Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x t + y w $ |
| $=$ | $x w - x t - y t + z t$ |
| $=$ | $x t - x u - y t + y v + z t$ |
| $=$ | $x^{2} + x y + y^{2} - y z$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 5 x^{6} y + x^{5} z^{2} + 3 x^{4} y z^{2} + 3 x^{2} y^{3} z^{2} - x y^{4} z^{2} + y^{3} z^{4} $ |
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This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:1/2:0:-1/2:1)$, $(0:0:0:0:0:1:0)$, $(0:0:0:0:0:0:1)$, $(0:0:0:0:1:0:0)$ |
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{1}{2}\cdot\frac{5942189561230941440z^{2}u^{10}+6848638156079151360z^{2}u^{9}v-92949498597580113920z^{2}u^{8}v^{2}-421001677153447954400z^{2}u^{7}v^{3}-936737988705091645520z^{2}u^{6}v^{4}-1419690239676239108360z^{2}u^{5}v^{5}-1707855390877328797100z^{2}u^{4}v^{6}-1818900768243951360590z^{2}u^{3}v^{7}-1845922883466450158225z^{2}u^{2}v^{8}-1849783409149197454440z^{2}uv^{9}-1850074963697817562880z^{2}v^{10}-740033908531614746812wtv^{10}+5243518869809416704wu^{11}+42045528501715419648wu^{10}v+154825715411233407232wu^{9}v^{2}+350954811264554675264wu^{8}v^{3}+560751655685090860832wu^{7}v^{4}+698571290439711311504wu^{6}v^{5}+747512231621991301064wu^{5}v^{6}+749635426557026345636wu^{4}v^{7}+743470889318188426094wu^{3}v^{8}+740654856077399479094wu^{2}v^{9}+740088966102952802624wuv^{10}+2960113639310012wv^{11}-53747712t^{12}+161243136t^{11}v-10037385216t^{10}v^{2}+39699403776t^{9}v^{3}-723548339712t^{8}v^{4}+3390320852352t^{7}v^{5}-26696403519840t^{6}v^{6}+128821939593408t^{5}v^{7}-602588788482888t^{4}v^{8}+2526331162257234t^{3}v^{9}-21663602115518441984t^{2}u^{10}-153173618739433424384t^{2}u^{9}v-506073636246261154176t^{2}u^{8}v^{2}-1056853581258892510080t^{2}u^{7}v^{3}-1610091751756451774496t^{2}u^{6}v^{4}-1983545645552813741952t^{2}u^{5}v^{5}-2155117892518246889760t^{2}u^{4}v^{6}-2208132612002316120336t^{2}u^{3}v^{7}-2218757900709117165516t^{2}u^{2}v^{8}-2220011237782595493638t^{2}uv^{9}-740033877368769381168t^{2}v^{10}+3406849392452971264tu^{11}+28004730588625006080tu^{10}v+117672313035189148160tu^{9}v^{2}+326503301001646545376tu^{8}v^{3}+655871638103462822896tu^{7}v^{4}+1013597494432309736056tu^{6}v^{5}+1282026615127607957524tu^{5}v^{6}+1420200613049410043818tu^{4}v^{7}+1467955121280324562423tu^{3}v^{8}+1478575852021382315867tu^{2}v^{9}+739933246315025873142tuv^{10}+1110047643992975109958tv^{11}-39182082048u^{12}-1188438147338680576u^{11}v-10718767642815958272u^{10}v^{2}-45009261828060391936u^{9}v^{3}-116609390054200257184u^{8}v^{4}-211783768341851792368u^{7}v^{5}-295581578932481051800u^{6}v^{6}-345030900823316531140u^{5}v^{7}-364387431986738581114u^{4}v^{8}-369237931433599239259u^{3}v^{9}-369959641943317557764u^{2}v^{10}-370014992739241026304uv^{11}-53747712v^{12}}{2033589438720z^{2}u^{10}-2655238561920z^{2}u^{9}v-29658550406720z^{2}u^{8}v^{2}-67011065040800z^{2}u^{7}v^{3}-89782289929600z^{2}u^{6}v^{4}-96172924681440z^{2}u^{5}v^{5}-96851314409580z^{2}u^{4}v^{6}-96863054612170z^{2}u^{3}v^{7}-96854981382395z^{2}u^{2}v^{8}-96867234528680z^{2}uv^{9}-96856815942640z^{2}v^{10}-38741483154164wtv^{10}+1794540552192wu^{11}+9976437674496wu^{10}v+24575077306368wu^{9}v^{2}+36672102260352wu^{8}v^{3}+40391732419616wu^{7}v^{4}+39610374986160wu^{6}v^{5}+38874255028856wu^{5}v^{6}+38748595672092wu^{4}v^{7}+38742089568218wu^{3}v^{8}+38746859825490wu^{2}v^{9}+38742235955360wuv^{10}-1243222892wv^{11}+6718464t^{10}v^{2}-26873856t^{9}v^{3}+41990400t^{8}v^{4}-56267136t^{7}v^{5}+143922096t^{6}v^{6}-401008320t^{5}v^{7}+1077316200t^{4}v^{8}-3009202650t^{3}v^{9}-7410729768192t^{2}u^{10}-34173143108352t^{2}u^{9}v-73081690974400t^{2}u^{8}v^{2}-102089112160640t^{2}u^{7}v^{3}-113706472308592t^{2}u^{6}v^{4}-116050812849248t^{2}u^{5}v^{5}-116232252298576t^{2}u^{4}v^{6}-116234849836544t^{2}u^{3}v^{7}-116227305497380t^{2}u^{2}v^{8}-116238378834338t^{2}uv^{9}-38740100525760t^{2}v^{10}+1164518657280tu^{11}+6713627518080tu^{10}v+21237249889856tu^{9}v^{2}+44082138043424tu^{8}v^{3}+64681938213760tu^{7}v^{4}+74805436434544tu^{6}v^{5}+77261380888652tu^{5}v^{6}+77486149594038tu^{4}v^{7}+77489415367293tu^{3}v^{8}+77485649266361tu^{2}v^{9}+38748846338930tuv^{10}+58113687119090tv^{11}-406717887744u^{11}v-2667060383616u^{10}v^{2}-7960052049344u^{9}v^{3}-14228670797408u^{8}v^{4}-18083778489152u^{7}v^{5}-19237756982528u^{6}v^{6}-19370654611028u^{5}v^{7}-19371955325006u^{4}v^{8}-19372151524169u^{3}v^{9}-19372203682844u^{2}v^{10}-19371363188528uv^{11}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
60.144.7.mf.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 5X^{6}Y+X^{5}Z^{2}+3X^{4}YZ^{2}+3X^{2}Y^{3}Z^{2}-XY^{4}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.
