| $\GL_2(\Z/38\Z)$-generators: |
$\begin{bmatrix}17&34\\26&11\end{bmatrix}$, $\begin{bmatrix}26&11\\37&15\end{bmatrix}$, $\begin{bmatrix}32&1\\17&37\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
38.80.2-38.a.1.1, 38.80.2-38.a.1.2, 76.80.2-38.a.1.1, 76.80.2-38.a.1.2, 76.80.2-38.a.1.3, 76.80.2-38.a.1.4, 114.80.2-38.a.1.1, 114.80.2-38.a.1.2, 152.80.2-38.a.1.1, 152.80.2-38.a.1.2, 152.80.2-38.a.1.3, 152.80.2-38.a.1.4, 152.80.2-38.a.1.5, 152.80.2-38.a.1.6, 190.80.2-38.a.1.1, 190.80.2-38.a.1.2, 228.80.2-38.a.1.1, 228.80.2-38.a.1.2, 228.80.2-38.a.1.3, 228.80.2-38.a.1.4, 266.80.2-38.a.1.1, 266.80.2-38.a.1.2 |
| Cyclic 38-isogeny field degree: |
$3$ |
| Cyclic 38-torsion field degree: |
$54$ |
| Full 38-torsion field degree: |
$18468$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 y^{2} w + y w^{2} - z^{2} w $ |
| $=$ | $2 y^{2} z + y z w - z^{3}$ |
| $=$ | $2 y^{3} + y^{2} w - y z^{2}$ |
| $=$ | $2 x y^{2} + x y w - x z^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 19 x^{2} y^{2} + 8 x^{2} y z - y z^{3} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + x^{3} y $ | $=$ | $ -4x^{4} + 16x^{2} - 19 $ |
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This modular curve has 2 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 embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -19xy^{2}-4y^{2}z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -y$ |
Maps to other modular curves
$j$-invariant map
of degree 40 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2\,\frac{79235168x^{6}w^{3}+161982144x^{4}w^{5}+113402944x^{2}w^{7}+955552000xyz^{7}-14954317440xyz^{5}w^{2}+57336793200xyz^{3}w^{4}-53158602336xyzw^{6}-1501942400xz^{7}w+12284940304xz^{5}w^{3}+7490403328xz^{3}w^{5}-28050431104xzw^{7}+152640000yz^{8}-3093396480yz^{6}w^{2}+12420716820yz^{4}w^{4}-15071859560yz^{2}w^{6}+3008007823yw^{8}-144633600z^{8}w+2069573764z^{6}w^{3}+11320518z^{4}w^{5}-6018944207z^{2}w^{7}+1505468192w^{9}}{438976x^{4}w^{5}-17024x^{2}w^{7}+238888xyz^{7}-813576xyz^{5}w^{2}+59026xyz^{3}w^{4}+40245xyzw^{6}+29316xz^{7}w+337496xz^{5}w^{3}-140149xz^{3}w^{5}+11520xzw^{7}+38160yz^{8}-48456yz^{6}w^{2}-69500yz^{4}w^{4}+7224yz^{2}w^{6}+600yw^{8}-24432z^{8}w+67628z^{6}w^{3}-6024z^{4}w^{5}-600z^{2}w^{7}}$ |
Hi
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Cover information
Click on a modular curve in the diagram to see information about it.
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The following modular covers realize this modular curve as a fiber product over $X(1)$.
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.
