Invariants
| Level: | $38$ | $\SL_2$-level: | $38$ | Newform level: | $1444$ | ||
| Index: | $3420$ | $\PSL_2$-index: | $3420$ | ||||
| Genus: | $241 = 1 + \frac{ 3420 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 90 }{2}$ | ||||||
| Cusps: | $90$ (none of which are rational) | Cusp widths | $38^{90}$ | Cusp orbits | $3\cdot6\cdot9\cdot18^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $101$ | ||||||
| $\Q$-gonality: | $57 \le \gamma \le 162$ | ||||||
| $\overline{\Q}$-gonality: | $57 \le \gamma \le 162$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 38.3420.241.7 |
Level structure
| $\GL_2(\Z/38\Z)$-generators: | $\begin{bmatrix}27&20\\34&11\end{bmatrix}$, $\begin{bmatrix}35&36\\11&35\end{bmatrix}$ |
| $\GL_2(\Z/38\Z)$-subgroup: | $D_6:C_{18}$ |
| 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: | $216$ |
Jacobian
| Conductor: | $2^{200}\cdot19^{470}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{50}\cdot2^{28}\cdot3^{15}\cdot4^{14}\cdot6^{3}\cdot8^{2}$ |
| Newforms: | 19.2.a.a$^{5}$, 38.2.a.a$^{3}$, 38.2.a.b$^{3}$, 76.2.a.a, 361.2.a.a$^{6}$, 361.2.a.b$^{4}$, 361.2.a.c$^{3}$, 361.2.a.d$^{3}$, 361.2.a.e$^{3}$, 361.2.a.f$^{3}$, 361.2.a.g$^{5}$, 361.2.a.h$^{4}$, 361.2.a.i$^{6}$, 722.2.a.a$^{4}$, 722.2.a.b$^{3}$, 722.2.a.c$^{4}$, 722.2.a.d$^{4}$, 722.2.a.e$^{3}$, 722.2.a.f$^{4}$, 722.2.a.g$^{4}$, 722.2.a.h$^{2}$, 722.2.a.i$^{2}$, 722.2.a.j$^{4}$, 722.2.a.k$^{3}$, 722.2.a.l$^{3}$, 722.2.a.m$^{4}$, 722.2.a.n$^{4}$, 1444.2.a.a$^{2}$, 1444.2.a.b, 1444.2.a.c$^{3}$, 1444.2.a.d, 1444.2.a.e$^{2}$, 1444.2.a.f, 1444.2.a.g$^{2}$, 1444.2.a.h, 1444.2.a.i$^{2}$ |
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 |
|---|---|---|---|---|---|---|
| 38.1140.79.b.1 | $38$ | $3$ | $3$ | $79$ | $37$ | $1^{38}\cdot2^{20}\cdot3^{12}\cdot4^{12}$ |
| 38.1140.81.c.1 | $38$ | $3$ | $3$ | $81$ | $42$ | $1^{32}\cdot2^{25}\cdot3^{10}\cdot4^{7}\cdot6^{2}\cdot8$ |
| 38.1710.111.a.1 | $38$ | $2$ | $2$ | $111$ | $49$ | $1^{26}\cdot2^{14}\cdot3^{6}\cdot4^{6}\cdot6^{3}\cdot8^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 38.6840.481.q.1 | $38$ | $2$ | $2$ | $481$ | $176$ | $1^{51}\cdot2^{45}\cdot3^{15}\cdot4^{7}\cdot6^{3}\cdot8$ |
| 38.6840.481.r.1 | $38$ | $2$ | $2$ | $481$ | $185$ | $1^{51}\cdot2^{45}\cdot3^{15}\cdot4^{7}\cdot6^{3}\cdot8$ |
| 38.10260.721.f.1 | $38$ | $3$ | $3$ | $721$ | $281$ | $1^{87}\cdot2^{69}\cdot3^{35}\cdot4^{21}\cdot6^{7}\cdot8^{3}$ |