Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 3 x z - 2 w^{2} $ |
| $=$ | $2 x^{2} - 2 x z - x t - z^{2} - z t - t^{2}$ |
| $=$ | $x^{2} + 2 x y + 2 x z - 2 y^{2} - 2 y z - z^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 81 x^{8} - 432 x^{7} y + 648 x^{7} z + 468 x^{6} y^{2} - 1728 x^{6} z^{2} - 216 x^{5} y^{3} + \cdots + 16 z^{8} $ |
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This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map
of degree 192 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{2^{12}\cdot3}\cdot\frac{6033028354670592xw^{22}t+73033308803432448xw^{20}t^{3}+394976763567931392xw^{18}t^{5}+1241357136477290496xw^{16}t^{7}+2504228939272028160xw^{14}t^{9}+3383473992789786624xw^{12}t^{11}+3106427474821275648xw^{10}t^{13}+1933274698356584448xw^{8}t^{15}+801165119611009536xw^{6}t^{17}+211608784278052416xw^{4}t^{19}+32264395373964456xw^{2}t^{21}+2166612279786045xt^{23}+527356485697536z^{2}w^{22}+11359813191598080z^{2}w^{20}t^{2}+81354996364345344z^{2}w^{18}t^{4}+308860561014128640z^{2}w^{16}t^{6}+722313165907427328z^{2}w^{14}t^{8}+1103966755059990528z^{2}w^{12}t^{10}+1126706299399692288z^{2}w^{10}t^{12}+767096432144695296z^{2}w^{8}t^{14}+342600171041793024z^{2}w^{6}t^{16}+96198779310551712z^{2}w^{4}t^{18}+15407021689377624z^{2}w^{2}t^{20}+1078848283379139z^{2}t^{22}+2508743547813888zw^{22}t+32033538714894336zw^{20}t^{3}+179492086226092032zw^{18}t^{5}+579661605233491968zw^{16}t^{7}+1196076373562621952zw^{14}t^{9}+1645835478062727168zw^{12}t^{11}+1531670656829755392zw^{10}t^{13}+961153044521674752zw^{8}t^{15}+399663214726924800zw^{6}t^{17}+105519076159406976zw^{4}t^{19}+16060868883396180zw^{2}t^{21}+1078848283379139zt^{23}+955387020836864w^{24}+22214639496462336w^{22}t^{2}+169429059061678080w^{20}t^{4}+687898167173185536w^{18}t^{6}+1737605977534365696w^{16}t^{8}+2915765565984866304w^{14}t^{10}+3346605506027372544w^{12}t^{12}+2651980302193987584w^{10}t^{14}+1446009267294088704w^{8}t^{16}+532440461290436352w^{6}t^{18}+126584849867790672w^{4}t^{20}+17570662064821080w^{2}t^{22}+1083306333603267t^{24}}{w^{8}(254803968xw^{14}t+4269932544xw^{12}t^{3}+18513764352xw^{10}t^{5}+33955946496xw^{8}t^{7}+31680585216xw^{6}t^{9}+15904050624xw^{4}t^{11}+4108708152xw^{2}t^{13}+429975135xt^{15}+12582912z^{2}w^{14}+635240448z^{2}w^{12}t^{2}+4436140032z^{2}w^{10}t^{4}+10666266624z^{2}w^{8}t^{6}+11862567936z^{2}w^{6}t^{8}+6741644256z^{2}w^{4}t^{10}+1911035592z^{2}w^{2}t^{12}+214997409z^{2}t^{14}+100663296zw^{14}t+1852440576zw^{12}t^{3}+8596942848zw^{10}t^{5}+16389900288zw^{8}t^{7}+15613917696zw^{6}t^{9}+7920104832zw^{4}t^{11}+2054354076zw^{2}t^{13}+214997409zt^{15}+22544384w^{16}+1212678144w^{14}t^{2}+9238167552w^{12}t^{4}+25461043200w^{10}t^{6}+34254420480w^{8}t^{8}+25324731648w^{6}t^{10}+10563715440w^{4}t^{12}+2341017288w^{2}t^{14}+214997409t^{16})}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
24.192.5.fc.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x+z-2w+2t$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 6t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 3y+3z+3t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 81X^{8}-432X^{7}Y+468X^{6}Y^{2}-216X^{5}Y^{3}+36X^{4}Y^{4}+648X^{7}Z-792X^{5}Y^{2}Z+648X^{4}Y^{3}Z-144X^{3}Y^{4}Z-1728X^{6}Z^{2}+2016X^{5}YZ^{2}-684X^{4}Y^{2}Z^{2}-504X^{3}Y^{3}Z^{2}+216X^{2}Y^{4}Z^{2}+1008X^{5}Z^{3}-1584X^{4}YZ^{3}+1968X^{3}Y^{2}Z^{3}-216X^{2}Y^{3}Z^{3}-144XY^{4}Z^{3}-120X^{4}Z^{4}-1056X^{3}YZ^{4}-848X^{2}Y^{2}Z^{4}+432XY^{3}Z^{4}+36Y^{4}Z^{4}+672X^{3}Z^{5}+1248X^{2}YZ^{5}-320XY^{2}Z^{5}-144Y^{3}Z^{5}-768X^{2}Z^{6}-64XYZ^{6}+208Y^{2}Z^{6}+192XZ^{7}-128YZ^{7}+16Z^{8} $ |
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.
