Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x y w + x z w - x w t + y^{2} w + y z w + y w^{2} - y w t $ |
| $=$ | $x^{2} y + x^{2} z - x^{2} t + x y w - y^{3} - y^{2} z - y^{2} w + y^{2} t$ |
| $=$ | $x^{2} y + x^{2} z - x^{2} t + x y^{2} + x y z + x y w - x y t$ |
| $=$ | $x y t + x z t - x t^{2} + y^{2} t + y z t + y w t - y t^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 8 x^{6} z + 40 x^{5} y^{2} + 20 x^{5} z^{2} + 120 x^{4} y^{2} z + 24 x^{4} z^{3} + 80 x^{3} y^{2} z^{2} + \cdots - 10 y^{2} z^{5} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 10x^{7} - 70x^{5} + 70x^{3} - 10x $ |
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This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:0:0:-1:1)$, $(0:-2:1:1:0)$, $(-2:0:0:1:0)$, $(2:-2:1:0:0)$ |
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{1}{5}\cdot\frac{34617876973xzt^{12}-680000000xw^{13}-5264000000xw^{12}t+14859600000xw^{11}t^{2}+334922000000xw^{10}t^{3}+839405720000xw^{9}t^{4}-789404320000xw^{8}t^{5}-3459377000000xw^{7}t^{6}-2340716652000xw^{6}t^{7}+2088615126000xw^{5}t^{8}+2760464324400xw^{4}t^{9}+149252453160xw^{3}t^{10}-764599768520xw^{2}t^{11}-122434644877xwt^{12}+18063164043xt^{13}-2325000000yzw^{12}-20592000000yzw^{11}t+32375400000yzw^{10}t^{2}+1338098800000yzw^{9}t^{3}+3497133030000yzw^{8}t^{4}-4010575440000yzw^{7}t^{5}-14841031460000yzw^{6}t^{6}-4896844344000yzw^{5}t^{7}+11864892244500yzw^{4}t^{8}+6157022517600yzw^{3}t^{9}-2417568760060yzw^{2}t^{10}-1099302832600yzwt^{11}+42349164770yzt^{12}+40000000yw^{13}-480000000yw^{12}t+20604100000yw^{11}t^{2}-98434800000yw^{10}t^{3}+27910420000yw^{9}t^{4}+1571549600000yw^{8}t^{5}+454813785000yw^{7}t^{6}-5859505240000yw^{6}t^{7}-3354000082000yw^{5}t^{8}+5404231721200yw^{4}t^{9}+2652174885110yw^{3}t^{10}-1948121531720yw^{2}t^{11}-401771335854ywt^{12}+190393722122yt^{13}+4000000z^{12}t^{2}+48000000z^{11}t^{3}+262000000z^{10}t^{4}+860000000z^{9}t^{5}+1890640000z^{8}t^{6}+2932160000z^{7}t^{7}+3287296000z^{6}t^{8}+2680272000z^{5}t^{9}+1573011600z^{4}t^{10}+645553600z^{3}t^{11}-2405000000z^{2}w^{12}-16164000000z^{2}w^{11}t+95799800000z^{2}w^{10}t^{2}+1216021600000z^{2}w^{9}t^{3}+1703054710000z^{2}w^{8}t^{4}-5749672880000z^{2}w^{7}t^{5}-10704964500000z^{2}w^{6}t^{6}+1110162552000z^{2}w^{5}t^{7}+10165394152500z^{2}w^{4}t^{8}+2310111485200z^{2}w^{3}t^{9}-2373790073620z^{2}w^{2}t^{10}-480896488800z^{2}wt^{11}+15637985154z^{2}t^{12}-1165000000zw^{13}-8892000000zw^{12}t-19092500000zw^{11}t^{2}+654553200000zw^{10}t^{3}+2032256650000zw^{9}t^{4}-1376146800000zw^{8}t^{5}-7938208115000zw^{7}t^{6}-2514568216000zw^{6}t^{7}+8117193186500zw^{5}t^{8}+4182896523600zw^{4}t^{9}-3148326431750zw^{3}t^{10}-1711012049320zw^{2}t^{11}+410739458222zwt^{12}+188381434352zt^{13}-1320000000w^{14}-9363000000w^{13}t+40783700000w^{12}t^{2}+703078500000w^{11}t^{3}+924222740000w^{10}t^{4}-4656081290000w^{9}t^{5}-6769716055000w^{8}t^{6}+8354419091000w^{7}t^{7}+15096056962000w^{6}t^{8}-3749367053700w^{5}t^{9}-11542757584130w^{4}t^{10}-200922365090w^{3}t^{11}+2964530547936w^{2}t^{12}+108437148650wt^{13}-161599279914t^{14}}{t^{8}(1475xzt^{4}+500xw^{5}-7000xw^{4}t+8384xw^{3}t^{2}-7750xw^{2}t^{3}+1496xwt^{4}-1500xt^{5}+4500yzw^{4}-24720yzw^{3}t+24976yzw^{2}t^{2}-17540yzwt^{3}+816yzt^{4}+800yw^{5}-5140yw^{4}t+4734yw^{3}t^{2}-1988yw^{2}t^{3}-4414ywt^{4}+883yt^{5}+80z^{4}t^{2}+320z^{3}t^{3}+2900z^{2}w^{4}-18840z^{2}w^{3}t+19732z^{2}w^{2}t^{2}-11840z^{2}wt^{3}-814z^{2}t^{4}-500zw^{5}-3280zw^{4}t+1490zw^{3}t^{2}-1664zw^{2}t^{3}-620zwt^{4}+1492zt^{5}+1800w^{6}-13500w^{5}t+16958w^{4}t^{2}+1990w^{3}t^{3}-15770w^{2}t^{4}+13010wt^{5}-1488t^{6})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
40.96.3.bf.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{2}{5}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 2y$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 40X^{5}Y^{2}+8X^{6}Z+120X^{4}Y^{2}Z+20X^{5}Z^{2}+80X^{3}Y^{2}Z^{2}+24X^{4}Z^{3}-40X^{2}Y^{2}Z^{3}+16X^{3}Z^{4}-50XY^{2}Z^{4}+6X^{2}Z^{5}-10Y^{2}Z^{5}+XZ^{6} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
40.96.3.bf.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -x^{3}-3x^{2}y-4xy^{2}-2y^{3}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -2x^{11}t-24x^{10}yt-124x^{9}y^{2}t-360x^{8}y^{3}t-608x^{7}y^{4}t-448x^{6}y^{5}t+448x^{5}y^{6}t+1664x^{4}y^{7}t+2208x^{3}y^{8}t+1664x^{2}y^{9}t+704xy^{10}t+128y^{11}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle x^{2}y+2xy^{2}+2y^{3}$ |
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.
