$\GL_2(\Z/60\Z)$-generators: |
$\begin{bmatrix}1&35\\36&47\end{bmatrix}$, $\begin{bmatrix}53&21\\54&5\end{bmatrix}$, $\begin{bmatrix}59&16\\12&19\end{bmatrix}$, $\begin{bmatrix}59&40\\36&11\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
60.192.3-60.y.1.1, 60.192.3-60.y.1.2, 60.192.3-60.y.1.3, 60.192.3-60.y.1.4, 60.192.3-60.y.1.5, 60.192.3-60.y.1.6, 60.192.3-60.y.1.7, 60.192.3-60.y.1.8, 120.192.3-60.y.1.1, 120.192.3-60.y.1.2, 120.192.3-60.y.1.3, 120.192.3-60.y.1.4, 120.192.3-60.y.1.5, 120.192.3-60.y.1.6, 120.192.3-60.y.1.7, 120.192.3-60.y.1.8 |
Cyclic 60-isogeny field degree: |
$12$ |
Cyclic 60-torsion field degree: |
$192$ |
Full 60-torsion field degree: |
$23040$ |
Embedded model Embedded model in $\mathbb{P}^{5}$
$ 0 $ | $=$ | $ x w - x u - y w + y u + z t $ |
| $=$ | $2 x^{2} + x y + 2 x u + y t + y u + z t + w t + t^{2}$ |
| $=$ | $2 x w - 2 x u + y w + y t - y u - w t - t^{2} - t u$ |
| $=$ | $x y - 2 x z - x w - x t - x u - y^{2} - y z + y w + y t + y u$ |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{8} - 4 x^{7} y - 10 x^{7} z + 6 x^{6} y^{2} + 8 x^{6} y z - 68 x^{6} z^{2} - 4 x^{5} y^{3} + \cdots + 9801 z^{8} $ |
Weierstrass model Weierstrass model
$ y^{2} $ | $=$ | $ -5x^{8} - 200x^{4} - 720 $ |
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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 2u$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}t$ |
Birational map from embedded model to Weierstrass model:
$\displaystyle X$ |
$=$ |
$\displaystyle \frac{3}{145}y^{6}-\frac{4}{145}y^{5}t-\frac{12}{145}y^{5}u-\frac{131}{435}y^{4}t^{2}-\frac{36}{145}y^{4}tu+\frac{12}{145}y^{4}u^{2}-\frac{2272}{1305}y^{3}t^{3}-\frac{104}{435}y^{3}t^{2}u+\frac{48}{145}y^{3}tu^{2}+\frac{141}{145}y^{2}t^{4}+\frac{8}{15}y^{2}t^{3}u+\frac{248}{435}y^{2}t^{2}u^{2}+\frac{1484}{3915}yt^{5}-\frac{20}{783}yt^{4}u+\frac{16}{435}yt^{3}u^{2}+\frac{11}{3915}t^{6}+\frac{244}{3915}t^{5}u+\frac{244}{3915}t^{4}u^{2}$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{9}{3536405}y^{22}t^{2}+\frac{93}{3536405}y^{21}t^{3}-\frac{54}{3536405}y^{21}t^{2}u+\frac{1064}{3536405}y^{20}t^{4}-\frac{672}{3536405}y^{20}t^{3}u+\frac{108}{3536405}y^{20}t^{2}u^{2}+\frac{4144}{3536405}y^{19}t^{5}-\frac{7816}{3536405}y^{19}t^{4}u+\frac{1512}{3536405}y^{19}t^{3}u^{2}-\frac{72}{3536405}y^{19}t^{2}u^{3}+\frac{35231}{6365529}y^{18}t^{6}-\frac{8316}{707281}y^{18}t^{5}u+\frac{17964}{3536405}y^{18}t^{4}u^{2}-\frac{216}{707281}y^{18}t^{3}u^{3}-\frac{57523}{10609215}y^{17}t^{7}-\frac{365842}{6365529}y^{17}t^{6}u+\frac{110844}{3536405}y^{17}t^{5}u^{2}-\frac{13056}{3536405}y^{17}t^{4}u^{3}+\frac{3417976}{286448805}y^{16}t^{8}-\frac{2507876}{31827645}y^{16}t^{7}u+\frac{571284}{3536405}y^{16}t^{6}u^{2}-\frac{86952}{3536405}y^{16}t^{5}u^{3}-\frac{64074928}{859346415}y^{15}t^{9}-\frac{4876736}{57289761}y^{15}t^{8}u+\frac{1372992}{3536405}y^{15}t^{7}u^{2}-\frac{467808}{3536405}y^{15}t^{6}u^{3}+\frac{3878072714}{2578039245}y^{14}t^{10}+\frac{952906928}{859346415}y^{14}t^{9}u+\frac{53693744}{95482935}y^{14}t^{8}u^{2}-\frac{1383136}{3536405}y^{14}t^{7}u^{3}+\frac{4751154898}{2578039245}y^{13}t^{11}-\frac{1019701100}{515607849}y^{13}t^{10}u-\frac{45797200}{19096587}y^{13}t^{9}u^{2}-\frac{219421504}{286448805}y^{13}t^{8}u^{3}+\frac{94288387264}{23202353205}y^{12}t^{12}-\frac{6761610736}{859346415}y^{12}t^{11}u-\frac{293594056}{57289761}y^{12}t^{10}u^{2}+\frac{79516832}{95482935}y^{12}t^{9}u^{3}-\frac{257805468944}{7734117735}y^{11}t^{13}-\frac{532321717136}{23202353205}y^{11}t^{12}u-\frac{354310256}{286448805}y^{11}t^{11}u^{2}+\frac{3651592048}{859346415}y^{11}t^{10}u^{3}+\frac{288037695362}{7734117735}y^{10}t^{14}+\frac{1523583926296}{23202353205}y^{10}t^{13}u+\frac{20187099608}{859346415}y^{10}t^{12}u^{2}+\frac{872042512}{171869283}y^{10}t^{11}u^{3}-\frac{2029129127126}{626463536535}y^{9}t^{15}-\frac{1189051182020}{41764235769}y^{9}t^{14}u-\frac{15126517768}{859346415}y^{9}t^{13}u^{2}-\frac{27293561536}{2578039245}y^{9}t^{12}u^{3}-\frac{494319564896}{41764235769}y^{8}t^{16}-\frac{10021745517224}{626463536535}y^{8}t^{15}u+\frac{17824750552}{69607059615}y^{8}t^{14}u^{2}+\frac{591894800}{515607849}y^{8}t^{13}u^{3}+\frac{95633221648}{41764235769}y^{7}t^{17}+\frac{4470193818496}{626463536535}y^{7}t^{16}u+\frac{8976120704}{69607059615}y^{7}t^{15}u^{2}+\frac{204067892896}{208821178845}y^{7}t^{14}u^{3}+\frac{416977834357}{208821178845}y^{6}t^{18}+\frac{2312387082736}{626463536535}y^{6}t^{17}u+\frac{37558898192}{23202353205}y^{6}t^{16}u^{2}+\frac{20679516832}{23202353205}y^{6}t^{15}u^{3}-\frac{73947667207}{208821178845}y^{5}t^{19}-\frac{134267322130}{125292707307}y^{5}t^{18}u-\frac{20686036112}{69607059615}y^{5}t^{17}u^{2}-\frac{13079245888}{69607059615}y^{5}t^{16}u^{3}-\frac{24207913192}{208821178845}y^{4}t^{20}-\frac{112524351824}{626463536535}y^{4}t^{19}u+\frac{6570128948}{69607059615}y^{4}t^{18}u^{2}+\frac{426866912}{41764235769}y^{4}t^{17}u^{3}-\frac{109261504}{69607059615}y^{3}t^{21}-\frac{11579878328}{626463536535}y^{3}t^{20}u-\frac{101091656}{1546823547}y^{3}t^{19}u^{2}-\frac{3668641112}{69607059615}y^{3}t^{18}u^{3}+\frac{207522623}{208821178845}y^{2}t^{22}-\frac{1213594052}{626463536535}y^{2}t^{21}u-\frac{900709436}{69607059615}y^{2}t^{20}u^{2}-\frac{635891432}{69607059615}y^{2}t^{19}u^{3}+\frac{14216411}{208821178845}yt^{23}-\frac{66599126}{626463536535}yt^{22}u-\frac{10430684}{13921411923}yt^{21}u^{2}-\frac{106255424}{208821178845}yt^{20}u^{3}+\frac{819896}{626463536535}t^{24}-\frac{1283084}{626463536535}t^{23}u-\frac{324764}{23202353205}t^{22}u^{2}-\frac{649528}{69607059615}t^{21}u^{3}$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{3}{145}y^{6}+\frac{1}{145}y^{5}t-\frac{12}{145}y^{5}u-\frac{22}{145}y^{4}t^{2}-\frac{36}{145}y^{4}tu+\frac{12}{145}y^{4}u^{2}-\frac{782}{1305}y^{3}t^{3}-\frac{104}{435}y^{3}t^{2}u+\frac{48}{145}y^{3}tu^{2}+\frac{2417}{3915}y^{2}t^{4}+\frac{8}{15}y^{2}t^{3}u+\frac{248}{435}y^{2}t^{2}u^{2}+\frac{379}{3915}yt^{5}-\frac{20}{783}yt^{4}u+\frac{16}{435}yt^{3}u^{2}-\frac{44}{3915}t^{6}+\frac{244}{3915}t^{5}u+\frac{244}{3915}t^{4}u^{2}$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{2\cdot5^{16}}\cdot\frac{119297146872260914470xt^{11}+1421889638660433884925xt^{10}u+6352032721480796481840xt^{9}u^{2}+14776955444949173394390xt^{8}u^{3}+19045233584182211515290xt^{7}u^{4}+10997379336501994824225xt^{6}u^{5}-3630747614834851200360xt^{5}u^{6}-10066177332265511715840xt^{4}u^{7}-5858766689883921510390xt^{3}u^{8}-1035976254803938439415xt^{2}u^{9}+60024313283589697680xtu^{10}+2583121012674262560xu^{11}-119588677353682609910yt^{11}-812886457750273910565yt^{10}u-3223762405856223509880yt^{9}u^{2}-9396482105676703468230yt^{8}u^{3}-19478716697873332697130yt^{7}u^{4}-27166079344406123905785yt^{6}u^{5}-24188277958395784344000yt^{5}u^{6}-12459671379571999893120yt^{4}u^{7}-2731421331716591299770yt^{3}u^{8}+193742649979645614255yt^{2}u^{9}+96119174236957177320ytu^{10}+11805745138237440yu^{11}+25036079958600712932z^{2}t^{10}+150535644386891641830z^{2}t^{9}u+470183793328983929040z^{2}t^{8}u^{2}+979311425904529685460z^{2}t^{7}u^{3}+1382790975313915589220z^{2}t^{6}u^{4}+1253078631816699433086z^{2}t^{5}u^{5}+654999574208652292320z^{2}t^{4}u^{6}+149717659976318482560z^{2}t^{3}u^{7}-3061469492587903860z^{2}t^{2}u^{8}-2601146825291427570z^{2}tu^{9}+3541723541471232z^{2}u^{10}+367920841419567779286zwt^{10}+2572267382270281686465zwt^{9}u+8035889530984749516420zwt^{8}u^{2}+14595092213717267296830zwt^{7}u^{3}+16375613309731445751810zwt^{6}u^{4}+10759043762559693139653zwt^{5}u^{5}+3028374855737932450860zwt^{4}u^{6}-598162961193096597120zwt^{3}u^{7}-536996748705259155030zwt^{2}u^{8}-56089835611874513235zwtu^{9}+1208826119907878436zwu^{10}+280014300289317106000zt^{11}+1752816734678503497714zt^{10}u+3872525230154029856535zt^{9}u^{2}+1740472854712945097580zt^{8}u^{3}-8503634010790730626830zt^{7}u^{4}-19558587530645609562810zt^{6}u^{5}-19212972619493354296653zt^{5}u^{6}-8900084888861518498860zt^{4}u^{7}-970192070423298596880zt^{3}u^{8}+501410984185337280030zt^{2}u^{9}+69587934172733888235ztu^{10}-1208826119907878436zu^{11}+211080205100328287370w^{2}t^{10}+1529266152159432417825w^{2}t^{9}u+5378876179512853261500w^{2}t^{8}u^{2}+11619865552580276133150w^{2}t^{7}u^{3}+16070659441136545180950w^{2}t^{6}u^{4}+13659831052341687464085w^{2}t^{5}u^{5}+6097206140259319428300w^{2}t^{4}u^{6}+549512387470226016000w^{2}t^{3}u^{7}-473718745033802845650w^{2}t^{2}u^{8}-68650640518021431075w^{2}tu^{9}+1152143601810384420w^{2}u^{10}+418746201236141849000wt^{11}+3008262379154667397140wt^{10}u+9604305322837877395440wt^{9}u^{2}+17582017961468365228440wt^{8}u^{3}+19305687643379040948840wt^{7}u^{4}+11443339574908750139940wt^{6}u^{5}+1559236725714936249360wt^{5}u^{6}-2029656455205228253440wt^{4}u^{7}-926012494797754114440wt^{3}u^{8}-38779505874822665340wt^{2}u^{9}+2514541135018012560wtu^{10}+95061300689360743970t^{12}+865781701189009061465t^{11}u+4089993664623311489580t^{10}u^{2}+11874129058810978079535t^{9}u^{3}+22762958465529812392110t^{8}u^{4}+30108166268928762864195t^{7}u^{5}+27873423664686383158080t^{6}u^{6}+17358548342527661716125t^{5}u^{7}+6051464971666212753090t^{4}u^{8}+219420921868253604765t^{3}u^{9}-517047182878739135040t^{2}u^{10}-67542422543481174795tu^{11}+1208830641810384420u^{12}}{t^{6}(2142xt^{5}+21294xt^{4}u+36624xt^{3}u^{2}+117xt^{2}u^{3}-29970xtu^{4}-5832xu^{5}-9086yt^{5}-35406yt^{4}u-68448yt^{3}u^{2}-73341yt^{2}u^{3}-34830ytu^{4}+1524z^{2}t^{4}+3780z^{2}t^{3}u+4104z^{2}t^{2}u^{2}+1782z^{2}tu^{3}+14166zwt^{4}+40302zwt^{3}u+38880zwt^{2}u^{2}+10449zwtu^{3}-2430zwu^{4}+5632zt^{5}+522zt^{4}u-41166zt^{3}u^{2}-61560zt^{2}u^{3}-23409ztu^{4}+2430zu^{5}+11262w^{2}t^{4}+36342w^{2}t^{3}u+45036w^{2}t^{2}u^{2}+20817w^{2}tu^{3}-2430w^{2}u^{4}+15752wt^{5}+47256wt^{4}u+42624wt^{3}u^{2}+6804wt^{2}u^{3}-5184wtu^{4}+6662t^{6}+29858t^{5}u+62334t^{4}u^{2}+74763t^{3}u^{3}+56268t^{2}u^{4}+19197tu^{5}-2430u^{6})}$ |
Hi
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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.