Invariants
| Level: | $37$ | $\SL_2$-level: | $37$ | Newform level: | $1369$ | ||
| Index: | $2109$ | $\PSL_2$-index: | $2109$ | ||||
| Genus: | $142 = 1 + \frac{ 2109 }{12} - \frac{ 9 }{4} - \frac{ 12 }{3} - \frac{ 57 }{2}$ | ||||||
| Cusps: | $57$ (none of which are rational) | Cusp widths | $37^{57}$ | Cusp orbits | $9\cdot12\cdot36$ | ||
| Elliptic points: | $9$ of order $2$ and $12$ of order $3$ | ||||||
| Analytic rank: | $83$ | ||||||
| $\Q$-gonality: | $38 \le \gamma \le 142$ | ||||||
| $\overline{\Q}$-gonality: | $38 \le \gamma \le 142$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 37.2109.142.1 |
| Sutherland (S) label: | 37S4 |
Level structure
| $\GL_2(\Z/37\Z)$-generators: | $\begin{bmatrix}2&25\\33&14\end{bmatrix}$, $\begin{bmatrix}4&14\\32&11\end{bmatrix}$ |
| $\GL_2(\Z/37\Z)$-subgroup: | $\SL(2,3):C_{36}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 37-isogeny field degree: | $6$ |
| Cyclic 37-torsion field degree: | $216$ |
| Full 37-torsion field degree: | $864$ |
Jacobian
| Conductor: | $37^{280}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{17}\cdot2^{4}\cdot3^{6}\cdot18\cdot27^{3}$ |
| Newforms: | 37.2.a.a$^{2}$, 37.2.a.b$^{2}$, 1369.2.a.a$^{2}$, 1369.2.a.b$^{2}$, 1369.2.a.c$^{2}$, 1369.2.a.d$^{3}$, 1369.2.a.e$^{2}$, 1369.2.a.f$^{2}$, 1369.2.a.g$^{2}$, 1369.2.a.h$^{2}$, 1369.2.a.i$^{2}$, 1369.2.a.j$^{2}$, 1369.2.a.k, 1369.2.a.l, 1369.2.a.m, 1369.2.a.n$^{2}$, 1369.2.a.o |
Rational points
This modular curve has real points and $\Q_p$ points for good $p < 8192$, but no known rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X(1)$ | $1$ | $2109$ | $2109$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 37.6327.436.a.1 | $37$ | $3$ | $3$ | $436$ | $227$ | $1^{22}\cdot2^{4}\cdot3^{16}\cdot18^{3}\cdot27^{6}$ |
| 37.8436.586.d.1 | $37$ | $4$ | $4$ | $586$ | $295$ | $1^{37}\cdot2^{10}\cdot3^{18}\cdot18^{5}\cdot27^{9}$ |