Invariants
| Level: | $38$ | $\SL_2$-level: | $2$ | ||||
| Index: | $6$ | $\PSL_2$-index: | $6$ | ||||
| Genus: | $0 = 1 + \frac{ 6 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 3 }{2}$ | ||||||
| Cusps: | $3$ (of which $1$ is rational) | Cusp widths | $2^{3}$ | Cusp orbits | $1\cdot2$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $1$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 2C0 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 38.6.0.2 |
Level structure
| $\GL_2(\Z/38\Z)$-generators: | $\begin{bmatrix}12&31\\9&20\end{bmatrix}$, $\begin{bmatrix}35&26\\8&11\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 38-isogeny field degree: | $20$ |
| Cyclic 38-torsion field degree: | $360$ |
| Full 38-torsion field degree: | $123120$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 1262 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 6 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{2^{11}}{19}\cdot\frac{x^{6}(1792x^{2}+232xy+7y^{2})^{3}}{x^{6}(16x+y)^{2}(1280x^{2}+144xy+5y^{2})^{2}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| $X_0(2)$ | $2$ | $2$ | $2$ | $0$ | $0$ |
| 38.2.0.a.1 | $38$ | $3$ | $3$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus |
|---|---|---|---|---|
| 38.120.8.e.1 | $38$ | $20$ | $20$ | $8$ |
| 38.1026.73.c.1 | $38$ | $171$ | $171$ | $73$ |
| 38.1140.81.c.1 | $38$ | $190$ | $190$ | $81$ |
| 38.1710.121.d.1 | $38$ | $285$ | $285$ | $121$ |
| 76.12.0.e.1 | $76$ | $2$ | $2$ | $0$ |
| 76.12.0.g.1 | $76$ | $2$ | $2$ | $0$ |
| 76.12.0.i.1 | $76$ | $2$ | $2$ | $0$ |
| 76.12.0.k.1 | $76$ | $2$ | $2$ | $0$ |
| 114.18.1.b.1 | $114$ | $3$ | $3$ | $1$ |
| 114.24.0.b.1 | $114$ | $4$ | $4$ | $0$ |
| 152.12.0.n.1 | $152$ | $2$ | $2$ | $0$ |
| 152.12.0.t.1 | $152$ | $2$ | $2$ | $0$ |
| 152.12.0.bc.1 | $152$ | $2$ | $2$ | $0$ |
| 152.12.0.bi.1 | $152$ | $2$ | $2$ | $0$ |
| 190.30.2.b.1 | $190$ | $5$ | $5$ | $2$ |
| 190.36.1.b.1 | $190$ | $6$ | $6$ | $1$ |
| 190.60.3.j.1 | $190$ | $10$ | $10$ | $3$ |
| 228.12.0.m.1 | $228$ | $2$ | $2$ | $0$ |
| 228.12.0.o.1 | $228$ | $2$ | $2$ | $0$ |
| 228.12.0.z.1 | $228$ | $2$ | $2$ | $0$ |
| 228.12.0.bb.1 | $228$ | $2$ | $2$ | $0$ |
| 266.48.2.n.1 | $266$ | $8$ | $8$ | $2$ |
| 266.126.7.d.1 | $266$ | $21$ | $21$ | $7$ |
| 266.168.9.j.1 | $266$ | $28$ | $28$ | $9$ |