Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x y w - x z w - y^{2} w - y^{2} t + z^{2} w + z^{2} t $ |
| $=$ | $x y w - y^{2} w - y^{2} t - y z w - y z t$ |
| $=$ | $x w t - y w t - y t^{2} - z w t - z t^{2}$ |
| $=$ | $x w^{2} - y w^{2} - y w t - z w^{2} - z w t$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{7} - 12 x^{6} z + 11 x^{5} y^{2} + 5 x^{5} z^{2} + 82 x^{4} y^{2} z + 6 x^{4} z^{3} + \cdots - 64 y^{2} z^{5} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{4} + x^{2}\right) y $ | $=$ | $ -2x^{6} - 7x^{4} - 7x^{2} - 4 $ |
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This modular curve has 2 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 36 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^6\cdot17\,\frac{59216389xz^{4}t+71570272xz^{2}t^{3}-87279445yz^{4}t-340502112yz^{2}t^{3}-756563968yw^{5}+2262825856yw^{4}t+6215837832yw^{3}t^{2}-2502896958yw^{2}t^{3}-5488440777ywt^{4}-841181620yt^{5}+3591403z^{5}t-95299195z^{3}t^{3}+595820544zw^{5}-9600784zw^{4}t-1830752387zw^{3}t^{2}+541248387zw^{2}t^{3}+502216131zwt^{4}-703622796zt^{5}}{44015567xz^{4}t-95882108xz^{2}t^{3}-21297855yz^{4}t-183078032yz^{2}t^{3}+21131008yw^{5}+29738624yw^{4}t-203188584yw^{3}t^{2}+51135846yw^{2}t^{3}+67485245ywt^{4}+12041340yt^{5}-21297855z^{5}t+162949471z^{3}t^{3}-14500352zw^{5}+18342544zw^{4}t+110977591zw^{3}t^{2}-68433779zw^{2}t^{3}-73505915zwt^{4}-12041340zt^{5}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
34.36.3.a.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 4X^{7}+11X^{5}Y^{2}+1156X^{3}Y^{4}-12X^{6}Z+82X^{4}Y^{2}Z+2312X^{2}Y^{4}Z+5X^{5}Z^{2}+311X^{3}Y^{2}Z^{2}-3468XY^{4}Z^{2}+6X^{4}Z^{3}-794X^{2}Y^{2}Z^{3}-11560Y^{4}Z^{3}+X^{3}Z^{4}-456XY^{2}Z^{4}-64Y^{2}Z^{5} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
34.36.3.a.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{51}{70}z^{3}w^{5}-\frac{17}{140}z^{3}w^{4}t+\frac{51}{40}z^{3}w^{3}t^{2}+\frac{323}{560}z^{3}w^{2}t^{3}-\frac{17}{20}z^{3}wt^{4}-\frac{17}{112}z^{3}t^{5}+\frac{1}{1190}zw^{7}-\frac{87}{1190}zw^{6}t-\frac{361}{2380}zw^{5}t^{2}+\frac{97}{238}zw^{4}t^{3}-\frac{1419}{19040}zw^{3}t^{4}-\frac{1753}{19040}zw^{2}t^{5}-\frac{11}{680}zwt^{6}-\frac{1}{1190}zt^{7}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{2}{10115}z^{2}w^{30}-\frac{506}{354025}z^{2}w^{29}t+\frac{194759}{49563500}z^{2}w^{28}t^{2}-\frac{14497131}{3469445000}z^{2}w^{27}t^{3}-\frac{88725013}{55511120000}z^{2}w^{26}t^{4}+\frac{196793297}{27755560000}z^{2}w^{25}t^{5}-\frac{37881413}{22204448000}z^{2}w^{24}t^{6}-\frac{1005150093}{111022240000}z^{2}w^{23}t^{7}+\frac{8633292293}{888177920000}z^{2}w^{22}t^{8}-\frac{281641177}{222044480000}z^{2}w^{21}t^{9}-\frac{2754730987}{888177920000}z^{2}w^{20}t^{10}+\frac{12646957}{11102224000}z^{2}w^{19}t^{11}+\frac{7903419141}{14210846720000}z^{2}w^{18}t^{12}-\frac{245514317}{1015060480000}z^{2}w^{17}t^{13}-\frac{365274943}{4060241920000}z^{2}w^{16}t^{14}+\frac{18681241}{812048384000}z^{2}w^{15}t^{15}+\frac{2797959531}{227373547520000}z^{2}w^{14}t^{16}+\frac{7734973}{28421693440000}z^{2}w^{13}t^{17}-\frac{17555879}{22737354752000}z^{2}w^{12}t^{18}-\frac{11944823}{56843386880000}z^{2}w^{11}t^{19}-\frac{6262853}{227373547520000}z^{2}w^{10}t^{20}-\frac{115903}{56843386880000}z^{2}w^{9}t^{21}-\frac{1167}{14210846720000}z^{2}w^{8}t^{22}-\frac{1}{710542336000}z^{2}w^{7}t^{23}+\frac{256}{2923235}w^{32}-\frac{68863}{102313225}w^{31}t+\frac{439294}{210644875}w^{30}t^{2}-\frac{3121454073}{1002669605000}w^{29}t^{3}+\frac{3496031909}{2005339210000}w^{28}t^{4}+\frac{23980910377}{32085427360000}w^{27}t^{5}-\frac{680749633}{1283417094400}w^{26}t^{6}-\frac{116698239481}{64170854720000}w^{25}t^{7}+\frac{63146140527}{32085427360000}w^{24}t^{8}+\frac{6366034089}{73338119680000}w^{23}t^{9}-\frac{448256452617}{513366837760000}w^{22}t^{10}+\frac{34737291521}{256683418880000}w^{21}t^{11}+\frac{45412659}{205346735104}w^{20}t^{12}-\frac{251358296569}{8213869404160000}w^{19}t^{13}-\frac{343210988363}{8213869404160000}w^{18}t^{14}-\frac{6561188957}{16427738808320000}w^{17}t^{15}+\frac{42461074393}{8213869404160000}w^{16}t^{16}+\frac{148688610657}{131421910466560000}w^{15}t^{17}-\frac{22230467731}{131421910466560000}w^{14}t^{18}-\frac{16931344309}{131421910466560000}w^{13}t^{19}-\frac{789797549}{26284382093312000}w^{12}t^{20}-\frac{33163423}{8213869404160000}w^{11}t^{21}-\frac{564267}{1642773880832000}w^{10}t^{22}-\frac{7551}{410693470208000}w^{9}t^{23}-\frac{73}{128341709440000}w^{8}t^{24}-\frac{1}{128341709440000}w^{7}t^{25}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{17}w^{8}-\frac{349}{2380}w^{7}t+\frac{63}{680}w^{6}t^{2}+\frac{137}{9520}w^{5}t^{3}-\frac{293}{19040}w^{4}t^{4}-\frac{69}{19040}w^{3}t^{5}-\frac{1}{4760}w^{2}t^{6}$ |
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.
