Invariants
| Level: | $38$ | $\SL_2$-level: | $38$ | Newform level: | $19$ | ||
| Index: | $40$ | $\PSL_2$-index: | $20$ | ||||
| Genus: | $1 = 1 + \frac{ 20 }{12} - \frac{ 0 }{4} - \frac{ 2 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $1\cdot19$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $2$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-19$) |
Other labels
| Cummins and Pauli (CP) label: | 19A1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 38.40.1.1 |
Level structure
| $\GL_2(\Z/38\Z)$-generators: | $\begin{bmatrix}9&24\\1&13\end{bmatrix}$, $\begin{bmatrix}15&31\\15&4\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 19.20.1.a.1 for the level structure with $-I$) |
| Cyclic 38-isogeny field degree: | $3$ |
| Cyclic 38-torsion field degree: | $54$ |
| Full 38-torsion field degree: | $18468$ |
Jacobian
| Conductor: | $19$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 19.2.a.a |
Models
Weierstrass model Weierstrass model
| $ y^{2} + y $ | $=$ | $ x^{3} + x^{2} - 9x - 15 $ |
Rational points
This modular curve has 2 rational cusps and 1 rational CM point, but no other known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve | CM | $j$-invariant | $j$-height | Weierstrass model | |
|---|---|---|---|---|---|
| no | $\infty$ | $0.000$ | $(0:1:0)$, $(5:-10:1)$ | ||
| 361.a1 | $-19$ | $-884736$ | $= -1 \cdot 2^{15} \cdot 3^{3}$ | $13.693$ | $(5:9:1)$ |
Maps to other modular curves
$j$-invariant map of degree 20 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{18x^{2}y^{5}+178x^{2}y^{4}z-1916x^{2}y^{3}z^{2}-9740x^{2}y^{2}z^{3}-5014x^{2}yz^{4}+90x^{2}z^{5}-141xy^{5}z-1122xy^{4}z^{2}+2722xy^{3}z^{3}+16168xy^{2}z^{4}+11283xyz^{5}+3858xz^{6}-y^{7}-13y^{6}z+582y^{5}z^{2}+3739y^{4}z^{3}+13703y^{3}z^{4}+50629y^{2}z^{5}+17260yz^{6}-12171z^{7}}{z^{6}(x-5z)}$ |
Modular covers
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 38.80.2-38.a.1.1 | $38$ | $2$ | $2$ | $2$ | $0$ | $1$ |
| 38.80.2-38.b.1.2 | $38$ | $2$ | $2$ | $2$ | $1$ | $1$ |
| 38.120.1-19.a.1.1 | $38$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 38.120.1-19.a.2.1 | $38$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 38.120.1-19.b.1.2 | $38$ | $3$ | $3$ | $1$ | $0$ | dimension zero |
| 38.120.4-38.a.1.3 | $38$ | $3$ | $3$ | $4$ | $0$ | $1^{3}$ |
| 38.760.22-19.a.1.1 | $38$ | $19$ | $19$ | $22$ | $8$ | $1^{3}\cdot2^{4}\cdot3^{2}\cdot4$ |
| 76.80.2-76.a.1.5 | $76$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 76.80.2-76.b.1.3 | $76$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 76.160.6-76.a.1.2 | $76$ | $4$ | $4$ | $6$ | $?$ | not computed |
| 114.80.2-114.a.1.3 | $114$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 114.80.2-114.b.1.1 | $114$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 114.120.5-57.a.1.6 | $114$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 114.160.5-57.a.1.5 | $114$ | $4$ | $4$ | $5$ | $?$ | not computed |
| 152.80.2-152.a.1.5 | $152$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 152.80.2-152.b.1.5 | $152$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 152.80.2-152.c.1.5 | $152$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 152.80.2-152.d.1.5 | $152$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 190.80.2-190.a.1.1 | $190$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 190.80.2-190.b.1.3 | $190$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 190.200.7-95.a.1.3 | $190$ | $5$ | $5$ | $7$ | $?$ | not computed |
| 190.240.9-95.a.1.4 | $190$ | $6$ | $6$ | $9$ | $?$ | not computed |
| 190.400.15-95.a.1.4 | $190$ | $10$ | $10$ | $15$ | $?$ | not computed |
| 228.80.2-228.a.1.6 | $228$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 228.80.2-228.b.1.6 | $228$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 266.80.2-266.a.1.2 | $266$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 266.80.2-266.b.1.1 | $266$ | $2$ | $2$ | $2$ | $?$ | not computed |
| 266.120.1-133.a.1.3 | $266$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 266.120.1-133.a.2.3 | $266$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 266.120.1-133.b.1.1 | $266$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 266.120.1-133.b.2.1 | $266$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 266.120.1-133.c.1.1 | $266$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 266.120.1-133.c.2.1 | $266$ | $3$ | $3$ | $1$ | $?$ | dimension zero |
| 266.320.11-133.a.1.6 | $266$ | $8$ | $8$ | $11$ | $?$ | not computed |