Canonical model in $\mathbb{P}^{ 2 }$
| $ 0 $ | $=$ | $ x^{3} y - 2 x^{2} y^{2} + 2 x^{2} y z + x y^{3} + 2 x y^{2} z - 2 x y z^{2} - x z^{3} - y z^{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 |
|---|
| $(1:0:0)$, $(1:0:1)$, $(0:1:1)$, $(0:1:0)$, $(1/2:1/2:1)$, $(1:1:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{x^{24}-24x^{22}z^{2}+24x^{21}z^{3}+960x^{20}z^{4}-3408x^{19}z^{5}-8036x^{18}z^{6}+61728x^{17}z^{7}+107676x^{16}z^{8}-1707472x^{15}z^{9}+4650912x^{14}z^{10}+2880552x^{13}z^{11}-36583946x^{12}z^{12}-25364352x^{11}z^{13}+785856672x^{10}z^{14}-3291072960x^{9}z^{15}+6760934922x^{8}z^{16}-1708475520x^{7}z^{17}-41749681752x^{6}z^{18}+183571148088x^{5}z^{19}-492844317780x^{4}z^{20}+891102120944x^{3}z^{21}+184549354x^{2}y^{22}-15032377948x^{2}y^{21}z+375171413988x^{2}y^{20}z^{2}-4545910135458x^{2}y^{19}z^{3}+32456493574592x^{2}y^{18}z^{4}-151422306632328x^{2}y^{17}z^{5}+492454732871512x^{2}y^{16}z^{6}-1173624238520592x^{2}y^{15}z^{7}+2159182035806000x^{2}y^{14}z^{8}-3269889273159132x^{2}y^{13}z^{9}+4344675961055720x^{2}y^{12}z^{10}-5195829706986324x^{2}y^{11}z^{11}+5489124844463922x^{2}y^{10}z^{12}-5069775022995252x^{2}y^{9}z^{13}+4211530833873328x^{2}y^{8}z^{14}-3174830601585634x^{2}y^{7}z^{15}+2027617271037293x^{2}y^{6}z^{16}-1028906727075248x^{2}y^{5}z^{17}+437274445456474x^{2}y^{4}z^{18}-142582293012377x^{2}y^{3}z^{19}-3648182306211x^{2}y^{2}z^{20}+20897734749273x^{2}yz^{21}-520414064700x^{2}z^{22}-167772140xy^{23}+11140064852xy^{22}z-221223836288xy^{21}z^{2}+2043832133906xy^{20}z^{3}-10240387055282xy^{19}z^{4}+27792904506904xy^{18}z^{5}-25843063724808xy^{17}z^{6}-87012721293056xy^{16}z^{7}+399511773866440xy^{15}z^{8}-862988912376860xy^{14}z^{9}+1344019039683780xy^{13}z^{10}-1813697812228628xy^{12}z^{11}+2213334813700400xy^{11}z^{12}-2273222755132044xy^{10}z^{13}+1917132691241576xy^{9}z^{14}-1455549825762350xy^{8}z^{15}+1004623807701424xy^{7}z^{16}-466341719047368xy^{6}z^{17}+52891371342866xy^{5}z^{18}+61049277765769xy^{4}z^{19}-74753561741653xy^{3}z^{20}+80909386626421xy^{2}z^{21}-29178126666837xyz^{22}-4269761997048xz^{23}+y^{24}-24y^{22}z^{2}+167772164y^{21}z^{3}-11643380312y^{20}z^{4}+255986204556y^{19}z^{5}-2799996389636y^{18}z^{6}+18375682684472y^{17}z^{7}-79957183770724y^{16}z^{8}+245805373643728y^{15}z^{9}-561719786300032y^{14}z^{10}+1006804357745720y^{13}z^{11}-1506966147693948y^{12}z^{12}+1991025779236416y^{11}z^{13}-2362481002580852y^{10}z^{14}+2478467097997820y^{9}z^{15}-2292393695297976y^{8}z^{16}+1915298893337204y^{7}z^{17}-1444406754031380y^{6}z^{18}+929511840149486y^{5}z^{19}-491333192000654y^{4}z^{20}+221257429561870y^{3}z^{21}-77261204319958y^{2}z^{22}+8280391917540yz^{23}+4247605514520z^{24}}{z^{4}(x^{20}-4x^{19}z-10x^{18}z^{2}+80x^{17}z^{3}-119x^{16}z^{4}-176x^{15}z^{5}+970x^{14}z^{6}-2524x^{13}z^{7}+7220x^{12}z^{8}-19808x^{11}z^{9}+37170x^{10}z^{10}-5700x^{9}z^{11}-337309x^{8}z^{12}+1966052x^{7}z^{13}-7789874x^{6}z^{14}+24440308x^{5}z^{15}-59739521x^{4}z^{16}+88815552x^{3}z^{17}+9437166x^{2}y^{18}-534769476x^{2}y^{17}z+9634048304x^{2}y^{16}z^{2}-86008371288x^{2}y^{15}z^{3}+455833295802x^{2}y^{14}z^{4}-1570112044722x^{2}y^{13}z^{5}+3678549347718x^{2}y^{12}z^{6}-5940416976738x^{2}y^{11}z^{7}+6447114124761x^{2}y^{10}z^{8}-4180247585853x^{2}y^{9}z^{9}+712036339448x^{2}y^{8}z^{10}+1309972041304x^{2}y^{7}z^{11}-1157116381355x^{2}y^{6}z^{12}+244842696116x^{2}y^{5}z^{13}+167129501444x^{2}y^{4}z^{14}-100396084913x^{2}y^{3}z^{15}+2739519366x^{2}y^{2}z^{16}+6854289318x^{2}yz^{17}+117573598x^{2}z^{18}-8388592xy^{19}+371192316xy^{18}z-5060756644xy^{17}z^{2}+31843858104xy^{16}z^{3}-100279945196xy^{15}z^{4}+105766863990xy^{14}z^{5}+349112740262xy^{13}z^{6}-1643183815570xy^{12}z^{7}+3202086167150xy^{11}z^{8}-3416017326885xy^{10}z^{9}+1544681028021xy^{9}z^{10}+819558090080xy^{8}z^{11}-1561021092422xy^{7}z^{12}+696630533344xy^{6}z^{13}+178587533110xy^{5}z^{14}-276989520255xy^{4}z^{15}+52963737287xy^{3}z^{16}+33245280754xy^{2}z^{17}-10649335998xyz^{18}-1636842856xz^{19}+y^{20}-4y^{19}z-10y^{18}z^{2}+8388672y^{17}z^{3}-396358211y^{16}z^{4}+6241442152y^{15}z^{5}-50164486008y^{14}z^{6}+244251994516y^{13}z^{7}-784676964292y^{12}z^{8}+1735587623104y^{11}z^{9}-2674868038216y^{10}z^{10}+2802740391648y^{9}z^{11}-1788050804376y^{8}z^{12}+345303461640y^{7}z^{13}+480620926392y^{6}z^{14}-447837930718y^{5}z^{15}+113238348174y^{4}z^{16}+48719862026y^{3}z^{17}-35984800120y^{2}z^{18}+3795046680yz^{19}+1471896950z^{20})}$ |
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.
