Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x y + x t - y t $ |
| $=$ | $x^{2} + x y - x w - x t - y^{2} - y z + y t - z^{2} + z t + w^{2} + w t$ |
| $=$ | $x z - x t + y^{2} + y z - y w - 2 y t + z^{2} - 2 z w - 2 z t + w t$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} z - 2 x^{5} z^{2} - x^{4} y^{3} - 3 x^{4} y^{2} z + 2 x^{4} y z^{2} + 2 x^{4} z^{3} + \cdots + y^{3} z^{4} $ |
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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 |
|---|
| $(1:0:1:1:0)$, $(-1:0:1:0:0)$, $(0:-1:1:1:0)$, $(0:0:1:-1:1)$, $(0:0:0:0:1)$, $(0:1:0:1:0)$ |
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{728xz^{23}-12352xz^{22}t+10835xz^{21}t^{2}+301143xz^{20}t^{3}-1045031xz^{19}t^{4}+1597892xz^{18}t^{5}-5406791xz^{17}t^{6}+28544920xz^{16}t^{7}-117298058xz^{15}t^{8}+431421161xz^{14}t^{9}-1435979737xz^{13}t^{10}+4050295544xz^{12}t^{11}-9761877166xz^{11}t^{12}+20296293655xz^{10}t^{13}-34963806579xz^{9}t^{14}+48860070362xz^{8}t^{15}-50485389305xz^{7}t^{16}+22185714465xz^{6}t^{17}+21135976840xz^{5}t^{18}-77123938479xz^{4}t^{19}+100904311511xz^{3}t^{20}+111323232686xz^{2}t^{21}+191456386799xzt^{22}-13122xw^{22}t-115183xw^{21}t^{2}+304706xw^{20}t^{3}+5609523xw^{19}t^{4}+7305937xw^{18}t^{5}-96041084xw^{17}t^{6}-311501351xw^{16}t^{7}+598578241xw^{15}t^{8}+4021272510xw^{14}t^{9}+1414924705xw^{13}t^{10}-23017940866xw^{12}t^{11}-34929390134xw^{11}t^{12}+50790330943xw^{10}t^{13}+143340135456xw^{9}t^{14}-16008501904xw^{8}t^{15}-183545575158xw^{7}t^{16}+172975794439xw^{6}t^{17}+307721031879xw^{5}t^{18}-762409218555xw^{4}t^{19}-1487061422500xw^{3}t^{20}-296741575048xw^{2}t^{21}+1093727929457xwt^{22}+702167602138xt^{23}-13122yz^{22}t+102061yz^{21}t^{2}+10896yz^{20}t^{3}-1263339yz^{19}t^{4}+1874385yz^{18}t^{5}+132503yz^{17}t^{6}+3047168yz^{16}t^{7}-2133281yz^{15}t^{8}-28905706yz^{14}t^{9}+83036780yz^{13}t^{10}-240127149yz^{12}t^{11}+705124322yz^{11}t^{12}-1761573355yz^{10}t^{13}+3642383544yz^{9}t^{14}-5960741601yz^{8}t^{15}+8767567725yz^{7}t^{16}-7210561949yz^{6}t^{17}-627074471yz^{5}t^{18}-274627811yz^{4}t^{19}-38216836650yz^{3}t^{20}-7132902940yz^{2}t^{21}+18954yzw^{21}t+166268yzw^{20}t^{2}+1291852yzw^{19}t^{3}+2316582yzw^{18}t^{4}-14723016yzw^{17}t^{5}-86101306yzw^{16}t^{6}-319754141yzw^{15}t^{7}-516473542yzw^{14}t^{8}+3292117183yzw^{13}t^{9}+17931159987yzw^{12}t^{10}+16681804817yzw^{11}t^{11}-88698704048yzw^{10}t^{12}-259322030904yzw^{9}t^{13}-74153987709yzw^{8}t^{14}+732569573884yzw^{7}t^{15}+1248195546783yzw^{6}t^{16}+224446141446yzw^{5}t^{17}-1648262855061yzw^{4}t^{18}-2228078032446yzw^{3}t^{19}-891790773765yzw^{2}t^{20}+460920740912yzwt^{21}+93571143305yzt^{22}-728yw^{23}-13833yw^{22}t+33665yw^{21}t^{2}+1247613yw^{20}t^{3}+3895531yw^{19}t^{4}-16347827yw^{18}t^{5}-98999043yw^{17}t^{6}-5877940yw^{16}t^{7}+697177954yw^{15}t^{8}+150953781yw^{14}t^{9}-3302699219yw^{13}t^{10}+8833102120yw^{12}t^{11}+47841696043yw^{11}t^{12}-25193563653yw^{10}t^{13}-369787455582yw^{9}t^{14}-435047693836yw^{8}t^{15}+787727665330yw^{7}t^{16}+2380008757588yw^{6}t^{17}+1386573501370yw^{5}t^{18}-2391612724486yw^{4}t^{19}-4828113459384yw^{3}t^{20}-3120608742233yw^{2}t^{21}+281188616352ywt^{22}+891382468370yt^{23}+729z^{24}-13850z^{23}t+24746z^{22}t^{2}+358726z^{21}t^{3}-1597170z^{20}t^{4}+1981139z^{19}t^{5}-908811z^{18}t^{6}+6120955z^{17}t^{7}-22393680z^{16}t^{8}+71318800z^{15}t^{9}-298188867z^{14}t^{10}+994100998z^{13}t^{11}-2613959775z^{12}t^{12}+5976806832z^{11}t^{13}-11459885806z^{10}t^{14}+17999815488z^{9}t^{15}-22019119937z^{8}t^{16}+15599241363z^{7}t^{17}-1339187459z^{6}t^{18}-23759743530z^{5}t^{19}+49704879600z^{4}t^{20}+37062236604z^{3}t^{21}+84461924916z^{2}t^{22}-2185zw^{23}-8748zw^{22}t+333449zw^{21}t^{2}+3147454zw^{20}t^{3}+6443832zw^{19}t^{4}-42682343zw^{18}t^{5}-258623803zw^{17}t^{6}-321030423zw^{16}t^{7}+1268708008zw^{15}t^{8}+5542386555zw^{14}t^{9}+10053856490zw^{13}t^{10}+5465006206zw^{12}t^{11}-53026942831zw^{11}t^{12}-238763844888zw^{10}t^{13}-339557987289zw^{9}t^{14}+416302891736zw^{8}t^{15}+2103469953505zw^{7}t^{16}+2313009974413zw^{6}t^{17}-1260395750858zw^{5}t^{18}-5630230095029zw^{4}t^{19}-5309360760756zw^{3}t^{20}-967777943638zw^{2}t^{21}+1402015635620zwt^{22}+504032870127zt^{23}+729w^{24}+14579w^{23}t-8002w^{22}t^{2}-1154886w^{21}t^{3}-5057963w^{20}t^{4}+10764386w^{19}t^{5}+119788968w^{18}t^{6}+159935167w^{17}t^{7}-754025878w^{16}t^{8}-2314903538w^{15}t^{9}-2552620w^{14}t^{10}+3973927683w^{13}t^{11}-4554032187w^{12}t^{12}+22130101342w^{11}t^{13}+181596979758w^{10}t^{14}+190145811819w^{9}t^{15}-633202543152w^{8}t^{16}-1722055096268w^{7}t^{17}-722239173090w^{6}t^{18}+2615646307674w^{5}t^{19}+4289549913547w^{4}t^{20}+1846498299162w^{3}t^{21}-1249202103616w^{2}t^{22}-1089517200375wt^{23}+t^{24}}{t^{7}(xz^{16}-34xz^{15}t+563xz^{14}t^{2}-6031xz^{13}t^{3}+46798xz^{12}t^{4}-278961xz^{11}t^{5}+1321314xz^{10}t^{6}-5070924xz^{9}t^{7}+15923782xz^{8}t^{8}-40994947xz^{7}t^{9}+86030589xz^{6}t^{10}-145010111xz^{5}t^{11}+191009017xz^{4}t^{12}-187400062xz^{3}t^{13}+125626058xz^{2}t^{14}-48852428xzt^{15}-xw^{14}t^{2}-9xw^{13}t^{3}-68xw^{12}t^{4}-178xw^{11}t^{5}-1956xw^{10}t^{6}-316xw^{9}t^{7}-17766xw^{8}t^{8}+23203xw^{7}t^{9}-45577xw^{6}t^{10}+189785xw^{5}t^{11}-45736xw^{4}t^{12}+680404xw^{3}t^{13}-284082xw^{2}t^{14}+1298622xwt^{15}+8335915xt^{16}-yz^{14}t^{2}+32yz^{13}t^{3}-495yz^{12}t^{4}+4913yz^{11}t^{5}-34992yz^{10}t^{6}+189328yz^{9}t^{7}-802777yz^{8}t^{8}+2709237yz^{7}t^{9}-7304031yz^{6}t^{10}+15605554yz^{5}t^{11}-25823161yz^{4}t^{12}+31528295yz^{3}t^{13}-25518311yz^{2}t^{14}+18yzw^{14}t-26yzw^{13}t^{2}+1338yzw^{12}t^{3}-2475yzw^{11}t^{4}+13869yzw^{10}t^{5}-41637yzw^{9}t^{6}+34410yzw^{8}t^{7}-181076yzw^{7}t^{8}+43632yzw^{6}t^{9}-346832yzw^{5}t^{10}+688632yzw^{4}t^{11}-1087827yzw^{3}t^{12}+1691221yzw^{2}t^{13}-449545yzwt^{14}+9828751yzt^{15}+yw^{16}+9yw^{15}t+83yw^{14}t^{2}+179yw^{13}t^{3}+2043yw^{12}t^{4}-138yw^{11}t^{5}+15877yw^{10}t^{6}-38903yw^{9}t^{7}+44327yw^{8}t^{8}-294446yw^{7}t^{9}+167034yw^{6}t^{10}-527494yw^{5}t^{11}+257880yw^{4}t^{12}+1206017yw^{3}t^{13}-1396859yw^{2}t^{14}+3098115ywt^{15}+1294231yt^{16}-z^{16}t+33z^{15}t^{2}-528z^{14}t^{3}+5440z^{13}t^{4}-40400z^{12}t^{5}+229230z^{11}t^{6}-1027010z^{10}t^{7}+3700154z^{9}t^{8}-10805953z^{8}t^{9}+25559614z^{7}t^{10}-48481922z^{6}t^{11}+72183019z^{5}t^{12}-81181118z^{4}t^{13}+64545308z^{3}t^{14}-32464918z^{2}t^{15}+36zw^{15}t+5zw^{14}t^{2}+1581zw^{13}t^{3}-1280zw^{12}t^{4}+15405zw^{11}t^{5}-45710zw^{10}t^{6}+58471zw^{9}t^{7}-358714zw^{8}t^{8}+269580zw^{7}t^{9}-855637zw^{6}t^{10}+793859zw^{5}t^{11}+237862zw^{4}t^{12}+341489zw^{3}t^{13}+1693028zw^{2}t^{14}+2174561zwt^{15}+8767668zt^{16}+w^{16}t-26w^{15}t^{2}+70w^{14}t^{3}-1374w^{13}t^{4}+3274w^{12}t^{5}-14977w^{11}t^{6}+60608w^{10}t^{7}-81976w^{9}t^{8}+370067w^{8}t^{9}-429097w^{7}t^{10}+805827w^{6}t^{11}-1156034w^{5}t^{12}+4791w^{4}t^{13}-376188w^{3}t^{14}-2169976w^{2}t^{15}-862478wt^{16})}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
21.192.5.c.3
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{6}Z-2X^{5}Z^{2}-X^{4}Y^{3}-3X^{4}Y^{2}Z+2X^{4}YZ^{2}+2X^{4}Z^{3}+4X^{3}Y^{3}Z+3X^{3}Y^{2}Z^{2}-4X^{3}YZ^{3}-X^{3}Z^{4}-X^{2}Y^{4}Z-7X^{2}Y^{3}Z^{2}+2X^{2}Y^{2}Z^{3}+3X^{2}YZ^{4}+5XY^{4}Z^{2}-3XY^{2}Z^{4}-Y^{5}Z^{2}+Y^{3}Z^{4} $ |
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.
