Invariants
| Level: | $42$ | $\SL_2$-level: | $42$ | Newform level: | $294$ | ||
| Index: | $672$ | $\PSL_2$-index: | $336$ | ||||
| Genus: | $21 = 1 + \frac{ 336 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $4$ are rational) | Cusp widths | $7^{4}\cdot14^{4}\cdot21^{4}\cdot42^{4}$ | Cusp orbits | $1^{4}\cdot3^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $4$ | ||||||
| $\Q$-gonality: | $5 \le \gamma \le 12$ | ||||||
| $\overline{\Q}$-gonality: | $5 \le \gamma \le 12$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-3,-12$) |
Other labels
| Cummins and Pauli (CP) label: | 42C21 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 42.672.21.5 |
Level structure
| $\GL_2(\Z/42\Z)$-generators: | $\begin{bmatrix}13&9\\0&29\end{bmatrix}$, $\begin{bmatrix}25&9\\18&17\end{bmatrix}$, $\begin{bmatrix}29&40\\12&19\end{bmatrix}$, $\begin{bmatrix}41&13\\6&29\end{bmatrix}$ |
| $\GL_2(\Z/42\Z)$-subgroup: | $D_6^2:C_6$ |
| Contains $-I$: | no $\quad$ (see 42.336.21.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: | $864$ |
Jacobian
| Conductor: | $2^{9}\cdot3^{15}\cdot7^{37}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{9}\cdot2^{6}$ |
| Newforms: | 14.2.a.a$^{2}$, 21.2.a.a$^{2}$, 42.2.a.a, 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 4 rational cusps and 2 rational CM points, but no other 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 | $6$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
| $X_{\mathrm{sp}}^+(7)$ | $7$ | $24$ | $12$ | $0$ | $0$ | 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 | $6$ | $28$ | $28$ | $0$ | $0$ | full Jacobian |
| 42.224.6-21.a.1.2 | $42$ | $3$ | $3$ | $6$ | $2$ | $1^{7}\cdot2^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 42.1344.41-42.a.1.8 | $42$ | $2$ | $2$ | $41$ | $5$ | $1^{20}$ |
| 42.1344.41-42.b.1.2 | $42$ | $2$ | $2$ | $41$ | $14$ | $1^{20}$ |
| 42.1344.41-42.c.1.8 | $42$ | $2$ | $2$ | $41$ | $5$ | $1^{20}$ |
| 42.1344.41-42.d.1.3 | $42$ | $2$ | $2$ | $41$ | $12$ | $1^{20}$ |
| 42.1344.45-42.a.1.8 | $42$ | $2$ | $2$ | $45$ | $6$ | $1^{12}\cdot2^{6}$ |
| 42.1344.45-42.g.1.6 | $42$ | $2$ | $2$ | $45$ | $13$ | $1^{12}\cdot2^{6}$ |
| 42.1344.45-42.h.1.4 | $42$ | $2$ | $2$ | $45$ | $9$ | $1^{12}\cdot2^{6}$ |
| 42.1344.45-42.i.1.2 | $42$ | $2$ | $2$ | $45$ | $12$ | $1^{12}\cdot2^{6}$ |
| 42.1344.45-42.n.1.3 | $42$ | $2$ | $2$ | $45$ | $13$ | $1^{24}$ |
| 42.1344.45-42.o.1.3 | $42$ | $2$ | $2$ | $45$ | $7$ | $1^{24}$ |
| 42.1344.45-42.q.1.1 | $42$ | $2$ | $2$ | $45$ | $11$ | $1^{24}$ |
| 42.1344.45-42.r.1.2 | $42$ | $2$ | $2$ | $45$ | $9$ | $1^{24}$ |
| 42.2016.61-42.b.1.2 | $42$ | $3$ | $3$ | $61$ | $9$ | $1^{28}\cdot2^{6}$ |
| 42.2016.69-42.a.1.7 | $42$ | $3$ | $3$ | $69$ | $14$ | $1^{20}\cdot2^{14}$ |