Invariants
| Level: | $38$ | $\SL_2$-level: | $38$ | Newform level: | $1444$ | ||
| Index: | $80$ | $\PSL_2$-index: | $40$ | ||||
| Genus: | $2 = 1 + \frac{ 40 }{12} - \frac{ 0 }{4} - \frac{ 4 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $2\cdot38$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $4$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-19$) |
Other labels
| Cummins and Pauli (CP) label: | 38A2 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 38.80.2.3 |
Level structure
| $\GL_2(\Z/38\Z)$-generators: | $\begin{bmatrix}6&25\\21&11\end{bmatrix}$, $\begin{bmatrix}29&16\\33&35\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 38.40.2.b.1 for the level structure with $-I$) |
| Cyclic 38-isogeny field degree: | $3$ |
| Cyclic 38-torsion field degree: | $54$ |
| Full 38-torsion field degree: | $9234$ |
Jacobian
| Conductor: | $2^{2}\cdot19^{3}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{2}$ |
| Newforms: | 19.2.a.a, 1444.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 19 y^{2} w + 2 z^{2} w + z w^{2} $ |
| $=$ | $19 y^{2} z + 2 z^{3} + z^{2} w$ | |
| $=$ | $19 y^{3} + 2 y z^{2} + y z w$ | |
| $=$ | $19 x y^{2} + 2 x z^{2} + x z w$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 19 x^{3} y + 8 x y z^{2} + y^{2} z^{2} - z^{4} $ |
Weierstrass model Weierstrass model
| $ y^{2} + y $ | $=$ | $ x^{6} + 16x^{4} + 76x^{2} + 90 $ |
Rational points
This modular curve has 2 rational cusps and 1 rational CM point, but no other known rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:0:0:1)$, $(1:0:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 40 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{47045881x^{5}w^{4}-5020788x^{3}w^{6}+238888000xz^{8}-17153600xz^{7}w-2253352080xz^{6}w^{2}-585430112xz^{5}w^{3}+4604903451xz^{4}w^{4}+2643681816xz^{3}w^{5}-1427196837xz^{2}w^{6}-877651491xzw^{7}+182150xw^{8}+38160000yz^{8}+21081600yz^{7}w-412292160yz^{6}w^{2}-150327360yz^{5}w^{3}+933802716yz^{4}w^{4}+453509962yz^{3}w^{5}-470640960yz^{2}w^{6}-188183524yzw^{7}+47045881yw^{8}}{6859x^{3}w^{6}-29861xz^{8}-48456xz^{7}w+22956xz^{6}w^{2}+23274xz^{5}w^{3}-10684xz^{4}w^{4}-1612xz^{3}w^{5}+1652xz^{2}w^{6}-324xzw^{7}+20xw^{8}-4770yz^{8}-4101yz^{7}w+4032yz^{6}w^{2}+496yz^{5}w^{3}-916yz^{4}w^{4}+244yz^{3}w^{5}-20yz^{2}w^{6}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 38.40.2.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 19x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ 19X^{3}Y+8XYZ^{2}+Y^{2}Z^{2}-Z^{4} $ |
Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 38.40.2.b.1 :
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 19xz^{2}+10y^{3}+4yz^{2}$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -y$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 19.40.1-19.a.1.1 | $19$ | $2$ | $2$ | $1$ | $0$ | $1$ |
| 38.40.1-19.a.1.2 | $38$ | $2$ | $2$ | $1$ | $0$ | $1$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 38.240.4-38.c.1.1 | $38$ | $3$ | $3$ | $4$ | $1$ | $2$ |
| 38.240.4-38.c.2.1 | $38$ | $3$ | $3$ | $4$ | $1$ | $2$ |
| 38.240.4-38.d.1.2 | $38$ | $3$ | $3$ | $4$ | $3$ | $1^{2}$ |
| 38.240.8-38.e.1.2 | $38$ | $3$ | $3$ | $8$ | $2$ | $1^{6}$ |
| 38.1520.53-38.c.1.1 | $38$ | $19$ | $19$ | $53$ | $21$ | $1^{7}\cdot2^{7}\cdot3^{2}\cdot4\cdot6^{2}\cdot8$ |
| 76.320.11-76.c.1.4 | $76$ | $4$ | $4$ | $11$ | $?$ | not computed |
| 114.240.10-114.c.1.2 | $114$ | $3$ | $3$ | $10$ | $?$ | not computed |
| 114.320.11-114.c.1.1 | $114$ | $4$ | $4$ | $11$ | $?$ | not computed |
| 190.400.14-190.c.1.1 | $190$ | $5$ | $5$ | $14$ | $?$ | not computed |
| 190.480.19-190.c.1.8 | $190$ | $6$ | $6$ | $19$ | $?$ | not computed |
| 266.240.4-266.i.1.2 | $266$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 266.240.4-266.i.2.4 | $266$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 266.240.4-266.l.1.3 | $266$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 266.240.4-266.l.2.4 | $266$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 266.240.4-266.n.1.2 | $266$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 266.240.4-266.n.2.4 | $266$ | $3$ | $3$ | $4$ | $?$ | not computed |