| $\GL_2(\Z/8\Z)$-generators: |
$\begin{bmatrix}1&4\\0&1\end{bmatrix}$, $\begin{bmatrix}5&0\\0&3\end{bmatrix}$, $\begin{bmatrix}7&0\\0&3\end{bmatrix}$, $\begin{bmatrix}7&4\\0&7\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: |
$C_2^4$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
8.192.1-8.b.1.1, 8.192.1-8.b.1.2, 8.192.1-8.b.1.3, 8.192.1-8.b.1.4, 8.192.1-8.b.1.5, 8.192.1-8.b.1.6, 16.192.1-8.b.1.1, 16.192.1-8.b.1.2, 16.192.1-8.b.1.3, 16.192.1-8.b.1.4, 24.192.1-8.b.1.1, 24.192.1-8.b.1.2, 24.192.1-8.b.1.3, 24.192.1-8.b.1.4, 24.192.1-8.b.1.5, 24.192.1-8.b.1.6, 40.192.1-8.b.1.1, 40.192.1-8.b.1.2, 40.192.1-8.b.1.3, 40.192.1-8.b.1.4, 40.192.1-8.b.1.5, 40.192.1-8.b.1.6, 48.192.1-8.b.1.1, 48.192.1-8.b.1.2, 48.192.1-8.b.1.3, 48.192.1-8.b.1.4, 56.192.1-8.b.1.1, 56.192.1-8.b.1.2, 56.192.1-8.b.1.3, 56.192.1-8.b.1.4, 56.192.1-8.b.1.5, 56.192.1-8.b.1.6, 80.192.1-8.b.1.1, 80.192.1-8.b.1.2, 80.192.1-8.b.1.3, 80.192.1-8.b.1.4, 88.192.1-8.b.1.1, 88.192.1-8.b.1.2, 88.192.1-8.b.1.3, 88.192.1-8.b.1.4, 88.192.1-8.b.1.5, 88.192.1-8.b.1.6, 104.192.1-8.b.1.1, 104.192.1-8.b.1.2, 104.192.1-8.b.1.3, 104.192.1-8.b.1.4, 104.192.1-8.b.1.5, 104.192.1-8.b.1.6, 112.192.1-8.b.1.1, 112.192.1-8.b.1.2, 112.192.1-8.b.1.3, 112.192.1-8.b.1.4, 120.192.1-8.b.1.1, 120.192.1-8.b.1.2, 120.192.1-8.b.1.3, 120.192.1-8.b.1.4, 120.192.1-8.b.1.5, 120.192.1-8.b.1.6, 136.192.1-8.b.1.1, 136.192.1-8.b.1.2, 136.192.1-8.b.1.3, 136.192.1-8.b.1.4, 136.192.1-8.b.1.5, 136.192.1-8.b.1.6, 152.192.1-8.b.1.1, 152.192.1-8.b.1.2, 152.192.1-8.b.1.3, 152.192.1-8.b.1.4, 152.192.1-8.b.1.5, 152.192.1-8.b.1.6, 168.192.1-8.b.1.1, 168.192.1-8.b.1.2, 168.192.1-8.b.1.3, 168.192.1-8.b.1.4, 168.192.1-8.b.1.5, 168.192.1-8.b.1.6, 176.192.1-8.b.1.1, 176.192.1-8.b.1.2, 176.192.1-8.b.1.3, 176.192.1-8.b.1.4, 184.192.1-8.b.1.1, 184.192.1-8.b.1.2, 184.192.1-8.b.1.3, 184.192.1-8.b.1.4, 184.192.1-8.b.1.5, 184.192.1-8.b.1.6, 208.192.1-8.b.1.1, 208.192.1-8.b.1.2, 208.192.1-8.b.1.3, 208.192.1-8.b.1.4, 232.192.1-8.b.1.1, 232.192.1-8.b.1.2, 232.192.1-8.b.1.3, 232.192.1-8.b.1.4, 232.192.1-8.b.1.5, 232.192.1-8.b.1.6, 240.192.1-8.b.1.1, 240.192.1-8.b.1.2, 240.192.1-8.b.1.3, 240.192.1-8.b.1.4, 248.192.1-8.b.1.1, 248.192.1-8.b.1.2, 248.192.1-8.b.1.3, 248.192.1-8.b.1.4, 248.192.1-8.b.1.5, 248.192.1-8.b.1.6, 264.192.1-8.b.1.1, 264.192.1-8.b.1.2, 264.192.1-8.b.1.3, 264.192.1-8.b.1.4, 264.192.1-8.b.1.5, 264.192.1-8.b.1.6, 272.192.1-8.b.1.1, 272.192.1-8.b.1.2, 272.192.1-8.b.1.3, 272.192.1-8.b.1.4, 280.192.1-8.b.1.1, 280.192.1-8.b.1.2, 280.192.1-8.b.1.3, 280.192.1-8.b.1.4, 280.192.1-8.b.1.5, 280.192.1-8.b.1.6, 296.192.1-8.b.1.1, 296.192.1-8.b.1.2, 296.192.1-8.b.1.3, 296.192.1-8.b.1.4, 296.192.1-8.b.1.5, 296.192.1-8.b.1.6, 304.192.1-8.b.1.1, 304.192.1-8.b.1.2, 304.192.1-8.b.1.3, 304.192.1-8.b.1.4, 312.192.1-8.b.1.1, 312.192.1-8.b.1.2, 312.192.1-8.b.1.3, 312.192.1-8.b.1.4, 312.192.1-8.b.1.5, 312.192.1-8.b.1.6, 328.192.1-8.b.1.1, 328.192.1-8.b.1.2, 328.192.1-8.b.1.3, 328.192.1-8.b.1.4, 328.192.1-8.b.1.5, 328.192.1-8.b.1.6 |
| Cyclic 8-isogeny field degree: |
$1$ |
| Cyclic 8-torsion field degree: |
$4$ |
| Full 8-torsion field degree: |
$16$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - y^{2} + z^{2} + w^{2} $ |
| $=$ | $x y + x w + 2 y^{2} - y z + z w$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{4} - 4 x^{3} y - 2 x^{3} z + 2 x^{2} y^{2} + 4 x^{2} y z + 2 x z^{3} + 2 y^{2} z^{2} + z^{4} $ |
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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:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
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{2^4}{3^4}\cdot\frac{627607928564xz^{22}w+6338867408186xz^{21}w^{2}+26924552628810xz^{20}w^{3}+74936933151796xz^{19}w^{4}+166575047964508xz^{18}w^{5}+313968457661940xz^{17}w^{6}+520944321014808xz^{16}w^{7}+779443261976832xz^{15}w^{8}+1059857305058160xz^{14}w^{9}+1306824721000712xz^{13}w^{10}+1451332992560936xz^{12}w^{11}+1435319473245936xz^{11}w^{12}+1242402655657936xz^{10}w^{13}+916206528473968xz^{9}w^{14}+552134406578256xz^{8}w^{15}+246157965360768xz^{7}w^{16}+56139076380432xz^{6}w^{17}-21774536205768xz^{5}w^{18}-30913308169144xz^{4}w^{19}-16291802361232xz^{3}w^{20}-5063704805424xz^{2}w^{21}-14457279200xzw^{22}+154459636096xw^{23}-94162812299y^{2}z^{22}-196858624y^{2}z^{21}w+6777360182040y^{2}z^{20}w^{2}+34265141161892y^{2}z^{19}w^{3}+97531112065156y^{2}z^{18}w^{4}+220339363152780y^{2}z^{17}w^{5}+404749374626478y^{2}z^{16}w^{6}+635853593695776y^{2}z^{15}w^{7}+869942737980612y^{2}z^{14}w^{8}+1044564653882944y^{2}z^{13}w^{9}+1101984778304384y^{2}z^{12}w^{10}+1016603241070416y^{2}z^{11}w^{11}+812318459987264y^{2}z^{10}w^{12}+537115122371008y^{2}z^{9}w^{13}+271676098429560y^{2}z^{8}w^{14}+71033237021760y^{2}z^{7}w^{15}-31560691896996y^{2}z^{6}w^{16}-55765829757168y^{2}z^{5}w^{17}-36747721573616y^{2}z^{4}w^{18}-13980710411056y^{2}z^{3}w^{19}-2624452941720y^{2}z^{2}w^{20}+491927439776y^{2}zw^{21}+137750373784y^{2}w^{22}+125537502052yz^{23}+1318124400026yz^{22}w+1946116104590yz^{21}w^{2}-10291696073992yz^{20}w^{3}-52347513650660yz^{19}w^{4}-151420382106392yz^{18}w^{5}-323658094888968yz^{17}w^{6}-548201917472376yz^{16}w^{7}-770789125415280yz^{15}w^{8}-902031261218072yz^{14}w^{9}-875254809289000yz^{13}w^{10}-669625106394688yz^{12}w^{11}-352466467227472yz^{11}w^{12}-18782356941536yz^{10}w^{13}+222040909041328yz^{9}w^{14}+321532363902192yz^{8}w^{15}+288794039180592yz^{7}w^{16}+188452600062456yz^{6}w^{17}+86970211497880yz^{5}w^{18}+22795483623920yz^{4}w^{19}-169448769616yz^{3}w^{20}-3307560099952yz^{2}w^{21}-908967938432yzw^{22}+5141182960yw^{23}-62762119218z^{24}-753145430616z^{23}w-721770740863z^{22}w^{2}+13682064503444z^{21}w^{3}+67155109624572z^{20}w^{4}+195691355614204z^{19}w^{5}+439959465270820z^{18}w^{6}+801743139557100z^{17}w^{7}+1245822131785794z^{16}w^{8}+1673773164765024z^{15}w^{9}+1965681376823700z^{14}w^{10}+2009231381664560z^{13}w^{11}+1776603027384032z^{12}w^{12}+1323668979754512z^{11}w^{13}+785768465491840z^{10}w^{14}+311283314329792z^{9}w^{15}+199409385096z^{8}w^{16}-132046793586816z^{7}w^{17}-134707233275700z^{6}w^{18}-83956254234048z^{5}w^{19}-34747027489264z^{4}w^{20}-8613786059632z^{3}w^{21}+52032694344z^{2}w^{22}+760998321856zw^{23}+68916756568w^{24}}{w^{4}(1775360xz^{18}w+9503872xz^{17}w^{2}+6528640xz^{16}w^{3}+51642880xz^{15}w^{4}+1716990436xz^{14}w^{5}+6135793666xz^{13}w^{6}+13425901450xz^{12}w^{7}+21919759324xz^{11}w^{8}+26288198140xz^{10}w^{9}+23102820456xz^{9}w^{10}+12881327924xz^{8}w^{11}+423046832xz^{7}w^{12}-7703015596xz^{6}w^{13}-9261546598xz^{5}w^{14}-6426428038xz^{4}w^{15}-2902556284xz^{3}w^{16}-846239524xz^{2}w^{17}-143327092xzw^{18}-9901736xw^{19}+1670848y^{2}z^{18}+15978240y^{2}z^{17}w+39477312y^{2}z^{16}w^{2}+65167104y^{2}z^{15}w^{3}-773957547y^{2}z^{14}w^{4}-209423808y^{2}z^{13}w^{5}-98960160y^{2}z^{12}w^{6}-229503492y^{2}z^{11}w^{7}-1379374578y^{2}z^{10}w^{8}-6901652228y^{2}z^{9}w^{9}-12483897894y^{2}z^{8}w^{10}-16903539936y^{2}z^{7}w^{11}-16748205843y^{2}z^{6}w^{12}-12196057536y^{2}z^{5}w^{13}-6739944624y^{2}z^{4}w^{14}-2590385820y^{2}z^{3}w^{15}-650970072y^{2}z^{2}w^{16}-105137532y^{2}zw^{17}-7778450y^{2}w^{18}-1775360yz^{19}-11279232yz^{18}w-16241536yz^{17}w^{2}-44595712yz^{16}w^{3}+1345875428yz^{15}w^{4}+4443571130yz^{14}w^{5}+10059477614yz^{13}w^{6}+20915821952yz^{12}w^{7}+33430136876yz^{11}w^{8}+47591613180yz^{10}w^{9}+56114210396yz^{9}w^{10}+54503295976yz^{8}w^{11}+43501434580yz^{7}w^{12}+26935037818yz^{6}w^{13}+12551104750yz^{5}w^{14}+3942613720yz^{4}w^{15}+560367820yz^{3}w^{16}-70052768yz^{2}w^{17}-42862984yzw^{18}-6290480yw^{19}-104512z^{18}w^{2}+6683392z^{17}w^{3}-758600786z^{16}w^{4}-3147979976z^{15}w^{5}-5456950871z^{14}w^{6}-10963099772z^{13}w^{7}-17269408448z^{12}w^{8}-27000584828z^{11}w^{9}-39204107858z^{10}w^{10}-48143629500z^{9}w^{11}-51081108148z^{8}w^{12}-44037374056z^{7}w^{13}-30598358167z^{6}w^{14}-16700370844z^{5}w^{15}-6722677612z^{4}w^{16}-1884900916z^{3}w^{17}-307770040z^{2}w^{18}-10689676zw^{19}+2534266w^{20})}$ |
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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.
