$\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}3&0\\2&13\end{bmatrix}$, $\begin{bmatrix}9&20\\2&37\end{bmatrix}$, $\begin{bmatrix}31&24\\28&31\end{bmatrix}$, $\begin{bmatrix}35&24\\6&9\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
40.192.1-40.c.2.1, 40.192.1-40.c.2.2, 40.192.1-40.c.2.3, 40.192.1-40.c.2.4, 40.192.1-40.c.2.5, 40.192.1-40.c.2.6, 40.192.1-40.c.2.7, 40.192.1-40.c.2.8, 80.192.1-40.c.2.1, 80.192.1-40.c.2.2, 80.192.1-40.c.2.3, 80.192.1-40.c.2.4, 120.192.1-40.c.2.1, 120.192.1-40.c.2.2, 120.192.1-40.c.2.3, 120.192.1-40.c.2.4, 120.192.1-40.c.2.5, 120.192.1-40.c.2.6, 120.192.1-40.c.2.7, 120.192.1-40.c.2.8, 240.192.1-40.c.2.1, 240.192.1-40.c.2.2, 240.192.1-40.c.2.3, 240.192.1-40.c.2.4, 280.192.1-40.c.2.1, 280.192.1-40.c.2.2, 280.192.1-40.c.2.3, 280.192.1-40.c.2.4, 280.192.1-40.c.2.5, 280.192.1-40.c.2.6, 280.192.1-40.c.2.7, 280.192.1-40.c.2.8 |
Cyclic 40-isogeny field degree: |
$12$ |
Cyclic 40-torsion field degree: |
$192$ |
Full 40-torsion field degree: |
$7680$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ y^{2} - y z - z^{2} - w^{2} $ |
| $=$ | $10 x^{2} + 3 y^{2} + 2 y z + 2 z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 6 x^{2} y^{2} + 20 x^{2} z^{2} + 4 y^{4} + 60 y^{2} z^{2} + 225 z^{4} $ |
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{5}w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^4}{5^2}\cdot\frac{11320312500000yz^{23}+49809375000000yz^{21}w^{2}+95090625000000yz^{19}w^{4}+103241250000000yz^{17}w^{6}+70220242968750yz^{15}w^{8}+31082920312500yz^{13}w^{10}+9016934062500yz^{11}w^{12}+1686155625000yz^{9}w^{14}+195043944375yz^{7}w^{16}+12918701250yz^{5}w^{18}+421687350yz^{3}w^{20}+4688460yzw^{22}+6996337890625z^{24}+35846484375000z^{22}w^{2}+80032148437500z^{20}w^{4}+102282343750000z^{18}w^{6}+82643753906250z^{16}w^{8}+44004021093750z^{14}w^{10}+15616353593750z^{12}w^{12}+3655861687500z^{10}w^{14}+547098911250z^{8}w^{16}+49340526875z^{6}w^{18}+2409394275z^{4}w^{20}+50923410z^{2}w^{22}+226981w^{24}}{w^{8}(15421875yz^{15}+43181250yz^{13}w^{2}+48116250yz^{11}w^{4}+27142500yz^{9}w^{6}+8140650yz^{7}w^{8}+1241100yz^{5}w^{10}+82068yz^{3}w^{12}+1512yzw^{14}+9531250z^{16}+33584375z^{14}w^{2}+47669375z^{12}w^{4}+34982750z^{10}w^{6}+14135075z^{8}w^{8}+3083450z^{6}w^{10}+327862z^{4}w^{12}+13068z^{2}w^{14}+81w^{16})}$ |
Hi
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Cover information
Click on a modular curve in the diagram to see information about it.
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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.