Embedded model Embedded model in $\mathbb{P}^{6}$
| $ 0 $ | $=$ | $ z^{2} v - z w v + z u v + w t v + t u v + u^{2} v $ |
| $=$ | $z^{2} u - z w u + z u^{2} + w t u + t u^{2} + u^{3}$ |
| $=$ | $z^{3} - z^{2} w + z^{2} u + z w t + z t u + z u^{2}$ |
| $=$ | $z^{2} t - z w t + z t u + w t^{2} + t^{2} u + t u^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 11 x^{7} + 53 x^{6} z + 4 x^{5} y^{2} + 81 x^{5} z^{2} + 30 x^{4} y^{2} z + 40 x^{4} z^{3} + \cdots - 11 z^{7} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -10x^{11} - 110x^{6} + 10x $ |
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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 to other modular curves
$j$-invariant map
of degree 120 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{3}\cdot\frac{558149400000xy^{10}-699522450000xy^{8}v^{2}+1461126510000xy^{6}v^{4}-3990432286500xy^{4}v^{6}+12316523419350xy^{2}v^{8}+1112359519506300000xu^{10}+546969634677930000xu^{8}v^{2}-108060717349269000xu^{6}v^{4}+3151734834390900xu^{4}v^{6}+789616896353400xu^{2}v^{8}-40842196858737xv^{10}+213194100000y^{11}-1651313550000y^{9}v^{2}+3827814915000y^{7}v^{4}-10476089076000y^{5}v^{6}+32325266464650y^{3}v^{8}+64614113317200000yu^{10}+301379392919520000yu^{8}v^{2}-38824583451546000yu^{6}v^{4}+1846283646738600yu^{4}v^{6}+331777489856100yu^{2}v^{8}-107173872075168yv^{10}+235394880044400000ztu^{9}+152592035376510000ztu^{7}v^{2}-18313915447974000ztu^{5}v^{4}+629191214755500ztu^{3}v^{6}+301477907455500ztuv^{8}-932436140350000000zu^{10}-1017832128660430000zu^{8}v^{2}+61905306433378000zu^{6}v^{4}+2887535523917300zu^{4}v^{6}-1440131422253700zu^{2}v^{8}-50489059569750zv^{10}-697041259518100000wtu^{9}-946669141734070000wtu^{7}v^{2}+136236512535514000wtu^{5}v^{4}-3523168518617200wtu^{3}v^{6}-941293224597450wtuv^{8}+219048544660100000wu^{10}+93568054674650000wu^{8}v^{2}-44702596253252000wu^{6}v^{4}+550580222719400wu^{4}v^{6}-46534006105350wu^{2}v^{8}+6563271950625wv^{10}+454443426279500000t^{2}u^{9}+177896571072410000t^{2}u^{7}v^{2}-41481218118626000t^{2}u^{5}v^{4}+1420907702138900t^{2}u^{3}v^{6}+237247384000650t^{2}uv^{8}-235394879556900000tu^{10}-35691997375260000tu^{8}v^{2}+16814304448584000tu^{6}v^{4}-5545762338177000tu^{4}v^{6}-180542073884250tu^{2}v^{8}+18252416536125tv^{10}+235394882106900000u^{11}+195564484337610000u^{9}v^{2}+58305179839446000u^{7}v^{4}-9634000334916000u^{5}v^{6}+571878966168600u^{3}v^{8}+78143326140225uv^{10}}{v^{10}(2x+3y+4z+8w-3t-11u)}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
40.120.5.cu.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}v$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 11X^{7}+4X^{5}Y^{2}+53X^{6}Z+30X^{4}Y^{2}Z+81X^{5}Z^{2}+140X^{3}Y^{2}Z^{2}+40X^{4}Z^{3}+360X^{2}Y^{2}Z^{3}-40X^{3}Z^{4}+470XY^{2}Z^{4}-81X^{2}Z^{5}+246Y^{2}Z^{5}-53XZ^{6}-11Z^{7} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
40.120.5.cu.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{2}{5}x-\frac{3}{5}y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{2}{625}x^{5}v-\frac{3}{125}x^{4}yv-\frac{14}{125}x^{3}y^{2}v-\frac{36}{125}x^{2}y^{3}v-\frac{47}{125}xy^{4}v-\frac{123}{625}y^{5}v$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -\frac{1}{5}x+\frac{1}{5}y$ |
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.
