| $\GL_2(\Z/38\Z)$-generators: |
$\begin{bmatrix}29&17\\19&18\end{bmatrix}$, $\begin{bmatrix}33&30\\27&29\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
38.240.4-38.c.2.1, 38.240.4-38.c.2.2, 76.240.4-38.c.2.1, 76.240.4-38.c.2.2, 114.240.4-38.c.2.1, 114.240.4-38.c.2.2, 152.240.4-38.c.2.1, 152.240.4-38.c.2.2, 152.240.4-38.c.2.3, 152.240.4-38.c.2.4, 190.240.4-38.c.2.1, 190.240.4-38.c.2.2, 228.240.4-38.c.2.1, 228.240.4-38.c.2.2, 266.240.4-38.c.2.1, 266.240.4-38.c.2.2 |
| Cyclic 38-isogeny field degree: |
$3$ |
| Cyclic 38-torsion field degree: |
$18$ |
| Full 38-torsion field degree: |
$6156$ |
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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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.
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y+z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 120 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{y^{20}-8y^{19}w+12y^{18}w^{2}+24y^{17}w^{3}+10y^{16}w^{4}-120y^{15}w^{5}-192y^{14}w^{6}-88y^{13}w^{7}+251y^{12}w^{8}+628y^{11}w^{9}+948y^{10}w^{10}+1368y^{9}w^{11}+2470y^{8}w^{12}+5472y^{7}w^{13}+12996y^{6}w^{14}+31464y^{5}w^{15}+77653y^{4}w^{16}+196004y^{3}w^{17}+504822y^{2}w^{18}+8yz^{19}+76yz^{17}w^{2}+380yz^{16}w^{3}+228yz^{15}w^{4}+2128yz^{14}w^{5}+4142yz^{13}w^{6}+4560yz^{12}w^{7}+5662yz^{11}w^{8}+5368yz^{10}w^{9}-15416yz^{9}w^{10}-83036yz^{8}w^{11}-157296yz^{7}w^{12}-171912yz^{6}w^{13}-171046yz^{5}w^{14}-88186yz^{4}w^{15}+646104yz^{3}w^{16}+2434314yz^{2}w^{17}+3515041yzw^{18}+1322368yw^{19}+z^{20}+12z^{19}w-38z^{18}w^{2}+304z^{17}w^{3}-285z^{16}w^{4}+1102z^{15}w^{5}+1102z^{14}w^{6}+1710z^{13}w^{7}+1501z^{12}w^{8}+1976z^{11}w^{9}-3028z^{10}w^{10}-30228z^{9}w^{11}-55378z^{8}w^{12}-52778z^{7}w^{13}-48463z^{6}w^{14}-44264z^{5}w^{15}+188892z^{4}w^{16}+870281z^{3}w^{17}+1322380z^{2}w^{18}+8zw^{19}+w^{20}}{w^{9}(yz^{10}+29yz^{9}w+115yz^{8}w^{2}+206yz^{7}w^{3}+186yz^{6}w^{4}+52yz^{5}w^{5}-75yz^{4}w^{6}-95yz^{3}w^{7}-47yz^{2}w^{8}-11yzw^{9}-yw^{10}+7z^{10}w+41z^{9}w^{2}+73z^{8}w^{3}+59z^{7}w^{4}+4z^{6}w^{5}-36z^{5}w^{6}-29z^{4}w^{7}-9z^{3}w^{8}-z^{2}w^{9})}$ |
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Cover information
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This modular curve minimally covers the modular curves listed below.
This modular curve is minimally covered by the modular curves in the database listed below.
