Embedded model Embedded model in $\mathbb{P}^{7}$
$ 0 $ | $=$ | $ x w + x t - u r $ |
| $=$ | $x y + x z - x w + u v$ |
| $=$ | $x y - x z - x w - y r - t v + u v$ |
| $=$ | $z^{2} - z w + w^{2} + u^{2}$ |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 2744 x^{12} + 17052 x^{11} z + 1216 x^{10} y^{2} + 58254 x^{10} z^{2} + 1120 x^{9} y^{2} z + \cdots + 2744 z^{12} $ |
Geometric Weierstrass model Geometric Weierstrass model
$ 100 w^{2} $ | $=$ | $ 297 x^{6} + 90 x^{4} y z + 297 x^{4} z^{2} - 60 x^{2} y z^{3} + 99 x^{2} z^{4} + 2 y z^{5} + 11 z^{6} $ |
$0$ | $=$ | $3 x^{2} + y^{2} + z^{2}$ |
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle v$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle u$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle r$ |
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}{571^{11}}\cdot\frac{12583175046111620294204841698527186925xv^{9}+184603357226461248011863498949008127400xv^{8}r+979561680833728558103956386797690820750xv^{7}r^{2}+3125796064758454475307346538263869574725xv^{6}r^{3}+7631871018659095435026902782361540140200xv^{5}r^{4}+12571395859700879626490200603812999124575xv^{4}r^{5}+10166604679369853793821326418847381293475xv^{3}r^{6}-408342095708997145638328073645922460500xv^{2}r^{7}-7464429136417889973759278902528964528850xvr^{8}-6045608925822379754734541898345443516200xr^{9}-328713254878157991725913971995468750yu^{9}+1239701580352387286687837968059140625yu^{7}r^{2}-1486737316165432809765603882146484375yu^{5}r^{4}+5819847460257608982967408085316796875yu^{3}r^{6}-500730492767740358509229541594275390625yur^{8}-2212124837453706414845017870099453125ztu^{6}r^{2}-10439850708580603224190881440232421875ztu^{4}r^{4}-24030782593629706686675335453124609375ztu^{2}r^{6}-336492519644082107593395743604644531250ztr^{8}+1938060850553105092192941749331328125zu^{7}r^{2}-443158765956298615207969303600781250zu^{5}r^{4}+56956181490922590250952628018452343750zu^{3}r^{6}-1320527400418250301775201278486714843750zur^{8}-3402335080922959685701002203274375000wtu^{6}r^{2}+10117548500012270623078567713544921875wtu^{4}r^{4}+88687745377015759060083794289671484375wtu^{2}r^{6}-1534285860365594737671833854185750000000wtr^{8}-11677935079495229183327438995781250wu^{7}r^{2}+12513730992932690895773626306062890625wu^{5}r^{4}-115361034363023981244468540412841015625wu^{3}r^{6}+2858443484231928200504828977132312500000wur^{8}+555722714796958283863938617946093750t^{2}u^{6}r^{2}+12809892169117035750741253954171875000t^{2}u^{4}r^{4}-112176459864387075815252617731380859375t^{2}u^{2}r^{6}+1308529080024968684691272821318892578125t^{2}r^{8}+328713254878157991725913971995468750tu^{9}-1645072015221423311369386792563046875tu^{7}r^{2}-18465107585076256859088398402695312500tu^{5}r^{4}+183033890738817407215876246360784765625tu^{3}r^{6}-2837507592614630475605706702907294921875tur^{8}-164356627439078995862956985997734375u^{10}-2029655967616014642302575652161875000u^{8}r^{2}+9981051982688294313024702396561328125u^{6}r^{4}-43766676203074473539402101512260156250u^{4}r^{6}+34107348425768374475540360978753906250u^{2}r^{8}-9187391257584045817371560772528393467v^{10}-64815240701338724537181704623974613655v^{9}r-218312837887082525152926310113144072915v^{8}r^{2}-821339948924547387110185250659567213560v^{7}r^{3}-2944825960353111660975880788467545718745v^{6}r^{4}-7418699546499505360890853867471310262816v^{5}r^{5}-14881332987494619049194226130147919546870v^{4}r^{6}-20650171658318311712492571282885593385435v^{3}r^{7}-19230145282150710331500469836203838213540v^{2}r^{8}-12478997702446620187440519768614231254280vr^{9}-4407182612780701963235242002401921362217r^{10}}{(2v^{2}+vr+2r^{2})^{2}(275xv^{5}+925xv^{4}r+1175xv^{3}r^{2}+1175xv^{2}r^{3}+925xvr^{4}+275xr^{5}+58v^{6}+537v^{5}r+1230v^{4}r^{2}+1475v^{3}r^{3}+1230v^{2}r^{4}+537vr^{5}+58r^{6})}$ |
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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.