Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ 2 y u + 2 w t + t^{2} $ |
| $=$ | $2 y^{2} - y z + w^{2} + w t$ |
| $=$ | $3 y z - y u - w t$ |
| $=$ | $3 z^{2} - 2 t^{2} - u^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 36 x^{8} + 216 x^{7} z + 240 x^{6} y^{2} + 252 x^{6} z^{2} - 288 x^{5} y^{2} z - 540 x^{5} z^{3} + \cdots + 9 y^{4} z^{4} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 7x^{8} + 8x^{7} + 112x^{6} - 112x^{5} - 56x^{4} + 224x^{3} + 448x^{2} - 64x + 112 $ |
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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 to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^6\,\frac{2173796352xyw^{10}-14967742464xyw^{8}u^{2}+10860604416xyw^{6}u^{4}-1480543488xyw^{4}u^{6}-85681872xyw^{2}u^{8}+27826428xyu^{10}+528528xzt^{10}+15634464xzt^{8}u^{2}-93354540xzt^{6}u^{4}+22337124xzt^{4}u^{6}+23847186xzt^{2}u^{8}+279552xzu^{10}+5434490880xw^{10}u-7291772928xw^{8}u^{3}+627678720xw^{6}u^{5}+250000128xw^{4}u^{7}+20427048xw^{2}u^{9}-7076160xt^{10}u+25005024xt^{8}u^{3}+55053648xt^{6}u^{5}-38066472xt^{4}u^{7}-6809016xt^{2}u^{9}-35831808yw^{10}u+79626240yw^{8}u^{3}-503801856yw^{6}u^{5}+748268928yw^{4}u^{7}-229526928yw^{2}u^{9}+11274330yu^{11}-155520zt^{8}u^{3}+254016zt^{6}u^{5}-9764064zt^{4}u^{7}+12431952zt^{2}u^{9}+5971968w^{12}-39813120w^{10}u^{2}+139885056w^{8}u^{4}-405468288w^{6}u^{6}+160062984w^{4}u^{8}+13589574w^{2}u^{10}-728t^{12}+51728t^{10}u^{2}-427286t^{8}u^{4}+7677242t^{6}u^{6}-9194609t^{4}u^{8}-4576450t^{2}u^{10}+93312u^{12}}{u^{2}(1990656xyw^{8}-8460288xyw^{6}u^{2}+2315520xyw^{4}u^{4}+115776xyw^{2}u^{6}-25788xyu^{8}+1968xzt^{8}+37176xzt^{6}u^{2}-90564xzt^{4}u^{4}-17646xzt^{2}u^{6}+3981312xw^{8}u-2073600xw^{6}u^{3}-435456xw^{4}u^{5}-9504xw^{2}u^{7}-20736xt^{8}u+22464xt^{6}u^{3}+73440xt^{4}u^{5}+3168xt^{2}u^{7}+19906560yw^{8}u-25712640yw^{6}u^{3}+4136832yw^{4}u^{5}-35136yw^{2}u^{7}-44982yu^{9}+207360zt^{6}u^{3}-107136zt^{4}u^{5}-43488zt^{2}u^{7}-3981312w^{10}+16381440w^{8}u^{2}-2429568w^{6}u^{4}-564768w^{4}u^{6}-41994w^{2}u^{8}+3896t^{10}-86692t^{8}u^{2}-86114t^{6}u^{4}+105745t^{4}u^{6}+13998t^{2}u^{8})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
24.96.3.fc.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}u$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 36X^{8}+240X^{6}Y^{2}-368X^{4}Y^{4}+128X^{2}Y^{6}+216X^{7}Z-288X^{5}Y^{2}Z+96X^{3}Y^{4}Z+252X^{6}Z^{2}-516X^{4}Y^{2}Z^{2}+24X^{2}Y^{4}Z^{2}-540X^{5}Z^{3}+1044X^{3}Y^{2}Z^{3}+72XY^{4}Z^{3}-1071X^{4}Z^{4}+504X^{2}Y^{2}Z^{4}+9Y^{4}Z^{4}-324X^{3}Z^{5}+54XY^{2}Z^{5}-27X^{2}Z^{6} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
24.96.3.fc.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle 12wt^{2}+6wu^{2}+6t^{3}+2t^{2}u+4tu^{2}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -8640xwt^{7}u^{3}-3456xwt^{6}u^{4}-7776xwt^{5}u^{5}-1728xwt^{4}u^{6}-1728xwt^{3}u^{7}-4320xt^{8}u^{3}-3456xt^{7}u^{4}-4752xt^{6}u^{5}-1728xt^{5}u^{6}-1296xt^{4}u^{7}-1728wt^{6}u^{5}-864wt^{5}u^{6}-864wt^{4}u^{7}-432wt^{3}u^{8}-864t^{7}u^{5}-792t^{6}u^{6}-576t^{5}u^{7}-360t^{4}u^{8}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle t^{2}u-tu^{2}$ |
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.
