Invariants
| Level: | $38$ | $\SL_2$-level: | $38$ | Newform level: | $1444$ | ||
| Index: | $1140$ | $\PSL_2$-index: | $1140$ | ||||
| Genus: | $81 = 1 + \frac{ 1140 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 30 }{2}$ | ||||||
| Cusps: | $30$ (of which $1$ is rational) | Cusp widths | $38^{30}$ | Cusp orbits | $1\cdot2\cdot9\cdot18$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $42$ | ||||||
| $\Q$-gonality: | $20 \le \gamma \le 54$ | ||||||
| $\overline{\Q}$-gonality: | $20 \le \gamma \le 54$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 38.1140.81.3 |
Level structure
| $\GL_2(\Z/38\Z)$-generators: | $\begin{bmatrix}18&25\\33&20\end{bmatrix}$, $\begin{bmatrix}36&7\\13&12\end{bmatrix}$ |
| $\GL_2(\Z/38\Z)$-subgroup: | $C_{18}\wr C_2$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 38-isogeny field degree: | $2$ |
| Cyclic 38-torsion field degree: | $36$ |
| Full 38-torsion field degree: | $648$ |
Jacobian
| Conductor: | $2^{72}\cdot19^{158}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{18}\cdot2^{3}\cdot3^{5}\cdot4^{7}\cdot6\cdot8$ |
| Newforms: | 19.2.a.a$^{2}$, 38.2.a.a, 38.2.a.b, 361.2.a.a$^{3}$, 361.2.a.b, 361.2.a.g$^{2}$, 361.2.a.h, 361.2.a.i$^{3}$, 722.2.a.a$^{2}$, 722.2.a.b, 722.2.a.c, 722.2.a.d, 722.2.a.e, 722.2.a.f$^{2}$, 722.2.a.g, 722.2.a.j, 722.2.a.k, 722.2.a.l, 722.2.a.m$^{2}$, 722.2.a.n$^{2}$, 1444.2.a.a, 1444.2.a.c, 1444.2.a.e, 1444.2.a.g, 1444.2.a.i |
Rational points
This modular curve has 1 rational cusp but no known non-cuspidal rational points.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 38.6.0.b.1 | $38$ | $190$ | $190$ | $0$ | $0$ | full Jacobian |
| 38.380.27.b.1 | $38$ | $3$ | $3$ | $27$ | $16$ | $1^{14}\cdot2^{2}\cdot3^{4}\cdot4^{6}$ |
| 38.570.36.a.1 | $38$ | $2$ | $2$ | $36$ | $22$ | $1^{9}\cdot2^{2}\cdot3^{2}\cdot4^{3}\cdot6\cdot8$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 38.2280.161.g.1 | $38$ | $2$ | $2$ | $161$ | $58$ | $1^{19}\cdot2^{20}\cdot3^{5}\cdot6$ |
| 38.2280.161.h.1 | $38$ | $2$ | $2$ | $161$ | $68$ | $1^{19}\cdot2^{20}\cdot3^{5}\cdot6$ |
| 38.3420.241.f.1 | $38$ | $3$ | $3$ | $241$ | $101$ | $1^{32}\cdot2^{25}\cdot3^{10}\cdot4^{7}\cdot6^{2}\cdot8$ |