| $\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}13&14\\13&31\end{bmatrix}$, $\begin{bmatrix}25&38\\24&15\end{bmatrix}$, $\begin{bmatrix}33&16\\21&35\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
80.192.1-40.cn.1.1, 80.192.1-40.cn.1.2, 80.192.1-40.cn.1.3, 80.192.1-40.cn.1.4, 80.192.1-40.cn.1.5, 80.192.1-40.cn.1.6, 80.192.1-40.cn.1.7, 80.192.1-40.cn.1.8, 240.192.1-40.cn.1.1, 240.192.1-40.cn.1.2, 240.192.1-40.cn.1.3, 240.192.1-40.cn.1.4, 240.192.1-40.cn.1.5, 240.192.1-40.cn.1.6, 240.192.1-40.cn.1.7, 240.192.1-40.cn.1.8 |
| Cyclic 40-isogeny field degree: |
$12$ |
| Cyclic 40-torsion field degree: |
$192$ |
| Full 40-torsion field degree: |
$7680$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x^{2} - 4 x y - 2 x z - 3 y^{2} + 2 y z - 2 z^{2} $ |
| $=$ | $5 x^{2} + 10 x y - 5 y^{2} + 2 w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} - 16 x^{3} y - 20 x^{2} y^{2} + 30 x^{2} z^{2} - 8 x y^{3} + 40 x y z^{2} - 79 y^{4} + \cdots + 25 z^{4} $ |
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This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{2}{5}w$ |
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{2^3}{5^3}\cdot\frac{24495350086273448855562888425740936187400000000000xz^{23}+76348381906706197608588177634272788008980000000000xz^{21}w^{2}+96122994494325701036996205286598461472910000000000xz^{19}w^{4}+64696980911708157060973682370221278675927000000000xz^{17}w^{6}+26846664375198585499943702031325729277000360000000xz^{15}w^{8}+7666824587199879938067900874945048557154948000000xz^{13}w^{10}+1548382482352999639487101606722139016273585200000xz^{11}w^{12}+223740729960507246159055471599108833106813400000xz^{9}w^{14}+22413313807451493800708253243730295700658218000xz^{7}w^{16}+1466022134745053074903221747229990352240127400xz^{5}w^{18}+53667322572029034294849663399154382048954300xz^{3}w^{20}+748200242719273488843351920591112747871006xzw^{22}+29815665064298214982026733351023766258500000000000y^{2}z^{22}+93346232968258739355698652759099024310650000000000y^{2}z^{20}w^{2}+118133434405130471892661549970459034885015000000000y^{2}z^{18}w^{4}+79881857904370600369031631734150924688079500000000y^{2}z^{16}w^{6}+33162961665077828318210657045072997170577300000000y^{2}z^{14}w^{8}+9391738507743930571100008169584300235413874000000y^{2}z^{12}w^{10}+1861581415671815874516942411573478668383464600000y^{2}z^{10}w^{12}+259236483616010703262959151552091874043663500000y^{2}z^{8}w^{14}+24286406393772303863771401363231335886798053000y^{2}z^{6}w^{16}+1396185031718125661278615450376622662827096500y^{2}z^{4}w^{18}+38741283882068455898778825356041411396030590y^{2}z^{2}w^{20}+228408624524139922644116322379178346130967y^{2}w^{22}-18244025523629252177813547488287318571400000000000yz^{23}-60231789412118347993419760449303459808980000000000yz^{21}w^{2}-82006403349261252416670686123276495168430000000000yz^{19}w^{4}-61118445430715480192232336194185835298471000000000yz^{17}w^{6}-28488784553614986150035153485940435143050280000000yz^{15}w^{8}-9091787090887601019743834155130581275300932000000yz^{13}w^{10}-2061593061321506536117723631028228080841937200000yz^{11}w^{12}-334329716083092380498323260859119550499277400000yz^{9}w^{14}-37863601878179494959455251796954473906999434000yz^{7}w^{16}-2801316103899826344791271563281328503223356200yz^{5}w^{18}-116610037071142341896187310028377742035487740yz^{3}w^{20}-1807834868489540400056077862131143380145086yzw^{22}+13823680622202449656518048665482720596600000000000z^{24}+41872536665843681892011216627718639644580000000000z^{22}w^{2}+50632871145283111069839117112908693645726000000000z^{20}w^{4}+32275452235723936945229453462208814978547000000000z^{18}w^{6}+12601586020719810204530171769537420505953600000000z^{16}w^{8}+3428722856459684751358971728252985611592276000000z^{14}w^{10}+665984659236678084261548386349390604576574000000z^{12}w^{12}+95515215326171294202560674639321561632528440000z^{10}w^{14}+9954600805135061876730069684121942345311966000z^{8}w^{16}+753446354008692035667780861004723582002278600z^{6}w^{18}+38186307453846478195161142837176246126778140z^{4}w^{20}+1108552477033136349111859052010169209594774z^{2}w^{22}+8955571535024644202463281025320396722290w^{24}}{907235188380498105761588460212627266200000000xz^{23}+10987295560169521983933877713049458240000000xz^{21}w^{2}-53018272300709514626409960143842077044000000xz^{19}w^{4}+2805768066382050412368482198548114376400000xz^{17}w^{6}+1022294293258001864957009397505502263600000xz^{15}w^{8}-131996232076331344827142744618009377468000xz^{13}w^{10}+1076528527975963804278768592285752392400xz^{11}w^{12}+1385783012797151040735556270622719748440xz^{9}w^{14}-266095597838951338415721171563775771886xz^{7}w^{16}+17395535563269286820286704182022879214xz^{5}w^{18}+2326428882520704339398767371263867472xz^{3}w^{20}-389475689145499181427545685193830072xzw^{22}+1104283891270304258593582716704583935500000000y^{2}z^{22}+28751106445309497021641225990624238320000000y^{2}z^{20}w^{2}-70333716481074775470298076193348987818000000y^{2}z^{18}w^{4}+2683190128338376447708327483169899308200000y^{2}z^{16}w^{6}+1726575136835023920163450149554465195800000y^{2}z^{14}w^{8}-188779807007411679049743138972380162366000y^{2}z^{12}w^{10}-11743830449171470037581388732887315003800y^{2}z^{10}w^{12}+4156196244530953227899133620055340583180y^{2}z^{8}w^{14}-372108197695545839319221558441622601575y^{2}z^{6}w^{16}-7403836814461562304380864471278581805y^{2}z^{4}w^{18}+4755405791292150785686947071112047220y^{2}z^{2}w^{20}-209975432747015001430335662673893460y^{2}w^{22}-675704649023305636215316573640271058200000000yz^{23}-132920449544948928390003865703592547840000000yz^{21}w^{2}+41044282353070895677767784031433163828000000yz^{19}w^{4}+6204322417004246513573115355272372913200000yz^{17}w^{6}-1443089390022800541596527976127732137520000yz^{15}w^{8}-96627149138413214879206773127111788356000yz^{13}w^{10}+34757467036977698024659621305549189591600yz^{11}w^{12}-1040922636784270282984642842766376613080yz^{9}w^{14}-429495370841166636554964695931557220594yz^{7}w^{16}+65246168014899956443062661449665701506yz^{5}w^{18}-3868794139171938394085786312501080272yz^{3}w^{20}+142677068668794303000590433944038392yzw^{22}+511988171192683320611779580203063725800000000z^{24}-38755788379798444307473925250348628560000000z^{22}w^{2}-24159204975994534694909103103866528500000000z^{20}w^{4}+4173838818771092312543318163438717312400000z^{18}w^{6}-1292460793117658091033339250422496960000z^{16}w^{8}-91417450735341940756877747567633384348000z^{14}w^{10}+18542692506906399936355560821534351861200z^{12}w^{12}-1257076896502499299402827022193907520520z^{10}w^{14}-246468627210314426706337720745478457002z^{8}w^{16}+60996496130339208429934612352759534114z^{6}w^{18}-2929709225619757255150501440630969592z^{4}w^{20}-453202552005246279476380559817519696z^{2}w^{22}+57138572186625410116438230985162752w^{24}}$ 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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.
