Invariants
| Level: | $42$ | $\SL_2$-level: | $42$ | Newform level: | $882$ | ||
| Index: | $2016$ | $\PSL_2$-index: | $1008$ | ||||
| Genus: | $69 = 1 + \frac{ 1008 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 32 }{2}$ | ||||||
| Cusps: | $32$ (of which $4$ are rational) | Cusp widths | $21^{16}\cdot42^{16}$ | Cusp orbits | $1^{4}\cdot2^{2}\cdot3^{4}\cdot6^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $14$ | ||||||
| $\Q$-gonality: | $14 \le \gamma \le 28$ | ||||||
| $\overline{\Q}$-gonality: | $14 \le \gamma \le 28$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 42.2016.69.3 |
Level structure
| $\GL_2(\Z/42\Z)$-generators: | $\begin{bmatrix}1&27\\6&31\end{bmatrix}$, $\begin{bmatrix}7&12\\30&35\end{bmatrix}$, $\begin{bmatrix}13&0\\0&41\end{bmatrix}$, $\begin{bmatrix}23&24\\0&19\end{bmatrix}$ |
| $\GL_2(\Z/42\Z)$-subgroup: | $C_6^2:D_4$ |
| Contains $-I$: | no $\quad$ (see 42.1008.69.a.1 for the level structure with $-I$) |
| Cyclic 42-isogeny field degree: | $2$ |
| Cyclic 42-torsion field degree: | $24$ |
| Full 42-torsion field degree: | $288$ |
Jacobian
| Conductor: | $2^{27}\cdot3^{90}\cdot7^{121}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{29}\cdot2^{20}$ |
| Newforms: | 14.2.a.a$^{3}$, 21.2.a.a$^{4}$, 42.2.a.a$^{2}$, 63.2.a.a$^{2}$, 63.2.a.b$^{2}$, 98.2.a.b$^{3}$, 126.2.a.a, 126.2.a.b, 147.2.a.c$^{4}$, 147.2.a.d$^{4}$, 147.2.a.e$^{4}$, 294.2.a.d$^{2}$, 294.2.a.e$^{2}$, 441.2.a.b$^{2}$, 441.2.a.d$^{2}$, 441.2.a.i$^{2}$, 441.2.a.j$^{2}$, 882.2.a.a, 882.2.a.c, 882.2.a.g, 882.2.a.j, 882.2.a.m, 882.2.a.n, 882.2.a.o |
Rational points
This modular curve has 4 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 |
|---|---|---|---|---|---|---|
| $X_0(2)$ | $2$ | $672$ | $336$ | $0$ | $0$ | full Jacobian |
| 21.672.21-21.a.1.2 | $21$ | $3$ | $3$ | $21$ | $5$ | $1^{22}\cdot2^{13}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 21.672.21-21.a.1.2 | $21$ | $3$ | $3$ | $21$ | $5$ | $1^{22}\cdot2^{13}$ |
| 42.72.0-6.a.1.1 | $42$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
| 42.672.21-42.a.1.11 | $42$ | $3$ | $3$ | $21$ | $4$ | $1^{20}\cdot2^{14}$ |
| 42.672.21-42.a.1.14 | $42$ | $3$ | $3$ | $21$ | $4$ | $1^{20}\cdot2^{14}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 42.4032.137-42.a.1.3 | $42$ | $2$ | $2$ | $137$ | $25$ | $1^{56}\cdot2^{6}$ |
| 42.4032.137-42.b.1.1 | $42$ | $2$ | $2$ | $137$ | $25$ | $1^{56}\cdot2^{6}$ |
| 42.4032.137-42.c.1.2 | $42$ | $2$ | $2$ | $137$ | $32$ | $1^{56}\cdot2^{6}$ |
| 42.4032.137-42.d.1.2 | $42$ | $2$ | $2$ | $137$ | $34$ | $1^{56}\cdot2^{6}$ |
| 42.4032.145-42.a.1.2 | $42$ | $2$ | $2$ | $145$ | $28$ | $1^{38}\cdot2^{17}\cdot4$ |
| 42.4032.145-42.db.1.2 | $42$ | $2$ | $2$ | $145$ | $35$ | $1^{38}\cdot2^{17}\cdot4$ |
| 42.4032.145-42.dr.1.2 | $42$ | $2$ | $2$ | $145$ | $34$ | $1^{70}\cdot2^{3}$ |
| 42.4032.145-42.ds.1.2 | $42$ | $2$ | $2$ | $145$ | $36$ | $1^{70}\cdot2^{3}$ |
| 42.4032.145-42.dy.1.2 | $42$ | $2$ | $2$ | $145$ | $30$ | $1^{70}\cdot2^{3}$ |
| 42.4032.145-42.dz.1.2 | $42$ | $2$ | $2$ | $145$ | $32$ | $1^{70}\cdot2^{3}$ |
| 42.4032.145-42.eg.1.1 | $42$ | $2$ | $2$ | $145$ | $38$ | $1^{38}\cdot2^{17}\cdot4$ |
| 42.4032.145-42.eh.1.2 | $42$ | $2$ | $2$ | $145$ | $35$ | $1^{38}\cdot2^{17}\cdot4$ |
| 42.6048.205-42.b.1.1 | $42$ | $3$ | $3$ | $205$ | $43$ | $1^{88}\cdot2^{24}$ |