Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x y + x w + x t + y z - y t $ |
| $=$ | $x z - x t + y^{2} + z t + w^{2} + 2 w t$ |
| $=$ | $2 x^{2} + 2 x z - y w - y t + z t$ |
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This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(-1:0:2:-1:1)$, $(-1:-1:1:0:0)$, $(0:0:0:0:1)$, $(0:0:0:-2:1)$, $(-1:1:1:0:0)$, $(0:0:1:0:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 84 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{8463329722368xz^{10}+16926659444736z^{11}+9074640384816058368xw^{10}+5712863477090881536zw^{10}-2100316326100992xz^{9}t-198888248475648z^{10}t+90746403848160583680xw^{9}t-4350248787948512256yw^{9}t+57128634770908815360zw^{9}t-20877634471996449792w^{10}t-140242584716550144xz^{8}t^{2}+3028108846915584z^{9}t^{2}+467475836402800115712xw^{8}t^{2}-39152239091536610304yw^{8}t^{2}+260039410942302388224zw^{8}t^{2}-208776344719964497920w^{9}t^{2}-3575034435951931392xz^{7}t^{3}-102704053873176576z^{8}t^{3}+1561892998866546917376xw^{7}t^{3}-187540639572201975936yw^{7}t^{3}+709228053036607537152zw^{7}t^{3}-1000410405862989603456w^{8}t^{3}+3986603878931633664xz^{6}t^{4}-1536762239387306496z^{7}t^{4}+3727823359760786889216xw^{6}t^{4}-581942680630063772544yw^{6}t^{4}+1263608763000733802880zw^{6}t^{4}-2992650973624768877568w^{7}t^{4}+5335592098597713792xz^{5}t^{5}-776790669064883328z^{6}t^{5}+6598596513029455715328xw^{5}t^{5}-1309562740916281056672yw^{5}t^{5}+1491504092096969689344zw^{5}t^{5}-6155827564663049451552w^{6}t^{5}-8294303416539003840xz^{4}t^{6}+8021167084602964800z^{5}t^{6}+8817277595806464151104xw^{4}t^{6}-2190070491031467675936yw^{4}t^{6}+1140522206476459543344zw^{4}t^{6}-9067622122413146683584w^{5}t^{6}-6953946260679421488xz^{3}t^{7}-10267245611504721072z^{4}t^{7}+8818263249307139582208xw^{3}t^{7}-2749471206038111332488yw^{3}t^{7}+545186617933052571840zw^{3}t^{7}-9667381641454708702008w^{4}t^{7}-22995896516860167288xz^{2}t^{8}-33569562594896423640z^{3}t^{8}+6393546827284112441152xw^{2}t^{8}-2473177564456744767768yw^{2}t^{8}+191376766620077166320zw^{2}t^{8}-7358158466571420231648w^{3}t^{8}+152955558807803063928xzt^{9}+183930389162923652968z^{2}t^{9}+3055956351661393333376xwt^{9}-1433044192710847139520ywt^{9}+92104460981284983904zwt^{9}-3760229487854348887448w^{2}t^{9}+589958255571897658480xt^{10}-399626052815577190320yt^{10}-190332202756320468160zt^{10}-1096880591496653880688wt^{10}-40850876073893363712t^{11}}{5814878889777984xw^{10}+4017110477357952zw^{10}-2821109907456xz^{9}t+58148788897779840xw^{9}t-3476741067074016yw^{9}t+40171104773579520zw^{9}t-6667297818670944w^{10}t+61750961307648xz^{8}t^{2}+2821109907456z^{9}t^{2}+259196823679556160xw^{8}t^{2}-31290669603666144yw^{8}t^{2}+170579742436218960zw^{8}t^{2}-66672978186709440w^{9}t^{2}-558897571897344xz^{7}t^{3}-10383251742720z^{8}t^{3}+678003655889733120xw^{7}t^{3}-124573359074350572yw^{7}t^{3}+400531424923843200zw^{7}t^{3}-283847126412076524w^{8}t^{3}+2880103490205696xz^{6}t^{4}-130311986153472z^{7}t^{4}+1147475996684017920xw^{6}t^{4}-287921014252019316yw^{6}t^{4}+566865155520599616zw^{6}t^{4}-670625534815585632w^{7}t^{4}-9500823824978916xz^{5}t^{5}+1263184852327452z^{6}t^{5}+1300403411578649088xw^{5}t^{5}-422382875377192236yw^{5}t^{5}+493249224974336640zw^{5}t^{5}-961700712466865616w^{6}t^{5}+21290085780348204xz^{4}t^{6}-4711648995102780z^{5}t^{6}+987194385974086200xw^{4}t^{6}-400889232902637396yw^{4}t^{6}+257326664955312762zw^{4}t^{6}-861870921529869216w^{5}t^{6}-32405623325324154xz^{3}t^{7}+9718703635683222z^{4}t^{7}+492444731064549600xw^{3}t^{7}-239769102558008427yw^{3}t^{7}+75663680885901288zw^{3}t^{7}-483036380612417613w^{4}t^{7}+30684998492708319xz^{2}t^{8}-12864641334791445z^{3}t^{8}+158404708547033624xw^{2}t^{8}-84049920323208081yw^{2}t^{8}+12239412144526114zw^{2}t^{8}-170541487575999156w^{3}t^{8}-14822792373780351xzt^{9}+10914007162967651z^{2}t^{9}+33450377033704048xwt^{9}-16345645697489304ywt^{9}+1288376538727028zwt^{9}-41193857506258585w^{2}t^{9}+6748165990573586xt^{10}-2396886692583618yt^{10}-4351279297989968zt^{10}-6772883979875018wt^{10}}$ |
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.
