Weierstrass model Weierstrass model
| $ y^{2} + y $ | $=$ | $ x^{3} - x^{2} $ |
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This modular curve has 1 known rational point but no rational cusps or CM points. The following are the known rational points on this modular curve (one row per $j$-invariant).
Maps to other modular curves
$j$-invariant map
of degree 60 from the Weierstrass model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 11^2\,\frac{525x^{2}y^{38}+42024x^{2}y^{37}z-15481143x^{2}y^{36}z^{2}-161422918x^{2}y^{35}z^{3}-1198863787x^{2}y^{34}z^{4}-5629802751x^{2}y^{33}z^{5}+34304494274x^{2}y^{32}z^{6}+623434526215x^{2}y^{31}z^{7}+6102329732442x^{2}y^{30}z^{8}+44514123033232x^{2}y^{29}z^{9}+249139116880874x^{2}y^{28}z^{10}+1145678755143163x^{2}y^{27}z^{11}+4375540900925985x^{2}y^{26}z^{12}+13287282042835840x^{2}y^{25}z^{13}+30604825069353563x^{2}y^{24}z^{14}+51186990424395768x^{2}y^{23}z^{15}+57350264477296785x^{2}y^{22}z^{16}+30671112450813681x^{2}y^{21}z^{17}-24108882111844587x^{2}y^{20}z^{18}-73789812096138532x^{2}y^{19}z^{19}-85798781290776176x^{2}y^{18}z^{20}-59761312918701811x^{2}y^{17}z^{21}-22963503773870450x^{2}y^{16}z^{22}+288088156243267x^{2}y^{15}z^{23}+6274642494385235x^{2}y^{14}z^{24}+4010140131036750x^{2}y^{13}z^{25}+1253913584551263x^{2}y^{12}z^{26}+123270141513913x^{2}y^{11}z^{27}-59674933178500x^{2}y^{10}z^{28}-27492026461090x^{2}y^{9}z^{29}-4819670054893x^{2}y^{8}z^{30}-238242581542x^{2}y^{7}z^{31}+46867685437x^{2}y^{6}z^{32}+7997474042x^{2}y^{5}z^{33}+436522807x^{2}y^{4}z^{34}+5614573x^{2}y^{3}z^{35}-199367x^{2}y^{2}z^{36}-3623x^{2}yz^{37}-19x^{2}z^{38}-43xy^{39}-37522xy^{38}z-1274844xy^{37}z^{2}+126574605xy^{36}z^{3}+1411425911xy^{35}z^{4}+17688469947xy^{34}z^{5}+156732549961xy^{33}z^{6}+1037229267928xy^{32}z^{7}+5930542193168xy^{31}z^{8}+26161020155943xy^{30}z^{9}+80893179310153xy^{29}z^{10}+124275368797516xy^{28}z^{11}-478806296207266xy^{27}z^{12}-4843352169497288xy^{26}z^{13}-21244849278591611xy^{25}z^{14}-60164000501544169xy^{24}z^{15}-118435837082846896xy^{23}z^{16}-162991942545371604xy^{22}z^{17}-146712025326859291xy^{21}z^{18}-58474620917491616xy^{20}z^{19}+50643386678407762xy^{19}z^{20}+110618719802035179xy^{18}z^{21}+98998306480247657xy^{17}z^{22}+50792060733686191xy^{16}z^{23}+10882827736341103xy^{15}z^{24}-4754428382854495xy^{14}z^{25}-5047004491381575xy^{13}z^{26}-1992695433703641xy^{12}z^{27}-343773974223971xy^{11}z^{28}+32750191869794xy^{10}z^{29}+30859675959422xy^{9}z^{30}+6615470632782xy^{8}z^{31}+507234254917xy^{7}z^{32}-32407296536xy^{6}z^{33}-8457589860xy^{5}z^{34}-520811821xy^{4}z^{35}-8667779xy^{3}z^{36}+169152xy^{2}z^{37}+3513xyz^{38}+19xz^{39}+y^{40}+460y^{39}z+1069527y^{38}z^{2}+17414854y^{37}z^{3}-363696564y^{36}z^{4}-5492189869y^{35}z^{5}-78255784512y^{34}z^{6}-747032534379y^{33}z^{7}-5593824987940y^{32}z^{8}-35177656641172y^{31}z^{9}-180900999764478y^{30}z^{10}-767251164528822y^{29}z^{11}-2687597398846131y^{28}z^{12}-7406861376284588y^{27}z^{13}-14541677489184627y^{26}z^{14}-15618732340007342y^{25}z^{15}+8792081006746392y^{24}z^{16}+74014230097291297y^{23}z^{17}+162407282815430145y^{22}z^{18}+218915858745106856y^{21}z^{19}+196202277701752698y^{20}z^{20}+106049142809435331y^{19}z^{21}+10580130139943434y^{18}z^{22}-38506229192962974y^{17}z^{23}-38901827501678874y^{16}z^{24}-19795835937541700y^{15}z^{25}-4996090930850539y^{14}z^{26}+382904105057540y^{13}z^{27}+823026804130143y^{12}z^{28}+310895325877019y^{11}z^{29}+52527831050893y^{10}z^{30}-139475856723y^{9}z^{31}-1816654580714y^{8}z^{32}-330587146929y^{7}z^{33}-21998962308y^{6}z^{34}+116863680y^{5}z^{35}+82893157y^{4}z^{36}+3364491y^{3}z^{37}+37547y^{2}z^{38}+220yz^{39}+z^{40}}{75x^{2}y^{38}+1425x^{2}y^{37}z-809913x^{2}y^{36}z^{2}-14736609x^{2}y^{35}z^{3}+391701440x^{2}y^{34}z^{4}+8123442010x^{2}y^{33}z^{5}+24232428473x^{2}y^{32}z^{6}-352943212820x^{2}y^{31}z^{7}-3481367663194x^{2}y^{30}z^{8}-13215686355950x^{2}y^{29}z^{9}-16644983130876x^{2}y^{28}z^{10}+65233615393290x^{2}y^{27}z^{11}+406658014726403x^{2}y^{26}z^{12}+1159988997351701x^{2}y^{25}z^{13}+2176652848731979x^{2}y^{24}z^{14}+2856509220930158x^{2}y^{23}z^{15}+2477360642800982x^{2}y^{22}z^{16}+853833077297068x^{2}y^{21}z^{17}-1349912836847696x^{2}y^{20}z^{18}-3067429348564397x^{2}y^{19}z^{19}-3621264949897651x^{2}y^{18}z^{20}-3099758577752959x^{2}y^{17}z^{21}-2083666333702535x^{2}y^{16}z^{22}-1133113430576902x^{2}y^{15}z^{23}-503922662394431x^{2}y^{14}z^{24}-183211658331015x^{2}y^{13}z^{25}-53861922427918x^{2}y^{12}z^{26}-12459227815798x^{2}y^{11}z^{27}-2122357302659x^{2}y^{10}z^{28}-212158758900x^{2}y^{9}z^{29}+7737808384x^{2}y^{8}z^{30}+8004233922x^{2}y^{7}z^{31}+1751390677x^{2}y^{6}z^{32}+231905346x^{2}y^{5}z^{33}+20075704x^{2}y^{4}z^{34}+1063150x^{2}y^{3}z^{35}+25134x^{2}y^{2}z^{36}-342x^{2}yz^{37}-24x^{2}z^{38}-2644xy^{38}z-50236xy^{37}z^{2}+9166268xy^{36}z^{3}+170569020xy^{35}z^{4}-1684205769xy^{34}z^{5}-45656853777xy^{33}z^{6}-241777114753xy^{32}z^{7}+449849755640xy^{31}z^{8}+10390917181054xy^{30}z^{9}+52067878625858xy^{29}z^{10}+129821607686218xy^{28}z^{11}+105203529763582xy^{27}z^{12}-438135740798217xy^{26}z^{13}-2040100946416005xy^{25}z^{14}-4758933003627617xy^{24}z^{15}-7615442163778164xy^{23}z^{16}-8940143624788743xy^{22}z^{17}-7649585578522721xy^{21}z^{18}-4254656929167602xy^{20}z^{19}-501111958413404xy^{19}z^{20}+1988412329201679xy^{18}z^{21}+2719951836746262xy^{17}z^{22}+2236452834659322xy^{16}z^{23}+1371795105694300xy^{15}z^{24}+665701057737739xy^{14}z^{25}+260794304576899xy^{13}z^{26}+82725295929909xy^{12}z^{27}+21031155042436xy^{11}z^{28}+4166228943981xy^{10}z^{29}+600216593859xy^{9}z^{30}+49596891896xy^{8}z^{31}-1725513316xy^{7}z^{32}-1299572551xy^{6}z^{33}-219997450xy^{5}z^{34}-21515109xy^{4}z^{35}-1252625xy^{3}z^{36}-34754xy^{2}z^{37}+149xyz^{38}+24xz^{39}-y^{40}-20y^{39}z+56064y^{38}z^{2}+1067686y^{37}z^{3}-70409350y^{36}z^{4}-1385936280y^{35}z^{5}+1132316034y^{34}z^{6}+159040118037y^{33}z^{7}+1210244888035y^{32}z^{8}+2890327462640y^{31}z^{9}-9682029152863y^{30}z^{10}-97363272545353y^{29}z^{11}-366711761275034y^{28}z^{12}-823738181303675y^{27}z^{13}-1086040353261490y^{26}z^{14}-300707700193945y^{25}z^{15}+2308050806076667y^{24}z^{16}+6498326367855149y^{23}z^{17}+10683500425217633y^{22}z^{18}+12864345421430356y^{21}z^{19}+12094956861009695y^{20}z^{20}+9082945659293384y^{19}z^{21}+5457538181097704y^{18}z^{22}+2570641030761620y^{17}z^{23}+886100055572410y^{16}z^{24}+166753344390197y^{15}z^{25}-33769085440970y^{14}z^{26}-46158318100169y^{13}z^{27}-23485281322123y^{12}z^{28}-8218248047625y^{11}z^{29}-2180578408040y^{10}z^{30}-449091410070y^{9}z^{31}-71369277119y^{8}z^{32}-8484557254y^{7}z^{33}-698925670y^{6}z^{34}-31216196y^{5}z^{35}+459151y^{4}z^{36}+161216y^{3}z^{37}+9134y^{2}z^{38}+169yz^{39}-z^{40}}$ |
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.
