Invariants
| Level: | $42$ | $\SL_2$-level: | $42$ | Newform level: | $42$ | ||
| Index: | $1152$ | $\PSL_2$-index: | $576$ | ||||
| Genus: | $25 = 1 + \frac{ 576 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 48 }{2}$ | ||||||
| Cusps: | $48$ (of which $12$ are rational) | Cusp widths | $1^{6}\cdot2^{6}\cdot3^{6}\cdot6^{6}\cdot7^{6}\cdot14^{6}\cdot21^{6}\cdot42^{6}$ | Cusp orbits | $1^{12}\cdot2^{6}\cdot6^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $6 \le \gamma \le 12$ | ||||||
| $\overline{\Q}$-gonality: | $6 \le \gamma \le 12$ | ||||||
| Rational cusps: | $12$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 42.1152.25.1 |
Level structure
| $\GL_2(\Z/42\Z)$-generators: | $\begin{bmatrix}1&17\\0&13\end{bmatrix}$, $\begin{bmatrix}1&19\\0&1\end{bmatrix}$, $\begin{bmatrix}1&22\\0&17\end{bmatrix}$ |
| $\GL_2(\Z/42\Z)$-subgroup: | $D_6\times F_7$ |
| Contains $-I$: | no $\quad$ (see 42.576.25.c.1 for the level structure with $-I$) |
| Cyclic 42-isogeny field degree: | $1$ |
| Cyclic 42-torsion field degree: | $1$ |
| Full 42-torsion field degree: | $504$ |
Jacobian
| Conductor: | $2^{15}\cdot3^{23}\cdot7^{25}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}\cdot2^{6}\cdot4^{2}$ |
| Newforms: | 14.2.a.a$^{2}$, 21.2.a.a$^{2}$, 21.2.e.a$^{2}$, 21.2.g.a$^{2}$, 42.2.a.a, 42.2.d.a, 42.2.e.a, 42.2.e.b, 42.2.f.a |
Rational points
This modular curve has 12 rational cusps but no known non-cuspidal 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 |
|---|---|---|---|---|---|---|
| $X_0(2)$ | $2$ | $384$ | $192$ | $0$ | $0$ | full Jacobian |
| $X_1(3)$ | $3$ | $144$ | $144$ | $0$ | $0$ | full Jacobian |
| $X_1(7)$ | $7$ | $24$ | $24$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_1(21)$ | $21$ | $3$ | $3$ | $5$ | $0$ | $1^{4}\cdot2^{4}\cdot4^{2}$ |
| 42.384.9-42.c.2.5 | $42$ | $3$ | $3$ | $9$ | $0$ | $2^{6}\cdot4$ |
| 42.576.13-42.a.1.4 | $42$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4^{2}$ |
| 42.576.13-42.a.1.16 | $42$ | $2$ | $2$ | $13$ | $0$ | $2^{2}\cdot4^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_1(2,42)$ | $42$ | $2$ | $2$ | $61$ | $0$ | $1^{6}\cdot2^{11}\cdot4^{2}$ |
| 42.2304.61-42.h.1.4 | $42$ | $2$ | $2$ | $61$ | $1$ | $1^{6}\cdot2^{11}\cdot4^{2}$ |
| 42.2304.61-42.l.1.4 | $42$ | $2$ | $2$ | $61$ | $1$ | $1^{6}\cdot2^{11}\cdot4^{2}$ |
| 42.2304.61-42.m.1.4 | $42$ | $2$ | $2$ | $61$ | $2$ | $1^{6}\cdot2^{11}\cdot4^{2}$ |
| 42.3456.97-42.a.2.2 | $42$ | $3$ | $3$ | $97$ | $0$ | $1^{8}\cdot2^{16}\cdot4^{6}\cdot8$ |
| 42.8064.241-42.c.2.4 | $42$ | $7$ | $7$ | $241$ | $5$ | $1^{24}\cdot2^{36}\cdot4^{10}\cdot8^{4}\cdot16^{3}$ |