Canonical model in $\mathbb{P}^{ 2 }$
| $ 0 $ | $=$ | $ x^{3} y - x^{3} z - x^{2} y^{2} + 2 x^{2} y z - x^{2} z^{2} - y^{3} z - y z^{3} $ |
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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 |
|---|
| $(0:0:1)$, $(0:1:0)$, $(1:1:0)$, $(-1:-1:1)$, $(-1:0:1)$, $(1:0:0)$ |
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{4096x^{36}-196608x^{32}z^{4}-786432x^{29}z^{7}+4030464x^{28}z^{8}-4325376x^{26}z^{10}+30670848x^{25}z^{11}-45973504x^{24}z^{12}-27525120x^{23}z^{13}+218234880x^{22}z^{14}-493486080x^{21}z^{15}+128188416x^{20}z^{16}+1568931840x^{19}z^{17}-4165140480x^{18}z^{18}+2830368768x^{17}z^{19}+10114301952x^{16}z^{20}-32597475328x^{15}z^{21}+29103489024x^{14}z^{22}+64926253056x^{13}z^{23}-243378290688x^{12}z^{24}+242085003264x^{11}z^{25}+441759301632x^{10}z^{26}-1781627879424x^{9}z^{27}+1797704908800x^{8}z^{28}+3256599183360x^{7}z^{29}-12956119072768x^{6}z^{30}+12288276824064x^{5}z^{31}+25712428843008x^{4}z^{32}-93849497108480x^{3}z^{33}-4072x^{2}y^{34}+6672x^{2}y^{33}z-128952x^{2}y^{32}z^{2}+227936x^{2}y^{31}z^{3}-2332912x^{2}y^{30}z^{4}+3548352x^{2}y^{29}z^{5}-20841232x^{2}y^{28}z^{6}+24629792x^{2}y^{27}z^{7}-116582160x^{2}y^{26}z^{8}+97908288x^{2}y^{25}z^{9}-378585200x^{2}y^{24}z^{10}+117738336x^{2}y^{23}z^{11}-584187440x^{2}y^{22}z^{12}-281642176x^{2}y^{21}z^{13}+434016816x^{2}y^{20}z^{14}-142596064x^{2}y^{19}z^{15}+4808685344x^{2}y^{18}z^{16}+9553223520x^{2}y^{17}z^{17}+19606648608x^{2}y^{16}z^{18}+48331630624x^{2}y^{15}z^{19}+83429331504x^{2}y^{14}z^{20}+134512933696x^{2}y^{13}z^{21}+251943256528x^{2}y^{12}z^{22}+363815406432x^{2}y^{11}z^{23}+393579413392x^{2}y^{10}z^{24}+788347614784x^{2}y^{9}z^{25}+1482169915632x^{2}y^{8}z^{26}-1745890784x^{2}y^{7}z^{27}-3908005528336x^{2}y^{6}z^{28}-6412157573952x^{2}y^{5}z^{29}-12371607853296x^{2}y^{4}z^{30}+1027811277408x^{2}y^{3}z^{31}+94645370357832x^{2}y^{2}z^{32}+95341853547024x^{2}yz^{33}-94595675328488x^{2}z^{34}-24xy^{35}-2312xy^{34}z-21448xy^{33}z^{2}-53240xy^{32}z^{3}+76656xy^{31}z^{4}-1798864xy^{30}z^{5}+916144xy^{29}z^{6}-13690896xy^{28}z^{7}+4036688xy^{27}z^{8}-86457008xy^{26}z^{9}-27796656xy^{25}z^{10}-310183216xy^{24}z^{11}-432433616xy^{23}z^{12}-1068494864xy^{22}z^{13}-2388948368xy^{21}z^{14}-3893511760xy^{20}z^{15}-8918097248xy^{19}z^{16}-14330514624xy^{18}z^{17}-24726844224xy^{17}z^{18}-39388488352xy^{16}z^{19}-50516965808xy^{15}z^{20}-49418832496xy^{14}z^{21}-23201320944xy^{13}z^{22}+121089714640xy^{12}z^{23}+491580359984xy^{11}z^{24}+1055022982320xy^{10}z^{25}+2098901957296xy^{9}z^{26}+4653002549168xy^{8}z^{27}+9699701352464xy^{7}z^{28}+20813846611280xy^{6}z^{29}+42749471912656xy^{5}z^{30}+55463876678800xy^{4}z^{31}+35412792889336xy^{3}z^{32}+38390402667464xy^{2}z^{33}+746178218248xyz^{34}+24xz^{35}+y^{36}-4348y^{35}z+11190y^{34}z^{2}-185388y^{33}z^{3}+156873y^{32}z^{4}-2295728y^{31}z^{5}+1820400y^{30}z^{6}-20543120y^{29}z^{7}+11731716y^{28}z^{8}-103095136y^{27}z^{9}+27375144y^{26}z^{10}-337674240y^{25}z^{11}-42354124y^{24}z^{12}-602208560y^{23}z^{13}-563761776y^{22}z^{14}-686741136y^{21}z^{15}-2086248962y^{20}z^{16}-3694080376y^{19}z^{17}-10200526012y^{18}z^{18}-30671188344y^{17}z^{19}-71418323458y^{16}z^{20}-170442675856y^{15}z^{21}-384418599536y^{14}z^{22}-767803455792y^{13}z^{23}-1491778454988y^{12}z^{24}-2933034418176y^{11}z^{25}-5170278713816y^{10}z^{26}-7964927859552y^{9}z^{27}-12429804436732y^{8}z^{28}-20083530692240y^{7}z^{29}-34362863335696y^{6}z^{30}-75383833757616y^{5}z^{31}-144748925786935y^{4}z^{32}-155552547263532y^{3}z^{33}-118815747724362y^{2}z^{34}-94595675328764yz^{35}+z^{36}}{z^{5}(4096x^{16}z^{15}+81920x^{13}z^{18}-114688x^{12}z^{19}+1105920x^{10}z^{21}-2588672x^{9}z^{22}+1204224x^{8}z^{23}+12697600x^{7}z^{24}-37584896x^{6}z^{25}+28868608x^{5}z^{26}+128208896x^{4}z^{27}-448430080x^{3}z^{28}-18x^{2}y^{29}+670x^{2}y^{28}z-3324x^{2}y^{27}z^{2}-1150x^{2}y^{26}z^{3}-11068x^{2}y^{25}z^{4}-6646x^{2}y^{24}z^{5}+20980x^{2}y^{23}z^{6}+71702x^{2}y^{22}z^{7}+249490x^{2}y^{21}z^{8}+513324x^{2}y^{20}z^{9}+980744x^{2}y^{19}z^{10}+1685140x^{2}y^{18}z^{11}+2584952x^{2}y^{17}z^{12}+3815060x^{2}y^{16}z^{13}+5240584x^{2}y^{15}z^{14}+6898988x^{2}y^{14}z^{15}+8777362x^{2}y^{13}z^{16}+10655766x^{2}y^{12}z^{17}+12825076x^{2}y^{11}z^{18}+14734858x^{2}y^{10}z^{19}+16258244x^{2}y^{9}z^{20}+19528578x^{2}y^{8}z^{21}+19428100x^{2}y^{7}z^{22}+15090334x^{2}y^{6}z^{23}+35225582x^{2}y^{5}z^{24}+17887232x^{2}y^{4}z^{25}-80191488x^{2}y^{3}z^{26}+260042752x^{2}y^{2}z^{27}+555958272x^{2}yz^{28}-502194176x^{2}z^{29}+19xy^{30}-823xy^{29}z+5544xy^{28}z^{2}-2704xy^{27}z^{3}+10498xy^{26}z^{4}-18962xy^{25}z^{5}-78896xy^{24}z^{6}-172552xy^{23}z^{7}-452783xy^{22}z^{8}-739269xy^{21}z^{9}-1229168xy^{20}z^{10}-1770336xy^{19}z^{11}-2176372xy^{18}z^{12}-2534028xy^{17}z^{13}-2284704xy^{16}z^{14}-1384080xy^{15}z^{15}+612293xy^{14}z^{16}+4266159xy^{13}z^{17}+9822728xy^{12}z^{18}+18265136xy^{11}z^{19}+30026258xy^{10}z^{20}+45471486xy^{9}z^{21}+67553936xy^{8}z^{22}+94743128xy^{7}z^{23}+121922359xy^{6}z^{24}+180846573xy^{5}z^{25}+249937920xy^{4}z^{26}+171376640xy^{3}z^{27}+173785088xy^{2}z^{28}+53764096xyz^{29}-y^{31}+153y^{30}z-2203y^{29}z^{2}+3337y^{28}z^{3}+1692y^{27}z^{4}+29960y^{26}z^{5}+72356y^{25}z^{6}+135989y^{24}z^{7}+254239y^{23}z^{8}+275359y^{22}z^{9}+208861y^{21}z^{10}-237118y^{20}z^{11}-1357216y^{19}z^{12}-3460736y^{18}z^{13}-7255456y^{17}z^{14}-13213246y^{16}z^{15}-22208547y^{15}z^{16}-35109985y^{14}z^{17}-52891361y^{13}z^{18}-76577995y^{12}z^{19}-107480412y^{11}z^{20}-147217144y^{10}z^{21}-195262820y^{9}z^{22}-256045815y^{8}z^{23}-335145115y^{7}z^{24}-408276839y^{6}z^{25}-515194881y^{5}z^{26}-770699264y^{4}z^{27}-806584320y^{3}z^{28}-522874880y^{2}z^{29}-502194176yz^{30})}$ |
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.
