| $\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}13&26\\2&37\end{bmatrix}$, $\begin{bmatrix}15&1\\12&29\end{bmatrix}$, $\begin{bmatrix}23&31\\4&15\end{bmatrix}$, $\begin{bmatrix}31&35\\14&37\end{bmatrix}$, $\begin{bmatrix}37&4\\24&7\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
40.144.3-40.eq.2.1, 40.144.3-40.eq.2.2, 40.144.3-40.eq.2.3, 40.144.3-40.eq.2.4, 40.144.3-40.eq.2.5, 40.144.3-40.eq.2.6, 40.144.3-40.eq.2.7, 40.144.3-40.eq.2.8, 40.144.3-40.eq.2.9, 40.144.3-40.eq.2.10, 40.144.3-40.eq.2.11, 40.144.3-40.eq.2.12, 40.144.3-40.eq.2.13, 40.144.3-40.eq.2.14, 40.144.3-40.eq.2.15, 40.144.3-40.eq.2.16, 120.144.3-40.eq.2.1, 120.144.3-40.eq.2.2, 120.144.3-40.eq.2.3, 120.144.3-40.eq.2.4, 120.144.3-40.eq.2.5, 120.144.3-40.eq.2.6, 120.144.3-40.eq.2.7, 120.144.3-40.eq.2.8, 120.144.3-40.eq.2.9, 120.144.3-40.eq.2.10, 120.144.3-40.eq.2.11, 120.144.3-40.eq.2.12, 120.144.3-40.eq.2.13, 120.144.3-40.eq.2.14, 120.144.3-40.eq.2.15, 120.144.3-40.eq.2.16, 280.144.3-40.eq.2.1, 280.144.3-40.eq.2.2, 280.144.3-40.eq.2.3, 280.144.3-40.eq.2.4, 280.144.3-40.eq.2.5, 280.144.3-40.eq.2.6, 280.144.3-40.eq.2.7, 280.144.3-40.eq.2.8, 280.144.3-40.eq.2.9, 280.144.3-40.eq.2.10, 280.144.3-40.eq.2.11, 280.144.3-40.eq.2.12, 280.144.3-40.eq.2.13, 280.144.3-40.eq.2.14, 280.144.3-40.eq.2.15, 280.144.3-40.eq.2.16 |
| Cyclic 40-isogeny field degree: |
$4$ |
| Cyclic 40-torsion field degree: |
$32$ |
| Full 40-torsion field degree: |
$10240$ |
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x z t + x w t - z^{2} t - z w t + w^{2} t $ |
| $=$ | $x^{2} t - x z t - x w t - y w t$ |
| $=$ | $x^{2} t + x w t + y z t$ |
| $=$ | $x z^{2} + x z w - z^{3} - z^{2} w + z w^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} + 9 x^{5} z + 23 x^{4} z^{2} - 2 x^{3} y^{2} z + 16 x^{3} z^{3} - 6 x^{2} y^{2} z^{2} + \cdots - 2 y^{2} z^{4} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -2x^{7} + 6x^{6} + 14x^{5} - 12x^{4} - 14x^{3} + 6x^{2} + 2x $ |
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This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:0:0:0:1)$, $(0:1:0:0:0)$, $(-1:2:0:1:0)$, $(-1:0:-2:1:0)$ |
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ |
$=$ |
$\displaystyle z+w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle z^{2}wt+2zw^{2}t+w^{3}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -w$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{2^3}\cdot\frac{1874880xw^{10}-1026512xw^{8}t^{2}-261519256xw^{6}t^{4}+63305496xw^{4}t^{6}+382152xw^{2}t^{8}-12365xt^{10}-56y^{7}t^{4}-560y^{5}t^{6}-770y^{3}t^{8}-3276000y^{2}w^{9}+3146096y^{2}w^{7}t^{2}+752950256y^{2}w^{5}t^{4}-18389096y^{2}w^{3}t^{6}+55880y^{2}wt^{8}+7565376yw^{10}-6636064yw^{8}t^{2}-1768163160yw^{6}t^{4}-81606876yw^{4}t^{6}+1558176yw^{2}t^{8}-1141yt^{10}+1041376z^{2}w^{9}-789184z^{2}w^{7}t^{2}-261706064z^{2}w^{5}t^{4}+13377556z^{2}w^{3}t^{6}-74296z^{2}wt^{8}+1069376zw^{10}-117184zw^{8}t^{2}-262324352zw^{6}t^{4}-137219820zw^{4}t^{6}+969762zw^{2}t^{8}+11217zt^{10}-1069376w^{11}+1234496w^{9}t^{2}+261145896w^{7}t^{4}-137801828w^{5}t^{6}+1792922w^{3}t^{8}-1512wt^{10}}{t^{2}w^{5}(128xw^{3}-38xwt^{2}-512y^{2}w^{2}+7y^{2}t^{2}+1152yw^{3}+70ywt^{2}+128z^{2}w^{2}-7z^{2}t^{2}+128zw^{3}+94zwt^{2}-128w^{4}+98w^{2}t^{2})}$ |
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.
