Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ 2 x^{2} y + x y z + x y w + 2 x y t + y z w + y z t + y w^{2} + y w t $ |
| $=$ | $x^{2} z + 2 y^{2} w + 2 y^{2} t + z w^{2} + 2 z w t + z t^{2}$ |
| $=$ | $ - 2 x^{2} z + 2 x y^{2} - x z^{2} + y^{2} z - y^{2} w - z w^{2} - z w t$ |
| $=$ | $2 x^{3} + x^{2} z + x^{2} w + x z w + x w^{2} - 2 x t^{2} - z w t - z t^{2} - w^{2} t - w t^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} - 4 x^{4} y^{2} - 2 x^{4} z^{2} + 4 x^{2} y^{4} + 12 x^{2} y^{2} z^{2} - 3 x^{2} z^{4} - 16 y^{4} z^{2} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 2x^{7} + 10x^{6} + 14x^{5} + 20x^{4} + 14x^{3} + 10x^{2} + 2x $ |
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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 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^2\cdot3\,\frac{562362555883029359xw^{13}+4871289640815376861xw^{12}t+20003534936476348841xw^{11}t^{2}+50919824130694893707xw^{10}t^{3}+88117735765999906261xw^{9}t^{4}+105618662422121790137xw^{8}t^{5}+82682667552547580371xw^{7}t^{6}+30496526148968142479xw^{6}t^{7}-13770872679334024517xw^{5}t^{8}-25999229031581562397xw^{4}t^{9}-15900915329826492539xw^{3}t^{10}-4783355280980045557xw^{2}t^{11}-463846092343532706xwt^{12}+51998697814228992xt^{13}-396718580736y^{14}+16133222283264y^{12}t^{2}-226900988259840y^{10}t^{4}+1210559584686336y^{8}t^{6}-1562035587527232y^{6}t^{8}+751827381061680y^{4}t^{10}-138496410682938y^{2}t^{12}-347276830245870237zw^{13}-2976902040258858348zw^{12}t-12370213071577115082zw^{11}t^{2}-33026388289766244012zw^{10}t^{3}-63841642037945088336zw^{9}t^{4}-95670679730627695914zw^{8}t^{5}-116013156679215345204zw^{7}t^{6}-115831431087733548054zw^{6}t^{7}-93513831067410393294zw^{5}t^{8}-57329576836582867578zw^{4}t^{9}-24017423190706478808zw^{3}t^{10}-5962542767152418016zw^{2}t^{11}-820842871304566779zwt^{12}-24312008927385306zt^{13}-96126474961895908w^{14}-908975564099714694w^{13}t-4257278889051489648w^{12}t^{2}-13145558125047612620w^{11}t^{3}-29769325637081256936w^{10}t^{4}-51736330014370870344w^{9}t^{5}-70748958320386683804w^{8}t^{6}-78086876885121993168w^{7}t^{7}-70859654332287876780w^{6}t^{8}-51277894383451196748w^{5}t^{9}-26141724090423251820w^{4}t^{10}-7179123761503752240w^{3}t^{11}-415678388633624692w^{2}t^{12}-13157358429921918wt^{13}+50779978334208t^{14}}{1146981842838709xw^{13}+14175458639399574xw^{12}t+79748227955496572xw^{11}t^{2}+269526841078828822xw^{10}t^{3}+607899333247095871xw^{9}t^{4}+960430040159299482xw^{8}t^{5}+1084206790848567515xw^{7}t^{6}+874085049776920358xw^{6}t^{7}+492810123854575989xw^{5}t^{8}+184751622628726840xw^{4}t^{9}+41237655379603988xw^{3}t^{10}+4035400784154148xw^{2}t^{11}-44684632180968xwt^{12}+264479053824y^{12}t^{2}+230194732032y^{10}t^{4}+119609654400y^{8}t^{6}+49613757120y^{6}t^{8}+18659575368y^{4}t^{10}+6854943240y^{2}t^{12}-745826217535119zw^{13}-9055867861161993zw^{12}t-50040100577964966zw^{11}t^{2}-166115400275968830zw^{10}t^{3}-368148260803673967zw^{9}t^{4}-572057071904783811zw^{8}t^{5}-636171538585163631zw^{7}t^{6}-506575064463415347zw^{6}t^{7}-283283832207934977zw^{5}t^{8}-106089797534137383zw^{4}t^{9}-23975830375780500zw^{3}t^{10}-2442495886243632zw^{2}t^{11}+43036794134700zwt^{12}+13345951792536zt^{13}-233771395371788w^{14}-2975854616631206w^{13}t-17303050047080374w^{12}t^{2}-60794939531047452w^{11}t^{3}-143871325530651136w^{10}t^{4}-241908504175611854w^{9}t^{5}-297028665693800278w^{8}t^{6}-269495014301420126w^{7}t^{7}-180782845939159726w^{6}t^{8}-88848947111922986w^{5}t^{9}-31367642968772130w^{4}t^{10}-7667374475940264w^{3}t^{11}-1195997578192928w^{2}t^{12}-92664940454472wt^{13}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
24.96.3.gz.4
:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{6}-4X^{4}Y^{2}+4X^{2}Y^{4}-2X^{4}Z^{2}+12X^{2}Y^{2}Z^{2}-16Y^{4}Z^{2}-3X^{2}Z^{4} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
24.96.3.gz.4
:
| $\displaystyle X$ |
$=$ |
$\displaystyle \frac{1}{2}z^{2}+\frac{7}{6}zw+zt+\frac{2}{3}w^{2}+\frac{2}{3}wt$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{2}{9}yz^{4}w^{3}-\frac{14}{27}yz^{3}w^{4}-\frac{4}{9}yz^{3}w^{3}t-\frac{4}{27}yz^{2}w^{5}-\frac{8}{27}yz^{2}w^{4}t+\frac{5}{27}yzw^{6}+\frac{4}{27}yzw^{5}t+\frac{2}{27}yw^{7}+\frac{8}{81}yw^{6}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}z^{2}+\frac{7}{6}zw+zt+\frac{1}{3}w^{2}+\frac{2}{3}wt$ |
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.
