| $\GL_2(\Z/20\Z)$-generators: |
$\begin{bmatrix}6&9\\5&14\end{bmatrix}$, $\begin{bmatrix}6&13\\15&12\end{bmatrix}$, $\begin{bmatrix}19&16\\5&1\end{bmatrix}$ |
| $\GL_2(\Z/20\Z)$-subgroup: |
$C_2^2.\GL(2,\mathbb{Z}/4)$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
20.240.5-20.u.1.1, 20.240.5-20.u.1.2, 20.240.5-20.u.1.3, 20.240.5-20.u.1.4, 40.240.5-20.u.1.1, 40.240.5-20.u.1.2, 40.240.5-20.u.1.3, 40.240.5-20.u.1.4, 60.240.5-20.u.1.1, 60.240.5-20.u.1.2, 60.240.5-20.u.1.3, 60.240.5-20.u.1.4, 120.240.5-20.u.1.1, 120.240.5-20.u.1.2, 120.240.5-20.u.1.3, 120.240.5-20.u.1.4, 140.240.5-20.u.1.1, 140.240.5-20.u.1.2, 140.240.5-20.u.1.3, 140.240.5-20.u.1.4, 220.240.5-20.u.1.1, 220.240.5-20.u.1.2, 220.240.5-20.u.1.3, 220.240.5-20.u.1.4, 260.240.5-20.u.1.1, 260.240.5-20.u.1.2, 260.240.5-20.u.1.3, 260.240.5-20.u.1.4, 280.240.5-20.u.1.1, 280.240.5-20.u.1.2, 280.240.5-20.u.1.3, 280.240.5-20.u.1.4 |
| Cyclic 20-isogeny field degree: |
$6$ |
| Cyclic 20-torsion field degree: |
$24$ |
| Full 20-torsion field degree: |
$384$ |
Embedded model Embedded model in $\mathbb{P}^{6}$
| $ 0 $ | $=$ | $ z^{2} v - z t v + z u v + w t v + w u v + u^{2} v $ |
| $=$ | $z^{2} u - z t u + z u^{2} + w t u + w u^{2} + u^{3}$ |
| $=$ | $z^{2} t - z t^{2} + z t u + w t^{2} + w t u + t u^{2}$ |
| $=$ | $x z v + x u v + y t v + y u v - z t v + u^{2} v$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 11 x^{7} + 53 x^{6} z - 2 x^{5} y^{2} + 81 x^{5} z^{2} - 15 x^{4} y^{2} z + 40 x^{4} z^{3} + \cdots - 11 z^{7} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 5x^{11} + 55x^{6} - 5x $ |
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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:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle v$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{2}{5}x-\frac{3}{5}y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{2}{625}x^{5}v+\frac{3}{125}x^{4}yv+\frac{14}{125}x^{3}y^{2}v+\frac{36}{125}x^{2}y^{3}v+\frac{47}{125}xy^{4}v+\frac{123}{625}y^{5}v$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -\frac{1}{5}x+\frac{1}{5}y$ |
Maps to other modular curves
$j$-invariant map
of degree 120 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{5814056250xy^{10}+14573384375xy^{8}v^{2}+60880271250xy^{6}v^{4}+332536023875xy^{4}v^{6}+2052753903225xy^{2}v^{8}+11587078328190625xu^{10}-11395200722456875xu^{8}v^{2}-4502529889552875xu^{6}v^{4}-262644569532575xu^{4}v^{6}+131602816058900xu^{2}v^{8}+13614065619579xv^{10}+2220771875y^{11}+34402365625y^{9}v^{2}+159492288125y^{7}v^{4}+873007423000y^{5}v^{6}+5387544410775y^{3}v^{8}+673063680387500yu^{10}-6278737352490000yu^{8}v^{2}-1617690977147750yu^{6}v^{4}-153856970561550yu^{4}v^{6}+55296248309350yu^{2}v^{8}+35724624025056yv^{10}-9297297070309375ztu^{9}+4760534218004375ztu^{7}v^{2}+3658992644163750ztu^{5}v^{4}+250361723075475ztu^{3}v^{6}-18131056181825ztuv^{8}+2206724919306250zu^{10}+10613478148731250zu^{8}v^{2}-1640460218827000zu^{6}v^{4}-529878511912550zu^{4}v^{6}-110693124466150zu^{2}v^{8}+16829686523250zv^{10}+4658754927971875wtu^{9}+9130915921098750wtu^{7}v^{2}+1456673368762000wtu^{5}v^{4}+4346824965325wtu^{3}v^{6}-27553424856775wtuv^{8}-2452029995384375wu^{10}+743583278651250wu^{8}v^{2}+700596018691000wu^{6}v^{4}+462146861514750wu^{4}v^{6}-30090345647375wu^{2}v^{8}-6084138845375wv^{10}+2452030000462500t^{2}u^{9}-3179000737010625t^{2}u^{7}v^{2}-763079810332250t^{2}u^{5}v^{4}-52432601229625t^{2}u^{3}v^{6}+50246317909250t^{2}uv^{8}-2452030016868750tu^{10}+1756844091620000tu^{8}v^{2}-134224088942750tu^{6}v^{4}+72527289951625tu^{4}v^{6}-47296898351000tu^{2}v^{8}-2187757316875tv^{10}+11749327092256250u^{11}-8834794308371250u^{9}v^{2}-1229610150853500u^{7}v^{4}+552471638167525u^{5}v^{6}+113444217209925u^{3}v^{8}-26047775380075uv^{10}}{v^{10}(2x+3y+4z-3w+8t-11u)}$ |
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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.
