Invariants
| Level: | $38$ | $\SL_2$-level: | $38$ | Newform level: | $722$ | ||
| Index: | $855$ | $\PSL_2$-index: | $855$ | ||||
| Genus: | $56 = 1 + \frac{ 855 }{12} - \frac{ 5 }{4} - \frac{ 0 }{3} - \frac{ 30 }{2}$ | ||||||
| Cusps: | $30$ (none of which are rational) | Cusp widths | $19^{15}\cdot38^{15}$ | Cusp orbits | $6^{2}\cdot9^{2}$ | ||
| Elliptic points: | $5$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $27$ | ||||||
| $\Q$-gonality: | $15 \le \gamma \le 42$ | ||||||
| $\overline{\Q}$-gonality: | $15 \le \gamma \le 42$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 38.855.56.1 |
Level structure
| $\GL_2(\Z/38\Z)$-generators: | $\begin{bmatrix}3&30\\14&33\end{bmatrix}$, $\begin{bmatrix}15&17\\4&19\end{bmatrix}$ |
| $\GL_2(\Z/38\Z)$-subgroup: | $C_{18}\times \GL(2,3)$ |
| 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: | $864$ |
Jacobian
| Conductor: | $2^{28}\cdot19^{108}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{15}\cdot2^{8}\cdot3^{3}\cdot4^{4}$ |
| Newforms: | 19.2.a.a$^{2}$, 38.2.a.a, 38.2.a.b, 361.2.a.a$^{2}$, 361.2.a.b$^{2}$, 361.2.a.c$^{2}$, 361.2.a.e$^{2}$, 361.2.a.g$^{2}$, 361.2.a.i$^{2}$, 722.2.a.a, 722.2.a.b, 722.2.a.c$^{2}$, 722.2.a.d, 722.2.a.e, 722.2.a.f, 722.2.a.g, 722.2.a.i, 722.2.a.j$^{2}$, 722.2.a.l, 722.2.a.m, 722.2.a.n |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no 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 |
|---|---|---|---|---|---|---|
| $X_0(2)$ | $2$ | $285$ | $285$ | $0$ | $0$ | full Jacobian |
| $X_{S_4}(19)$ | $19$ | $3$ | $3$ | $14$ | $8$ | $1^{12}\cdot2^{6}\cdot3^{2}\cdot4^{3}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_0(2)$ | $2$ | $285$ | $285$ | $0$ | $0$ | full Jacobian |
| $X_{S_4}(19)$ | $19$ | $3$ | $3$ | $14$ | $8$ | $1^{12}\cdot2^{6}\cdot3^{2}\cdot4^{3}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 38.1710.121.a.1 | $38$ | $2$ | $2$ | $121$ | $50$ | $1^{17}\cdot2^{8}\cdot3^{2}\cdot4^{3}\cdot6\cdot8$ |
| 38.1710.121.d.1 | $38$ | $2$ | $2$ | $121$ | $52$ | $1^{17}\cdot2^{8}\cdot3^{2}\cdot4^{3}\cdot6\cdot8$ |
| 38.2565.166.a.1 | $38$ | $3$ | $3$ | $166$ | $74$ | $1^{18}\cdot2^{12}\cdot3^{12}\cdot4^{8}$ |
| 38.3420.226.c.1 | $38$ | $4$ | $4$ | $226$ | $86$ | $1^{33}\cdot2^{30}\cdot3^{15}\cdot4^{8}$ |