Invariants
| Level: | $38$ | $\SL_2$-level: | $38$ | Newform level: | $1444$ | ||
| Index: | $5130$ | $\PSL_2$-index: | $5130$ | ||||
| Genus: | $361 = 1 + \frac{ 5130 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 135 }{2}$ | ||||||
| Cusps: | $135$ (none of which are rational) | Cusp widths | $38^{135}$ | Cusp orbits | $9\cdot18^{7}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $148$ | ||||||
| $\Q$-gonality: | $86 \le \gamma \le 240$ | ||||||
| $\overline{\Q}$-gonality: | $86 \le \gamma \le 240$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 38.5130.361.3 |
Level structure
| $\GL_2(\Z/38\Z)$-generators: | $\begin{bmatrix}19&25\\20&19\end{bmatrix}$, $\begin{bmatrix}21&31\\36&33\end{bmatrix}$ |
| $\GL_2(\Z/38\Z)$-subgroup: | $C_9\times \SD_{16}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 38-isogeny field degree: | $4$ |
| Cyclic 38-torsion field degree: | $72$ |
| Full 38-torsion field degree: | $144$ |
Jacobian
| Conductor: | $2^{298}\cdot19^{706}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{68}\cdot2^{40}\cdot3^{25}\cdot4^{21}\cdot6^{5}\cdot8^{3}$ |
| Newforms: | 19.2.a.a$^{6}$, 38.2.a.a$^{4}$, 38.2.a.b$^{4}$, 76.2.a.a$^{2}$, 361.2.a.a$^{9}$, 361.2.a.b$^{6}$, 361.2.a.c$^{5}$, 361.2.a.d$^{4}$, 361.2.a.e$^{5}$, 361.2.a.f$^{4}$, 361.2.a.g$^{8}$, 361.2.a.h$^{7}$, 361.2.a.i$^{9}$, 722.2.a.a$^{6}$, 722.2.a.b$^{4}$, 722.2.a.c$^{5}$, 722.2.a.d$^{5}$, 722.2.a.e$^{4}$, 722.2.a.f$^{6}$, 722.2.a.g$^{5}$, 722.2.a.h$^{3}$, 722.2.a.i$^{3}$, 722.2.a.j$^{5}$, 722.2.a.k$^{5}$, 722.2.a.l$^{5}$, 722.2.a.m$^{6}$, 722.2.a.n$^{6}$, 1444.2.a.a$^{2}$, 1444.2.a.b$^{2}$, 1444.2.a.c$^{3}$, 1444.2.a.d$^{2}$, 1444.2.a.e$^{3}$, 1444.2.a.f, 1444.2.a.g$^{3}$, 1444.2.a.h$^{2}$, 1444.2.a.i$^{3}$ |
Rational points
This modular curve has no $\Q_p$ points for $p=5,17,61,\ldots,353$, 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 |
|---|---|---|---|---|---|---|
| 38.1026.73.c.1 | $38$ | $5$ | $5$ | $73$ | $40$ | $1^{58}\cdot2^{37}\cdot3^{20}\cdot4^{14}\cdot6^{4}\cdot8^{2}$ |
| 38.1710.115.a.1 | $38$ | $3$ | $3$ | $115$ | $50$ | $1^{46}\cdot2^{28}\cdot3^{16}\cdot4^{14}\cdot6^{4}\cdot8^{2}$ |
| 38.1710.121.d.1 | $38$ | $3$ | $3$ | $121$ | $52$ | $1^{36}\cdot2^{24}\cdot3^{20}\cdot4^{14}\cdot6^{4}\cdot8^{2}$ |
| 38.1710.121.e.1 | $38$ | $3$ | $3$ | $121$ | $53$ | $1^{52}\cdot2^{28}\cdot3^{20}\cdot4^{18}$ |
| 38.2565.166.a.1 | $38$ | $2$ | $2$ | $166$ | $74$ | $1^{35}\cdot2^{20}\cdot3^{10}\cdot4^{9}\cdot6^{5}\cdot8^{3}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 38.10260.721.e.1 | $38$ | $2$ | $2$ | $721$ | $270$ | $1^{69}\cdot2^{57}\cdot3^{25}\cdot4^{14}\cdot6^{5}\cdot8^{2}$ |
| 38.10260.721.f.1 | $38$ | $2$ | $2$ | $721$ | $281$ | $1^{69}\cdot2^{57}\cdot3^{25}\cdot4^{14}\cdot6^{5}\cdot8^{2}$ |