Invariants
| Level: | $42$ | $\SL_2$-level: | $42$ | Newform level: | $882$ | ||
| Index: | $1512$ | $\PSL_2$-index: | $756$ | ||||
| Genus: | $52 = 1 + \frac{ 756 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $21^{12}\cdot42^{12}$ | Cusp orbits | $3^{4}\cdot6^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $14$ | ||||||
| $\Q$-gonality: | $10 \le \gamma \le 21$ | ||||||
| $\overline{\Q}$-gonality: | $10 \le \gamma \le 21$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 42.1512.52.3 |
Level structure
| $\GL_2(\Z/42\Z)$-generators: | $\begin{bmatrix}7&18\\24&35\end{bmatrix}$, $\begin{bmatrix}17&6\\12&11\end{bmatrix}$, $\begin{bmatrix}19&21\\0&19\end{bmatrix}$, $\begin{bmatrix}29&21\\6&13\end{bmatrix}$ |
| $\GL_2(\Z/42\Z)$-subgroup: | $C_{48}:C_2^3$ |
| Contains $-I$: | no $\quad$ (see 42.756.52.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: | $384$ |
Jacobian
| Conductor: | $2^{20}\cdot3^{68}\cdot7^{104}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{16}\cdot2^{18}$ |
| Newforms: | 98.2.a.b$^{3}$, 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 no $\Q_p$ points for $p=31,79$, 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 |
|---|---|---|---|---|---|---|
| $X_0(2)$ | $2$ | $504$ | $252$ | $0$ | $0$ | full Jacobian |
| 21.504.16-21.a.1.1 | $21$ | $3$ | $3$ | $16$ | $5$ | $1^{12}\cdot2^{12}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 21.504.16-21.a.1.1 | $21$ | $3$ | $3$ | $16$ | $5$ | $1^{12}\cdot2^{12}$ |
| 42.72.0-6.a.1.1 | $42$ | $21$ | $21$ | $0$ | $0$ | full Jacobian |
| 42.504.16-42.a.1.11 | $42$ | $3$ | $3$ | $16$ | $4$ | $1^{12}\cdot2^{12}$ |
| 42.504.16-42.a.1.15 | $42$ | $3$ | $3$ | $16$ | $4$ | $1^{12}\cdot2^{12}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 42.3024.103-42.a.1.2 | $42$ | $2$ | $2$ | $103$ | $25$ | $1^{43}\cdot2^{4}$ |
| 42.3024.103-42.b.1.1 | $42$ | $2$ | $2$ | $103$ | $22$ | $1^{43}\cdot2^{4}$ |
| 42.3024.103-42.c.1.2 | $42$ | $2$ | $2$ | $103$ | $26$ | $1^{43}\cdot2^{4}$ |
| 42.3024.103-42.d.1.2 | $42$ | $2$ | $2$ | $103$ | $34$ | $1^{43}\cdot2^{4}$ |
| 42.3024.109-42.a.1.2 | $42$ | $2$ | $2$ | $109$ | $27$ | $1^{21}\cdot2^{16}\cdot4$ |
| 42.3024.109-42.cc.1.1 | $42$ | $2$ | $2$ | $109$ | $30$ | $1^{21}\cdot2^{16}\cdot4$ |
| 42.3024.109-42.ch.1.2 | $42$ | $2$ | $2$ | $109$ | $32$ | $1^{53}\cdot2^{2}$ |
| 42.3024.109-42.ci.1.2 | $42$ | $2$ | $2$ | $109$ | $30$ | $1^{53}\cdot2^{2}$ |
| 42.3024.109-42.cp.1.2 | $42$ | $2$ | $2$ | $109$ | $25$ | $1^{53}\cdot2^{2}$ |
| 42.3024.109-42.cq.1.2 | $42$ | $2$ | $2$ | $109$ | $31$ | $1^{53}\cdot2^{2}$ |
| 42.3024.109-42.cr.1.2 | $42$ | $2$ | $2$ | $109$ | $32$ | $1^{21}\cdot2^{16}\cdot4$ |
| 42.3024.109-42.cs.1.1 | $42$ | $2$ | $2$ | $109$ | $33$ | $1^{21}\cdot2^{16}\cdot4$ |