Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x^{2} y + x^{2} z + y^{2} w $ |
| $=$ | $x^{2} t + x z t + x w t - y w t$ |
| $=$ | $x^{2} z + x z^{2} + x z w - y z w$ |
| $=$ | $x^{2} w + x z w + x w^{2} - y w^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{4} z + 60 x^{3} y^{2} - 7 x^{3} z^{2} - 90 x^{2} y^{2} z + 5 x^{2} z^{3} + 15 x y^{2} z^{2} + \cdots + 15 y^{2} z^{3} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 15x^{7} + 45x^{6} - 105x^{5} - 90x^{4} + 105x^{3} + 45x^{2} - 15x $ |
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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 |
|---|
| $(0:0:0:0:1)$, $(0:1:0:0:0)$, $(-1:-1:1:0:0)$, $(-1:0:0:1:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{10556761706437500xw^{10}+42088399104436875xw^{8}t^{2}+23127724275849000xw^{6}t^{4}-1253288127456900xw^{4}t^{6}+5367451007280xw^{2}t^{8}+42133502016xt^{10}-199065600000y^{11}-96215040000y^{9}t^{2}+409964544000y^{7}t^{4}-172375142400y^{5}t^{6}+24845813760y^{3}t^{8}-49994499960646875yw^{10}+121633561009299375yw^{8}t^{2}+67841648811589500yw^{6}t^{4}-1025845464297600yw^{4}t^{6}+3932287882560yw^{2}t^{8}-1455441152yt^{10}-2312193600000z^{11}-2630698560000z^{9}t^{2}-158879232000z^{7}t^{4}-341295552000z^{5}t^{6}+169297013760z^{3}t^{8}-176887522312762500z^{2}w^{9}+49992393271134375z^{2}w^{7}t^{2}+24558117446549250z^{2}w^{5}t^{4}-272748320663400z^{2}w^{3}t^{6}+1059891163680z^{2}wt^{8}-39932934688687500zw^{10}+125422850311610625zw^{8}t^{2}+73833886510313625zw^{6}t^{4}-1148993091944550zw^{4}t^{6}+3227082583320zw^{2}t^{8}+20514457888zt^{10}+10554592591237500w^{11}+42755262086407500w^{9}t^{2}+23761305746709750w^{7}t^{4}-857615752917450w^{5}t^{6}+3824533740120w^{3}t^{8}+20702538208wt^{10}}{t^{2}(134632833750xw^{8}+212660606250xw^{6}t^{2}-14153525100xw^{4}t^{4}+207127680xw^{2}t^{6}-389440xt^{8}+372115873125yw^{8}+525591828000yw^{6}t^{2}-14784026400yw^{4}t^{4}+57525120yw^{2}t^{6}+256yt^{8}+134632833750z^{2}w^{7}+39699578250z^{2}w^{5}t^{2}-877903200z^{2}w^{3}t^{4}+2860320z^{2}wt^{6}+403898501250zw^{8}+583948993500zw^{6}t^{2}-22387368150zw^{4}t^{4}+194621280zw^{2}t^{6}-198496zt^{8}+134632833750w^{9}+214779448125w^{7}t^{2}-10229703750w^{5}t^{4}+116499360w^{3}t^{6}-194656wt^{8})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
60.72.3.xu.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{15}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 2y$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 60X^{3}Y^{2}+2X^{4}Z-90X^{2}Y^{2}Z-7X^{3}Z^{2}+15XY^{2}Z^{2}+5X^{2}Z^{3}+15Y^{2}Z^{3}-XZ^{4} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
60.72.3.xu.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x-y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -x^{3}t+3x^{2}yt-xy^{2}t-2y^{3}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -y$ |
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.
