| $\GL_2(\Z/8\Z)$-generators: |
$\begin{bmatrix}1&4\\0&1\end{bmatrix}$, $\begin{bmatrix}1&4\\2&3\end{bmatrix}$, $\begin{bmatrix}5&0\\0&1\end{bmatrix}$, $\begin{bmatrix}7&4\\0&7\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: |
$C_2^4$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
8.192.1-8.e.1.1, 8.192.1-8.e.1.2, 8.192.1-8.e.1.3, 8.192.1-8.e.1.4, 24.192.1-8.e.1.1, 24.192.1-8.e.1.2, 24.192.1-8.e.1.3, 24.192.1-8.e.1.4, 40.192.1-8.e.1.1, 40.192.1-8.e.1.2, 40.192.1-8.e.1.3, 40.192.1-8.e.1.4, 56.192.1-8.e.1.1, 56.192.1-8.e.1.2, 56.192.1-8.e.1.3, 56.192.1-8.e.1.4, 88.192.1-8.e.1.1, 88.192.1-8.e.1.2, 88.192.1-8.e.1.3, 88.192.1-8.e.1.4, 104.192.1-8.e.1.1, 104.192.1-8.e.1.2, 104.192.1-8.e.1.3, 104.192.1-8.e.1.4, 120.192.1-8.e.1.1, 120.192.1-8.e.1.2, 120.192.1-8.e.1.3, 120.192.1-8.e.1.4, 136.192.1-8.e.1.1, 136.192.1-8.e.1.2, 136.192.1-8.e.1.3, 136.192.1-8.e.1.4, 152.192.1-8.e.1.1, 152.192.1-8.e.1.2, 152.192.1-8.e.1.3, 152.192.1-8.e.1.4, 168.192.1-8.e.1.1, 168.192.1-8.e.1.2, 168.192.1-8.e.1.3, 168.192.1-8.e.1.4, 184.192.1-8.e.1.1, 184.192.1-8.e.1.2, 184.192.1-8.e.1.3, 184.192.1-8.e.1.4, 232.192.1-8.e.1.1, 232.192.1-8.e.1.2, 232.192.1-8.e.1.3, 232.192.1-8.e.1.4, 248.192.1-8.e.1.1, 248.192.1-8.e.1.2, 248.192.1-8.e.1.3, 248.192.1-8.e.1.4, 264.192.1-8.e.1.1, 264.192.1-8.e.1.2, 264.192.1-8.e.1.3, 264.192.1-8.e.1.4, 280.192.1-8.e.1.1, 280.192.1-8.e.1.2, 280.192.1-8.e.1.3, 280.192.1-8.e.1.4, 296.192.1-8.e.1.1, 296.192.1-8.e.1.2, 296.192.1-8.e.1.3, 296.192.1-8.e.1.4, 312.192.1-8.e.1.1, 312.192.1-8.e.1.2, 312.192.1-8.e.1.3, 312.192.1-8.e.1.4, 328.192.1-8.e.1.1, 328.192.1-8.e.1.2, 328.192.1-8.e.1.3, 328.192.1-8.e.1.4 |
| Cyclic 8-isogeny field degree: |
$2$ |
| Cyclic 8-torsion field degree: |
$4$ |
| Full 8-torsion field degree: |
$16$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - x y - x z + y z - y w - z w - w^{2} $ |
| $=$ | $3 x^{2} - x y - 5 x z - 4 x w + y^{2} - y z + 3 y w + z^{2} + 3 z w + 3 w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 4 x^{3} y + 8 x^{3} z - 10 x^{2} y^{2} + 16 x^{2} y z + 8 x^{2} z^{2} + 12 x y^{3} + \cdots - 20 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:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
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}{3^8}\cdot\frac{17050576141257369480xz^{23}-2925160654792711320xz^{22}w-993780699888524213736xz^{21}w^{2}-144620241770506118328xz^{20}w^{3}+20332208027585002749480xz^{19}w^{4}+8557244451987269008824xz^{18}w^{5}-186234625049072894166744xz^{17}w^{6}-113649053792048458958520xz^{16}w^{7}+829803483965021221319760xz^{15}w^{8}+581853298857820847068752xz^{14}w^{9}-1868944959186373623435984xz^{13}w^{10}-1325324392093623441029040xz^{12}w^{11}+2162795695509519687630672xz^{11}w^{12}+1424610560795997999620016xz^{10}w^{13}-1280092212858928141508976xz^{9}w^{14}-731355445123601863957488xz^{8}w^{15}+374879239760891895002280xz^{7}w^{16}+173130346219697108292168xz^{6}w^{17}-50159618602923001835592xz^{5}w^{18}-17008979077770515579928xz^{4}w^{19}+2579704116066387251976xz^{3}w^{20}+536455615652026019480xz^{2}w^{21}-32949096833365448888xzw^{22}-2328153301857943000xw^{23}-18084862090495889100y^{2}z^{22}+948660312755973167856y^{2}z^{20}w^{2}-16957467947091471824520y^{2}z^{18}w^{4}+131823917606467494578844y^{2}z^{16}w^{6}-486055835790680923034760y^{2}z^{14}w^{8}+885972890978479426644720y^{2}z^{12}w^{10}-810987879949769292836256y^{2}z^{10}w^{12}+368735398448372830468296y^{2}z^{8}w^{14}-79335087764491680349644y^{2}z^{6}w^{16}+7209790231421182400640y^{2}z^{4}w^{18}-212944728856557743352y^{2}z^{2}w^{20}+873057488196766460y^{2}w^{22}+19119148039734408720yz^{23}-72339448361983556400yz^{22}w-903539925623422121976yz^{21}w^{2}+3794641251023892671424yz^{20}w^{3}+13582727866597940899560yz^{19}w^{4}-67829871788365887298080yz^{18}w^{5}-77413210163862094990944yz^{17}w^{6}+527295670425869978315376yz^{16}w^{7}+142308187616340624749760yz^{15}w^{8}-1944223343162723692139040yz^{14}w^{9}+96999177229414770146544yz^{13}w^{10}+3543891563913917706578880yz^{12}w^{11}-540819935609981101958160yz^{11}w^{12}-3243951519799077171345024yz^{10}w^{13}+542621415962182480572384yz^{9}w^{14}+1474941593793491321873184yz^{8}w^{15}-216209064231908534302992yz^{7}w^{16}-317340351057966721398576yz^{6}w^{17}+35740038140080637034312yz^{5}w^{18}+28839160925684729602560yz^{4}w^{19}-2153814658353271765272yz^{3}w^{20}-851778915426230973408yz^{2}w^{21}+31202981856971915968yzw^{22}+3492229952787065840yw^{23}-6028293934745396109z^{24}-14125415486464658160z^{23}w+413525424446302198068z^{22}w^{2}+1138400941659030332064z^{21}w^{3}-9417409716172457400390z^{20}w^{4}-28889452479572271758304z^{19}w^{5}+86825623833268677506016z^{18}w^{6}+299883678841121353125264z^{17}w^{7}-322814684208639741771951z^{16}w^{8}-1411656782822842068388512z^{15}w^{9}+355618939659306836192280z^{14}w^{10}+3194269351279997064465024z^{13}w^{11}+396648234086172629080860z^{12}w^{12}-3587406256305517687250688z^{11}w^{13}-1048038048656796416846256z^{10}w^{14}+2011447657982530005466464z^{9}w^{15}+707425397494067947958325z^{8}w^{16}-548009585980589003294448z^{7}w^{17}-190226738560126402611756z^{6}w^{18}+67168597680693517415520z^{5}w^{19}+20009817390983351883882z^{4}w^{20}-3116159731718413271456z^{3}w^{21}-656032342654652522256z^{2}w^{22}+35277250135223391888zw^{23}+2910191627322556231w^{24}}{224532000xz^{22}w-268272000xz^{21}w^{2}+26179225920xz^{20}w^{3}-46257052320xz^{19}w^{4}+1136041663968xz^{18}w^{5}-3038561964432xz^{17}w^{6}+32333269301496xz^{16}w^{7}+2470963170165552xz^{15}w^{8}+363115876465224xz^{14}w^{9}-34661676793324392xz^{13}w^{10}-22261272356121336xz^{12}w^{11}+136970737433867736xz^{11}w^{12}+101148969099606632xz^{10}w^{13}-199562917019506520xz^{9}w^{14}-135504188290520368xz^{8}w^{15}+115582487447487960xz^{7}w^{16}+63899547586479504xz^{6}w^{17}-25967613639953232xz^{5}w^{18}-10377437871713616xz^{4}w^{19}+1965710721820320xz^{3}w^{20}+468413372858400xz^{2}w^{21}-33059881728000xzw^{22}-2582803260000xw^{23}+5467500y^{2}z^{22}+2297589300y^{2}z^{20}w^{2}+196686344718y^{2}z^{18}w^{4}+8873618730444y^{2}z^{16}w^{6}-2456821601661672y^{2}z^{14}w^{8}+25100907367878990y^{2}z^{12}w^{10}-72809865365785854y^{2}z^{10}w^{12}+76734597537304964y^{2}z^{8}w^{14}-30994152466496832y^{2}z^{6}w^{16}+4509716804419626y^{2}z^{4}w^{18}-187406769654900y^{2}z^{2}w^{20}+968551222500y^{2}w^{22}-10935000yz^{23}+21870000yz^{22}w-4326906600yz^{21}w^{2}+9190357200yz^{20}w^{3}-347115637116yz^{19}w^{4}+786745378872yz^{18}w^{5}-14708675496456yz^{17}w^{6}+35494474921776yz^{16}w^{7}+2442680033157792yz^{15}w^{8}-9827286406646688yz^{14}w^{9}-15540137942433588yz^{13}w^{10}+100403629471515960yz^{12}w^{11}+8648993297703972yz^{11}w^{12}-291239461463143416yz^{10}w^{13}+46093721944896592yz^{9}w^{14}+306938390149219856yz^{8}w^{15}-53594182514494296yz^{7}w^{16}-123976609865987328yz^{6}w^{17}+16948180031113980yz^{5}w^{18}+18038867217678504yz^{4}w^{19}-1590897182510520yz^{3}w^{20}-749627078619600yz^{2}w^{21}+31122779283000yzw^{22}+3874204890000yw^{23}-5467500z^{24}-224532000z^{23}w-1996512300z^{22}w^{2}-25910953920z^{21}w^{3}-136063472598z^{20}w^{4}-1089784611648z^{19}w^{5}-4537573274664z^{18}w^{6}-29294707337064z^{17}w^{7}-1028334486244257z^{16}w^{8}-2834079046630776z^{15}w^{9}+17754284750721030z^{14}w^{10}+56922949149445728z^{13}w^{11}-51199005043892172z^{12}w^{12}-238119706533474368z^{11}w^{13}-9304029925530420z^{10}w^{14}+335067105310026888z^{9}w^{15}+103390770338818831z^{8}w^{16}-179482035033967464z^{7}w^{17}-65899760322142890z^{6}w^{18}+36345051511666848z^{5}w^{19}+11999775116538720z^{4}w^{20}-2434124094678720z^{3}w^{21}-570844958665500z^{2}w^{22}+35642684988000zw^{23}+3228504075000w^{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.
