| $\GL_2(\Z/32\Z)$-generators: |
$\begin{bmatrix}3&24\\0&17\end{bmatrix}$, $\begin{bmatrix}7&29\\0&7\end{bmatrix}$, $\begin{bmatrix}13&22\\0&31\end{bmatrix}$, $\begin{bmatrix}27&5\\0&31\end{bmatrix}$, $\begin{bmatrix}31&12\\0&23\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
32.192.1-32.f.2.1, 32.192.1-32.f.2.2, 32.192.1-32.f.2.3, 32.192.1-32.f.2.4, 32.192.1-32.f.2.5, 32.192.1-32.f.2.6, 32.192.1-32.f.2.7, 32.192.1-32.f.2.8, 32.192.1-32.f.2.9, 32.192.1-32.f.2.10, 32.192.1-32.f.2.11, 32.192.1-32.f.2.12, 32.192.1-32.f.2.13, 32.192.1-32.f.2.14, 32.192.1-32.f.2.15, 32.192.1-32.f.2.16, 64.192.1-32.f.2.1, 64.192.1-32.f.2.2, 64.192.1-32.f.2.3, 64.192.1-32.f.2.4, 64.192.1-32.f.2.5, 64.192.1-32.f.2.6, 64.192.1-32.f.2.7, 64.192.1-32.f.2.8, 96.192.1-32.f.2.1, 96.192.1-32.f.2.2, 96.192.1-32.f.2.3, 96.192.1-32.f.2.4, 96.192.1-32.f.2.5, 96.192.1-32.f.2.6, 96.192.1-32.f.2.7, 96.192.1-32.f.2.8, 96.192.1-32.f.2.9, 96.192.1-32.f.2.10, 96.192.1-32.f.2.11, 96.192.1-32.f.2.12, 96.192.1-32.f.2.13, 96.192.1-32.f.2.14, 96.192.1-32.f.2.15, 96.192.1-32.f.2.16, 160.192.1-32.f.2.1, 160.192.1-32.f.2.2, 160.192.1-32.f.2.3, 160.192.1-32.f.2.4, 160.192.1-32.f.2.5, 160.192.1-32.f.2.6, 160.192.1-32.f.2.7, 160.192.1-32.f.2.8, 160.192.1-32.f.2.9, 160.192.1-32.f.2.10, 160.192.1-32.f.2.11, 160.192.1-32.f.2.12, 160.192.1-32.f.2.13, 160.192.1-32.f.2.14, 160.192.1-32.f.2.15, 160.192.1-32.f.2.16, 192.192.1-32.f.2.1, 192.192.1-32.f.2.2, 192.192.1-32.f.2.3, 192.192.1-32.f.2.4, 192.192.1-32.f.2.5, 192.192.1-32.f.2.6, 192.192.1-32.f.2.7, 192.192.1-32.f.2.8, 224.192.1-32.f.2.1, 224.192.1-32.f.2.2, 224.192.1-32.f.2.3, 224.192.1-32.f.2.4, 224.192.1-32.f.2.5, 224.192.1-32.f.2.6, 224.192.1-32.f.2.7, 224.192.1-32.f.2.8, 224.192.1-32.f.2.9, 224.192.1-32.f.2.10, 224.192.1-32.f.2.11, 224.192.1-32.f.2.12, 224.192.1-32.f.2.13, 224.192.1-32.f.2.14, 224.192.1-32.f.2.15, 224.192.1-32.f.2.16, 320.192.1-32.f.2.1, 320.192.1-32.f.2.2, 320.192.1-32.f.2.3, 320.192.1-32.f.2.4, 320.192.1-32.f.2.5, 320.192.1-32.f.2.6, 320.192.1-32.f.2.7, 320.192.1-32.f.2.8 |
| Cyclic 32-isogeny field degree: |
$1$ |
| Cyclic 32-torsion field degree: |
$16$ |
| Full 32-torsion field degree: |
$4096$ |
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x $ |
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This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
Maps to other modular curves
$j$-invariant map
of degree 96 from the Weierstrass model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{y(196824x^{2}y^{46}-18957511824x^{2}y^{45}z+26147830922646x^{2}y^{44}z^{2}-5179104784553856x^{2}y^{43}z^{3}+302310601450533456x^{2}y^{42}z^{4}-7753265099442497544x^{2}y^{41}z^{5}+112453912379043056829x^{2}y^{40}z^{6}-1078898796788828968576x^{2}y^{39}z^{7}+7539028401429782972760x^{2}y^{38}z^{8}-40812886031226962244912x^{2}y^{37}z^{9}+178486132258056460562350x^{2}y^{36}z^{10}-649632168452575148900352x^{2}y^{35}z^{11}+2011663809997187005953168x^{2}y^{34}z^{12}-5389881514541402623466896x^{2}y^{33}z^{13}+12661035217634027475644850x^{2}y^{32}z^{14}-26351652302046298809480192x^{2}y^{31}z^{15}+49013605737796238822759904x^{2}y^{30}z^{16}-82044903263482487558841408x^{2}y^{29}z^{17}+124319572439863422838564440x^{2}y^{28}z^{18}-171346424507622106997159424x^{2}y^{27}z^{19}+215670488417420241099795264x^{2}y^{26}z^{20}-248718121596667142134652808x^{2}y^{25}z^{21}+263493802134850144929773541x^{2}y^{24}z^{22}-256963706471742237257770368x^{2}y^{23}z^{23}+231026424003708365873406456x^{2}y^{22}z^{24}-191671996259330914700737840x^{2}y^{21}z^{25}+146818147438004664496316226x^{2}y^{20}z^{26}-103845665430760927672772736x^{2}y^{19}z^{27}+67811202571263273243540912x^{2}y^{18}z^{28}-40845750645656395717828368x^{2}y^{17}z^{29}+22643284999775908015317186x^{2}y^{16}z^{30}-11502347632310424296104448x^{2}y^{15}z^{31}+5322062651779978117737336x^{2}y^{14}z^{32}-2231295766530433651321296x^{2}y^{13}z^{33}+847340058852842488093230x^{2}y^{12}z^{34}-293608202370716846973312x^{2}y^{11}z^{35}+93865938328477611738768x^{2}y^{10}z^{36}-27699669117896804445848x^{2}y^{9}z^{37}+7354710599616110428695x^{2}y^{8}z^{38}-1668652520730385831296x^{2}y^{7}z^{39}+302813094925894270152x^{2}y^{6}z^{40}-40713838517228899728x^{2}y^{5}z^{41}+3661419177575452233x^{2}y^{4}z^{42}-184246537834509920x^{2}y^{3}z^{43}+3381410572074792x^{2}y^{2}z^{44}-25563645345048x^{2}yz^{45}+68719476735x^{2}z^{46}-744xy^{47}+852488652xy^{46}z-3074282266976xy^{45}z^{2}+1044300071816688xy^{44}z^{3}-86826577303766352xy^{43}z^{4}+2836998280683333785xy^{42}z^{5}-48898675737473478368xy^{41}z^{6}+534628258864414577520xy^{40}z^{7}-4150065840752552016520xy^{39}z^{8}+24552906585382464222872xy^{38}z^{9}-116064872990828885974752xy^{37}z^{10}+453126178252576102722576xy^{36}z^{11}-1496811736856819015027408xy^{35}z^{12}+4260878501769449229959838xy^{34}z^{13}-10602383606263503371386624xy^{33}z^{14}+23323918874813081197254528xy^{32}z^{15}-45779815601371712014319136xy^{31}z^{16}+80776026376277349438705552xy^{30}z^{17}-128920590328471608912339840xy^{29}z^{18}+187081784593070290696284864xy^{28}z^{19}-247895596376518421654179136xy^{27}z^{20}+301002470061110628802749237xy^{26}z^{21}-335883964838003324943195936xy^{25}z^{22}+345246011876882779179935376xy^{24}z^{23}-327463995715005914729604776xy^{23}z^{24}+286989026055375306091153344xy^{22}z^{25}-232601219532436919008518240xy^{21}z^{26}+174406178501645881064134704xy^{20}z^{27}-120943370618125425815939184xy^{19}z^{28}+77481076920213583018742678xy^{18}z^{29}-45779992622736154589982592xy^{17}z^{30}+24908688543998752828498368xy^{16}z^{31}-12471387212339337584878024xy^{15}z^{32}+5743765947760724268218604xy^{14}z^{33}-2425886515453520340133344xy^{13}z^{34}+929759694922468102045488xy^{12}z^{35}-316475660937462869773456xy^{11}z^{36}+92612541594420257845155xy^{10}z^{37}-22358499862099995227168xy^{9}z^{38}+4235951201456862872592xy^{8}z^{39}-590321931647202915672xy^{7}z^{40}+54921202465120448952xy^{6}z^{41}-2855821336788814240xy^{5}z^{42}+54102569156542680xy^{4}z^{43}-421800148206312xy^{3}z^{44}+1168231104513xy^{2}z^{45}+y^{48}-21449120y^{47}z+281683645248y^{46}z^{2}-180577209616976y^{45}z^{3}+22465837208939684y^{44}z^{4}-960688612971482560y^{43}z^{5}+19856279565710080536y^{42}z^{6}-246184518388004941240y^{41}z^{7}+2091791474951036488072y^{40}z^{8}-13231758296759649061536y^{39}z^{9}+65774783114224492667040y^{38}z^{10}-266712038546382233209552y^{37}z^{11}+906252289671252219011236y^{36}z^{12}-2632807535108543736064960y^{35}z^{13}+6641839600900659390978768y^{34}z^{14}-14728952996606530443150896y^{33}z^{15}+28996203030331384950054162y^{32}z^{16}-51083020491554417986831488y^{31}z^{17}+81066325660387057588409856y^{30}z^{18}-116519349643725772507794752y^{29}z^{19}+152371367283470616640389648y^{28}z^{20}-181953950084223473205526272y^{27}z^{21}+199009345802661572609340600y^{26}z^{22}-199834325405697858195457528y^{25}z^{23}+184559619706655471616349892y^{24}z^{24}-156972227279309226780579360y^{23}z^{25}+123033821333366600452605792y^{22}z^{26}-88867967016057921042952944y^{21}z^{27}+59113859154287947269714044y^{20}z^{28}-36171774679050271320732224y^{19}z^{29}+20339258150515967176313616y^{18}z^{30}-10502660213714702125608880y^{17}z^{31}+4974470623591817862226299y^{16}z^{32}-2151307663873487685904160y^{15}z^{33}+839524504843181654625984y^{14}z^{34}-289284970739664894397328y^{13}z^{35}+85305451357257244361844y^{12}z^{36}-20692339790328161657024y^{11}z^{37}+3933186484945416713352y^{10}z^{38}-549608644000891066664y^{9}z^{39}+51259810448265808944y^{8}z^{40}-2671577872764690656y^{7}z^{41}+50721440263783008y^{6}z^{42}-396255460355976y^{5}z^{43}+1100364116406y^{4}z^{44}-21449120y^{3}z^{45}+196824y^{2}z^{46}-744yz^{47}+z^{48})}{y^{2}(1722x^{2}y^{44}z-4047952x^{2}y^{42}z^{3}+1005495541x^{2}y^{40}z^{5}-39133154032x^{2}y^{38}z^{7}-142077164834x^{2}y^{36}z^{9}+9268801044480x^{2}y^{34}z^{11}-24786191360166x^{2}y^{32}z^{13}-413415546861440x^{2}y^{30}z^{15}+3443957213388328x^{2}y^{28}z^{17}-12789795596767808x^{2}y^{26}z^{19}+29312567147384997x^{2}y^{24}z^{21}-46286992096778128x^{2}y^{22}z^{23}+53215099221196766x^{2}y^{20}z^{25}-45960121110799280x^{2}y^{18}z^{27}+30352481146764618x^{2}y^{16}z^{29}-15454486186081984x^{2}y^{14}z^{31}+6068552540392066x^{2}y^{12}z^{33}-1823488491506192x^{2}y^{10}z^{35}+411926018012143x^{2}y^{8}z^{37}-67688684554640x^{2}y^{6}z^{39}+7584912243034x^{2}y^{4}z^{41}-515396075460x^{2}y^{2}z^{43}+34359738367x^{2}z^{45}+xy^{46}-31460xy^{44}z^{2}+31610034xy^{42}z^{4}-4274104692xy^{40}z^{6}+83842334597xy^{38}z^{8}+1087638156428xy^{36}z^{10}-19167925011166xy^{34}z^{12}-4005270474400xy^{32}z^{14}+1006797500433044xy^{30}z^{16}-6331408382717136xy^{28}z^{18}+20581574301642184xy^{26}z^{20}-43146752876675532xy^{24}z^{22}+63706230153749209xy^{22}z^{24}-69412233564969348xy^{20}z^{26}+57338293780163686xy^{18}z^{28}-36459938870517072xy^{16}z^{30}+17965210422564237xy^{14}z^{32}-6853848684997908xy^{12}z^{34}+2007124117148346xy^{10}z^{36}-442982926939820xy^{8}z^{38}+71369471559335xy^{6}z^{40}-7988639170620xy^{4}z^{42}+566935683073xy^{2}z^{44}-60y^{46}z+409042y^{44}z^{3}-195622120y^{42}z^{5}+14039636259y^{40}z^{7}-72841734556y^{38}z^{9}-3719923270366y^{36}z^{11}+23718933438936y^{34}z^{13}+139857853095806y^{32}z^{15}-1783674549166640y^{30}z^{17}+7742946454382024y^{28}z^{19}-19581846357417632y^{26}z^{21}+33238354211639243y^{24}z^{23}-40451017352039788y^{22}z^{25}+36606966722819574y^{20}z^{27}-25147856041700120y^{18}z^{29}+13246717006628462y^{16}z^{31}-5358324031298764y^{14}z^{33}+1652858059594778y^{12}z^{35}-382252092866312y^{10}z^{37}+64282775897449y^{8}z^{39}-7473243125300y^{6}z^{41}+532575946405y^{4}z^{43}-60y^{2}z^{45}+z^{47})}$ 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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.
