Invariants
| Level: | $42$ | $\SL_2$-level: | $42$ | Newform level: | $1764$ | ||
| Index: | $2016$ | $\PSL_2$-index: | $1008$ | ||||
| Genus: | $67 = 1 + \frac{ 1008 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 36 }{2}$ | ||||||
| Cusps: | $36$ (none of which are rational) | Cusp widths | $14^{18}\cdot42^{18}$ | Cusp orbits | $6^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $22$ | ||||||
| $\Q$-gonality: | $13 \le \gamma \le 32$ | ||||||
| $\overline{\Q}$-gonality: | $13 \le \gamma \le 32$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 42.2016.67.36 |
Level structure
| $\GL_2(\Z/42\Z)$-generators: | $\begin{bmatrix}7&38\\24&25\end{bmatrix}$, $\begin{bmatrix}23&40\\0&19\end{bmatrix}$ |
| $\GL_2(\Z/42\Z)$-subgroup: | $C_{48}:C_6$ |
| Contains $-I$: | no $\quad$ (see 42.1008.67.b.1 for the level structure with $-I$) |
| Cyclic 42-isogeny field degree: | $8$ |
| Cyclic 42-torsion field degree: | $96$ |
| Full 42-torsion field degree: | $288$ |
Jacobian
| Conductor: | $2^{54}\cdot3^{86}\cdot7^{134}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{43}\cdot2^{12}$ |
| 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$^{2}$, 441.2.a.a$^{3}$, 441.2.a.c$^{6}$, 441.2.a.f$^{3}$, 588.2.a.a, 882.2.a.b$^{2}$, 882.2.a.d$^{2}$, 882.2.a.f$^{2}$, 882.2.a.i$^{4}$, 882.2.a.k$^{2}$, 882.2.a.l$^{2}$, 1764.2.a.b, 1764.2.a.c, 1764.2.a.g, 1764.2.a.i, 1764.2.a.j$^{2}$, 1764.2.a.k |
Rational points
This modular curve has no $\Q_p$ points for $p=5,13,17,\ldots,401$, 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 |
|---|---|---|---|---|---|---|
| $X(2)$ | $2$ | $336$ | $168$ | $0$ | $0$ | full Jacobian |
| 21.336.9-21.b.1.4 | $21$ | $6$ | $6$ | $9$ | $6$ | $1^{38}\cdot2^{10}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 42.672.23-42.b.1.1 | $42$ | $3$ | $3$ | $23$ | $6$ | $1^{28}\cdot2^{8}$ |
| 42.1008.31-42.b.1.1 | $42$ | $2$ | $2$ | $31$ | $14$ | $1^{24}\cdot2^{6}$ |
| 42.1008.31-42.b.1.8 | $42$ | $2$ | $2$ | $31$ | $14$ | $1^{24}\cdot2^{6}$ |
| 42.1008.34-42.a.1.1 | $42$ | $2$ | $2$ | $34$ | $6$ | $1^{33}$ |
| 42.1008.34-42.a.1.8 | $42$ | $2$ | $2$ | $34$ | $6$ | $1^{33}$ |
| 42.1008.34-42.h.1.2 | $42$ | $2$ | $2$ | $34$ | $10$ | $1^{21}\cdot2^{6}$ |
| 42.1008.34-42.h.1.7 | $42$ | $2$ | $2$ | $34$ | $10$ | $1^{21}\cdot2^{6}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 42.6048.217-42.d.1.1 | $42$ | $3$ | $3$ | $217$ | $65$ | $1^{90}\cdot2^{28}\cdot4$ |