix}41&6\\30&1\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 42.252.16.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: | $1152$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{12}\cdot7^{32}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}\cdot2^{6}$ |
| Newforms: | 98.2.a.b$^{2}$, 147.2.a.c$^{2}$, 147.2.a.d$^{2}$, 147.2.a.e$^{2}$, 294.2.a.d, 294.2.a.e |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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 |
|---|---|---|---|---|---|---|
| 6.24.0-6.a.1.3 | $6$ | $21$ | $21$ | $0$ | $0$ | full Jacobian |
| $X_{\mathrm{ns}}^+(7)$ | $7$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 6.24.0-6.a.1.3 | $6$ | $21$ | $21$ | $0$ | $0$ | full Jacobian |
| 42.168.5-21.a.1.6 | $42$ | $3$ | $3$ | $5$ | $2$ | $1^{3}\cdot2^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 42.1008.31-42.a.1.1 | $42$ | $2$ | $2$ | $31$ | $5$ | $1^{15}$ |
| 42.1008.31-42.b.1.1 | $42$ | $2$ | $2$ | $31$ | $14$ | $1^{15}$ |
| 42.1008.31-42.c.1.2 | $42$ | $2$ | $2$ | $31$ | $5$ | $1^{15}$ |
| 42.1008.31-42.d.1.1 | $42$ | $2$ | $2$ | $31$ | $9$ | $1^{15}$ |
| 42.1008.34-42.a.1.8 | $42$ | $2$ | $2$ | $34$ | $6$ | $1^{6}\cdot2^{6}$ |
| 42.1008.34-42.c.1.6 | $42$ | $2$ | $2$ | $34$ | $12$ | $1^{6}\cdot2^{6}$ |
| 42.1008.34-42.d.1.6 | $42$ | $2$ | $2$ | $34$ | $8$ | $1^{6}\cdot2^{6}$ |
| 42.1008.34-42.e.1.8 | $42$ | $2$ | $2$ | $34$ | $10$ | $1^{6}\cdot2^{6}$ |
| 42.1008.34-42.f.1.6 | $42$ | $2$ | $2$ | $34$ | $11$ | $1^{18}$ |
| 42.1008.34-42.g.1.2 | $42$ | $2$ | $2$ | $34$ | $6$ | $1^{18}$ |
| 42.1008.34-42.h.1.2 | $42$ | $2$ | $2$ | $34$ | $10$ | $1^{18}$ |
| 42.1008.34-42.i.1.4 | $42$ | $2$ | $2$ | $34$ | $9$ | $1^{18}$ |
| 42.1512.52-42.a.1.8 | $42$ | $3$ | $3$ | $52$ | $14$ | $1^{12}\cdot2^{12}$ |