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Properties

Label 38.240.4-38.c.2.2
Level $38$
Index $240$
Genus $4$
Analytic rank $1$
Cusps $6$
$\Q$-cusps $3$

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Invariants

Level: $38$ $\SL_2$-level: $38$ Newform level: $1444$
Index: $240$ $\PSL_2$-index:$120$
Genus: $4 = 1 + \frac{ 120 }{12} - \frac{ 0 }{4} - \frac{ 12 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (of which $3$ are rational) Cusp widths $2^{3}\cdot38^{3}$ Cusp orbits $1^{3}\cdot3$
Elliptic points: $0$ of order $2$ and $12$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $3 \le \gamma \le 4$
$\overline{\Q}$-gonality: $3$
Rational cusps: $3$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 38B4
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 38.240.4.2

Level structure

$\GL_2(\Z/38\Z)$-generators: $\begin{bmatrix}29&14\\11&9\end{bmatrix}$, $\begin{bmatrix}37&35\\6&3\end{bmatrix}$
Contains $-I$: no $\quad$ (see 38.120.4.c.2 for the level structure with $-I$)
Cyclic 38-isogeny field degree: $3$
Cyclic 38-torsion field degree: $9$
Full 38-torsion field degree: $3078$

Jacobian

Conductor: $2^{6}\cdot19^{7}$
Simple: no
Squarefree: yes
Decomposition: $1^{2}\cdot2$
Newforms: 19.2.a.a, 1444.2.a.a, 1444.2.e.b

Models

Canonical model in $\mathbb{P}^{ 3 }$

$ 0 $ $=$ $ 19 x^{2} - y z + y w - z w $
$=$ $y^{2} z - 2 y z w - y w^{2} - z^{2} w$
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Singular plane model Singular plane model

$ 0 $ $=$ $ - 6859 x^{6} - 361 x^{4} z^{2} + 19 x^{2} y^{3} z - 95 x^{2} y^{2} z^{2} + 95 x^{2} y z^{3} + \cdots + 7 y z^{5} $
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Rational points

This modular curve has 3 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:0:1:0)$, $(0:0:0:1)$, $(0:1:0:0)$

Maps to other modular curves

$j$-invariant map of degree 120 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{y^{20}-8y^{19}w+12y^{18}w^{2}+24y^{17}w^{3}+10y^{16}w^{4}-120y^{15}w^{5}-192y^{14}w^{6}-88y^{13}w^{7}+251y^{12}w^{8}+628y^{11}w^{9}+948y^{10}w^{10}+1368y^{9}w^{11}+2470y^{8}w^{12}+5472y^{7}w^{13}+12996y^{6}w^{14}+31464y^{5}w^{15}+77653y^{4}w^{16}+196004y^{3}w^{17}+504822y^{2}w^{18}+8yz^{19}+76yz^{17}w^{2}+380yz^{16}w^{3}+228yz^{15}w^{4}+2128yz^{14}w^{5}+4142yz^{13}w^{6}+4560yz^{12}w^{7}+5662yz^{11}w^{8}+5368yz^{10}w^{9}-15416yz^{9}w^{10}-83036yz^{8}w^{11}-157296yz^{7}w^{12}-171912yz^{6}w^{13}-171046yz^{5}w^{14}-88186yz^{4}w^{15}+646104yz^{3}w^{16}+2434314yz^{2}w^{17}+3515041yzw^{18}+1322368yw^{19}+z^{20}+12z^{19}w-38z^{18}w^{2}+304z^{17}w^{3}-285z^{16}w^{4}+1102z^{15}w^{5}+1102z^{14}w^{6}+1710z^{13}w^{7}+1501z^{12}w^{8}+1976z^{11}w^{9}-3028z^{10}w^{10}-30228z^{9}w^{11}-55378z^{8}w^{12}-52778z^{7}w^{13}-48463z^{6}w^{14}-44264z^{5}w^{15}+188892z^{4}w^{16}+870281z^{3}w^{17}+1322380z^{2}w^{18}+8zw^{19}+w^{20}}{w^{9}(yz^{10}+29yz^{9}w+115yz^{8}w^{2}+206yz^{7}w^{3}+186yz^{6}w^{4}+52yz^{5}w^{5}-75yz^{4}w^{6}-95yz^{3}w^{7}-47yz^{2}w^{8}-11yzw^{9}-yw^{10}+7z^{10}w+41z^{9}w^{2}+73z^{8}w^{3}+59z^{7}w^{4}+4z^{6}w^{5}-36z^{5}w^{6}-29z^{4}w^{7}-9z^{3}w^{8}-z^{2}w^{9})}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 38.120.4.c.2 :

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle y+z$
$\displaystyle Z$ $=$ $\displaystyle w$

Equation of the image curve:

$0$ $=$ $ -6859X^{6}-361X^{4}Z^{2}+19X^{2}Y^{3}Z-95X^{2}Y^{2}Z^{2}+95X^{2}YZ^{3}-133X^{2}Z^{4}+Y^{4}Z^{2}-7Y^{3}Z^{3}+10Y^{2}Z^{4}+7YZ^{5} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
19.120.1-19.a.2.2 $19$ $2$ $2$ $1$ $0$ $1\cdot2$
38.80.2-38.b.1.1 $38$ $3$ $3$ $2$ $1$ $2$
38.120.1-19.a.2.2 $38$ $2$ $2$ $1$ $0$ $1\cdot2$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
38.720.22-38.j.2.2 $38$ $3$ $3$ $22$ $1$ $6\cdot12$
38.720.22-38.k.2.2 $38$ $3$ $3$ $22$ $1$ $6\cdot12$
38.720.22-38.m.2.2 $38$ $3$ $3$ $22$ $2$ $1^{6}\cdot2^{2}\cdot4^{2}$
38.720.22-38.n.2.2 $38$ $3$ $3$ $22$ $1$ $6\cdot12$
38.4560.157-38.d.1.1 $38$ $19$ $19$ $157$ $21$ $1^{7}\cdot2^{14}\cdot3^{2}\cdot4^{8}\cdot6^{4}\cdot8^{2}\cdot12^{2}\cdot16$