Invariants
| Level: | $38$ | $\SL_2$-level: | $38$ | Newform level: | $1444$ | ||
| Index: | $1710$ | $\PSL_2$-index: | $1710$ | ||||
| Genus: | $121 = 1 + \frac{ 1710 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 45 }{2}$ | ||||||
| Cusps: | $45$ (none of which are rational) | Cusp widths | $38^{45}$ | Cusp orbits | $6^{3}\cdot9\cdot18$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $52$ | ||||||
| $\Q$-gonality: | $29 \le \gamma \le 84$ | ||||||
| $\overline{\Q}$-gonality: | $29 \le \gamma \le 84$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 38.1710.121.5 |
Level structure
| $\GL_2(\Z/38\Z)$-generators: | $\begin{bmatrix}19&7\\24&5\end{bmatrix}$, $\begin{bmatrix}25&1\\32&13\end{bmatrix}$ |
| $\GL_2(\Z/38\Z)$-subgroup: | $C_9\times \GL(2,3)$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 38-isogeny field degree: | $8$ |
| Cyclic 38-torsion field degree: | $144$ |
| Full 38-torsion field degree: | $432$ |
Jacobian
| Conductor: | $2^{102}\cdot19^{234}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{32}\cdot2^{16}\cdot3^{5}\cdot4^{7}\cdot6\cdot8$ |
| Newforms: | 19.2.a.a$^{3}$, 38.2.a.a$^{2}$, 38.2.a.b$^{2}$, 76.2.a.a, 361.2.a.a$^{3}$, 361.2.a.b$^{3}$, 361.2.a.c$^{2}$, 361.2.a.d, 361.2.a.e$^{2}$, 361.2.a.f, 361.2.a.g$^{2}$, 361.2.a.h, 361.2.a.i$^{3}$, 722.2.a.a$^{2}$, 722.2.a.b$^{2}$, 722.2.a.c$^{3}$, 722.2.a.d$^{3}$, 722.2.a.e$^{2}$, 722.2.a.f$^{2}$, 722.2.a.g$^{3}$, 722.2.a.h, 722.2.a.i, 722.2.a.j$^{3}$, 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.b, 1444.2.a.c$^{2}$, 1444.2.a.d, 1444.2.a.e, 1444.2.a.g, 1444.2.a.i |
Rational points
This modular curve has no $\Q_p$ points for $p=5,73,149$, and therefore no 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 | $38$ | $285$ | $285$ | $0$ | $0$ | full Jacobian |
| 38.570.37.b.1 | $38$ | $3$ | $3$ | $37$ | $19$ | $1^{24}\cdot2^{12}\cdot3^{4}\cdot4^{6}$ |
| 38.855.56.a.1 | $38$ | $2$ | $2$ | $56$ | $27$ | $1^{17}\cdot2^{8}\cdot3^{2}\cdot4^{3}\cdot6\cdot8$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 38.5130.361.c.1 | $38$ | $3$ | $3$ | $361$ | $148$ | $1^{36}\cdot2^{24}\cdot3^{20}\cdot4^{14}\cdot6^{4}\cdot8^{2}$ |
| 38.6840.481.r.1 | $38$ | $4$ | $4$ | $481$ | $185$ | $1^{69}\cdot2^{57}\cdot3^{25}\cdot4^{14}\cdot6^{5}\cdot8^{2}$ |