Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ y^{2} + y u + t^{2} $ |
| $=$ | $y^{2} + y z - w t$ |
| $=$ | $y w + y t + z w - z t - t u$ |
| $=$ | $y w + y t + z t + w u$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{8} + 7 x^{7} y + 14 x^{6} y^{2} + 23 x^{6} z^{2} + 7 x^{5} y^{3} + 81 x^{5} y z^{2} + 2 x^{4} y^{4} + \cdots + y^{4} z^{4} $ |
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Geometric Weierstrass model Geometric Weierstrass model
| $ 36 w^{2} $ | $=$ | $ 117 x^{4} - 36 x^{3} z + 102 x^{2} z^{2} + 12 x z^{3} + 13 z^{4} $ |
| $0$ | $=$ | $3 x^{2} + y^{2} + z^{2}$ |
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This modular curve has no real points and no $\Q_p$ points for $p=23$, and therefore no 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
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{745472xyt^{10}-9405712xyt^{8}u^{2}+17771600xyt^{6}u^{4}-10337672xyt^{4}u^{6}+2375776xyt^{2}u^{8}-212456xyu^{10}+318824xzt^{10}-2945344xzt^{8}u^{2}+4094648xzt^{6}u^{4}-2087936xzt^{4}u^{6}+424192xzt^{2}u^{8}-728xzu^{10}+1556328xwt^{9}u-5675232xwt^{7}u^{3}+5819544xwt^{5}u^{5}-2485920xwt^{3}u^{7}+431472xwtu^{9}-2171032xt^{10}u+8268104xt^{8}u^{3}-6775480xt^{6}u^{5}+1710736xt^{4}u^{7}-219008xt^{2}u^{9}-728xu^{11}-84419yt^{10}u+315022yt^{8}u^{3}-1462481yt^{6}u^{5}+1340963yt^{4}u^{7}+886478yt^{2}u^{9}-239104yu^{11}+142596zt^{10}u-61191zt^{8}u^{3}-728622zt^{6}u^{5}+216774zt^{4}u^{7}+333972zt^{2}u^{9}-59843wt^{11}+233125wt^{9}u^{2}+206263wt^{7}u^{4}-1180240wt^{5}u^{6}-103354wt^{3}u^{8}+334700wtu^{10}-63939t^{12}+273252t^{10}u^{2}-201555t^{8}u^{4}+998067t^{6}u^{6}+257280t^{4}u^{8}-248580t^{2}u^{10}-729u^{12}}{6xyt^{8}u^{2}-224xyt^{6}u^{4}+355xyt^{4}u^{6}-143xyt^{2}u^{8}+15xyu^{10}-5xzt^{10}+18xzt^{8}u^{2}-44xzt^{6}u^{4}+87xzt^{4}u^{6}-29xzt^{2}u^{8}-17xwt^{9}u+32xwt^{7}u^{3}-102xwt^{5}u^{5}+116xwt^{3}u^{7}-29xwtu^{9}-17xt^{10}u-65xt^{8}u^{3}+184xt^{6}u^{5}-99xt^{4}u^{7}+15xt^{2}u^{9}-79yt^{10}u+2188yt^{8}u^{3}-6934yt^{6}u^{5}+6555yt^{4}u^{7}-2303yt^{2}u^{9}+267yu^{11}+529zt^{10}u-3181zt^{8}u^{3}+4308zt^{6}u^{5}-1921zt^{4}u^{7}+265zt^{2}u^{9}-79wt^{11}+1918wt^{9}u^{2}-6098wt^{7}u^{4}+5964wt^{5}u^{6}-2186wt^{3}u^{8}+265wtu^{10}-79t^{12}+1977t^{10}u^{2}-6567t^{8}u^{4}+6442t^{6}u^{6}-2301t^{4}u^{8}+267t^{2}u^{10}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
24.96.3.fs.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle u$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{8}+7X^{7}Y+14X^{6}Y^{2}+7X^{5}Y^{3}+2X^{4}Y^{4}-5X^{3}Y^{5}+X^{2}Y^{6}+23X^{6}Z^{2}+81X^{5}YZ^{2}+34X^{4}Y^{2}Z^{2}+13X^{3}Y^{3}Z^{2}-9X^{2}Y^{4}Z^{2}+2XY^{5}Z^{2}+121X^{4}Z^{4}+116X^{3}YZ^{4}+18X^{2}Y^{2}Z^{4}-4XY^{3}Z^{4}+Y^{4}Z^{4}+144X^{2}Z^{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.
