Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ - x v + z t $ |
| $=$ | $y v + z^{2}$ |
| $=$ | $z^{2} - w t - t u$ |
| $=$ | $x z + y t$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{4} y^{6} + 2 x^{4} y^{4} z^{2} - x^{4} y^{2} z^{4} + 5 x^{2} y^{8} - 16 x^{2} y^{6} z^{2} + \cdots + 80 y^{8} z^{2} $ |
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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 |
|---|
| $(-4:0:0:-1:4:1:0)$, $(0:0:0:-1/4:0:1/4:1)$, $(0:0:0:1/4:0:-1/4:1)$, $(0:0:0:1/3:0:1:0)$, $(0:0:0:-1/5:0:1:0)$, $(-4:0:0:-1:-4:1:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 144 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{3^{10}\cdot5^9}\cdot\frac{1185583142658824xwu^{10}+898589839788228654870xwu^{8}v^{2}+6112344879720940321447200xwu^{6}v^{4}+3615616128898548081436657500xwu^{4}v^{6}+275822809601442103782176175000xwu^{2}v^{8}+1565063397263598093237181443750xwv^{10}-11523068278676xu^{9}v^{2}-12371623587523980240xu^{7}v^{4}-89185614979166780297400xu^{5}v^{6}-36285274192511135151090000xu^{3}v^{8}-1288838924606229014022412500xuv^{10}+1254032836891133456ywu^{9}v+53211246497138137208640ywu^{7}v^{3}+101185768730393982811996650ywu^{5}v^{5}+22226371031000872696601677500ywu^{3}v^{7}+563249555183278549232921081250ywuv^{9}+17446866664923111856ytu^{9}v+218684828707375475839440ytu^{7}v^{3}+170480968421206807381143150ytu^{5}v^{5}+14575165442693793838029183750ytu^{3}v^{7}+65085237228196567009382006250ytuv^{9}-236182110692955880yu^{10}v-27017931545617292147520yu^{8}v^{3}-73649990364370973662102200yu^{6}v^{5}-19332791167059099120993963750yu^{4}v^{7}-534146552647764976106300362500yu^{2}v^{9}+98376906815901126188089706250yv^{11}+1017744582768167280zu^{8}v^{3}+26237716145983516564800zu^{6}v^{5}+27761704507961582667536250zu^{4}v^{7}+2900742375725055236585362500zu^{2}v^{9}+19876691682837355077969806250zv^{11}-17249330320222w^{2}u^{10}+261921302135411598810w^{2}u^{8}v^{2}+3305615117763546670274100w^{2}u^{6}v^{4}+2626423787476983344881972500w^{2}u^{4}v^{6}+236892079952511022977916106250w^{2}u^{2}v^{8}+1490790693601581661502586900000w^{2}v^{10}+2316541096324wu^{11}-33890241285103664892wu^{9}v^{2}-412883114766521880539880wu^{7}v^{4}-318257188606703795146973550wu^{5}v^{6}-27647105779570456357096061250wu^{3}v^{8}-150976161197946607826045756250wuv^{10}+14762246893895680t^{3}u^{9}+1689711209008699127700t^{3}u^{7}v^{2}+4616852000878126337967000t^{3}u^{5}v^{4}+1219174731030976225088512500t^{3}u^{3}v^{6}+34362813922754166321742125000t^{3}uv^{8}-242111680135382630400t^{2}u^{8}v^{2}-1748455885992236692880250t^{2}u^{6}v^{4}-1078988493069690751958400000t^{2}u^{4}v^{6}-84746963099974569597407390625t^{2}u^{2}v^{8}-490639119902537808925841540625t^{2}v^{10}-236195950302330880tu^{11}-27017917715227369035664tu^{9}v^{2}-73648920128012251890662460tu^{7}v^{4}-19330778969519954519324800350tu^{5}v^{6}-533768665154875780266506610000tu^{3}v^{8}+106256021198737608654197456250tuv^{10}+1153300781250u^{12}-17446942782774674356u^{10}v^{2}-218857147303163302418370u^{8}v^{4}-171351588423371192101267200u^{6}v^{6}-14959218433617850677825645000u^{4}v^{8}-87331253239890614992542328125u^{2}v^{10}-98127823980507561785094496875v^{12}}{v(2209920xwu^{8}v+83900633320xwu^{6}v^{3}+167959254749000xwu^{4}v^{5}+37102989189750920xwu^{2}v^{7}+568094826316600320xwv^{9}-20480xu^{7}v^{3}-1275919360xu^{5}v^{5}-2161594733120xu^{3}v^{7}-262845203217920xuv^{9}+512ywu^{9}+336077280ywu^{7}v^{2}+2565236543050ywu^{5}v^{4}+1748841460037500ywu^{3}v^{6}+124516318839246450ywuv^{8}+23040ytu^{9}+2336293280ytu^{7}v^{2}+6970974014790ytu^{5}v^{4}+1906269991139910ytu^{3}v^{6}+28674561860616730ytuv^{8}-122549760yu^{8}v^{2}-1587534272160yu^{6}v^{4}-1406269576799950yu^{4}v^{6}-116633826452655300yu^{2}v^{8}+35709359690391090yv^{10}+512zu^{8}v^{2}+213606880zu^{6}v^{4}+981739345610zu^{4}v^{6}+346330622968100zu^{2}v^{8}+7214944601640010zv^{10}+345600w^{2}u^{8}v+35159308800w^{2}u^{6}v^{3}+106054638702000w^{2}u^{4}v^{5}+29917807575624000w^{2}u^{2}v^{7}+541134935259972010w^{2}v^{9}-45568wu^{9}v-4466742560wu^{7}v^{3}-13065891448750wu^{5}v^{5}-3565186001470550wu^{3}v^{7}-58837153961660790wuv^{9}+7660800t^{3}u^{7}v+99367121100t^{3}u^{5}v^{3}+88331679059000t^{3}u^{3}v^{5}+7413332059486900t^{3}uv^{7}-575488t^{2}u^{8}v-23319094350t^{2}u^{6}v^{3}-49059858381500t^{2}u^{4}v^{5}-11270107083741825t^{2}u^{2}v^{7}-178094731556880325t^{2}v^{9}-122549760tu^{9}v-1587528452640tu^{7}v^{3}-1406216714215530tu^{5}v^{5}-116601153175015100tu^{3}v^{7}+37557389420097430tuv^{9}-23040u^{10}v-2336776608u^{8}v^{3}-6984938730260u^{6}v^{5}-1927213558781810u^{4}v^{7}-32041622529518125u^{2}v^{9}-35618946311376065v^{11})}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
40.144.7.fw.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x+w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{4}y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{4}z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ -X^{4}Y^{6}+2X^{4}Y^{4}Z^{2}-X^{4}Y^{2}Z^{4}+5X^{2}Y^{8}-16X^{2}Y^{6}Z^{2}+2X^{2}Y^{4}Z^{4}+8X^{2}Y^{2}Z^{6}+X^{2}Z^{8}+80Y^{8}Z^{2} $ |
This modular curve minimally covers the modular curves listed below.
