Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x z w + x z t + y z t $ |
| $=$ | $x w t + x t^{2} + y t^{2}$ |
| $=$ | $x w^{2} + x w t + y w t$ |
| $=$ | $x^{2} w + x^{2} t + x y t$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} y - x^{2} y^{2} z - 12 x^{2} y z^{2} - 45 x^{2} z^{3} + 9 y z^{4} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{4} + 1\right) y $ | $=$ | $ -6x^{6} - 5x^{4} - 54x^{2} + 20 $ |
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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 |
|---|
| $(0:0:0:0:1)$, $(-1:1:0:0:0)$, $(0:0:1:0:0)$, $(0:1:0:0:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{170859375xy^{10}+91174359375xy^{8}t^{2}-2716128929250xy^{6}t^{4}+4204793200995xy^{4}t^{6}+1727385470028xy^{2}t^{8}-88848484174xt^{10}-303750000y^{11}+3929765625y^{9}w^{2}-18724921875y^{9}wt+89535375000y^{9}t^{2}-518327859375y^{7}w^{2}t^{2}+830784583125y^{7}wt^{3}-2743379523000y^{7}t^{4}+6447326186250y^{5}w^{2}t^{4}-642078784350y^{5}wt^{5}+4788594285120y^{5}t^{6}-913804196400y^{3}w^{2}t^{6}-2329161285087y^{3}wt^{7}+1441510484688y^{3}t^{8}-1153429858657yw^{2}t^{8}-723540728847ywt^{9}-303298969946yt^{10}-75937500z^{11}-132890625z^{9}w^{2}-3701953125z^{9}wt-3794343750z^{9}t^{2}-58644843750z^{7}w^{2}t^{2}-61461601875z^{7}wt^{3}-215232028875z^{7}t^{4}-87937785000z^{5}w^{2}t^{4}-245323688400z^{5}wt^{5}-58204430370z^{5}t^{6}-84661052025z^{3}w^{2}t^{6}-23721269403z^{3}wt^{7}-20708285130z^{3}t^{8}+15051009495zw^{2}t^{8}+268325929195zwt^{9}+214325485772zt^{10}}{t(105300000xy^{8}t-50868000xy^{6}t^{3}-23119200xy^{4}t^{5}+230544xy^{2}t^{7}+48185xt^{9}-20250000y^{9}w+105300000y^{9}t-333450000y^{7}w^{2}t-25920000y^{7}wt^{2}-79218000y^{7}t^{3}-163044000y^{5}w^{2}t^{3}-55994400y^{5}wt^{4}-42523200y^{5}t^{5}-4569360y^{3}w^{2}t^{5}-5943825y^{3}wt^{6}-2258256y^{3}t^{7}-17277yw^{2}t^{7}-280855ywt^{8}-125788yt^{9}-5062500z^{9}w-5315625z^{9}t+10715625z^{7}w^{2}t+11407500z^{7}wt^{2}-5629500z^{7}t^{3}-2769750z^{5}w^{2}t^{3}+4030650z^{5}wt^{4}+1241100z^{5}t^{5}-2766840z^{3}w^{2}t^{5}-783735z^{3}wt^{6}+491085z^{3}t^{7}+140228zw^{2}t^{7}+376656zwt^{8}+173973zt^{9})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
60.72.3.b.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{4}Y-X^{2}Y^{2}Z-12X^{2}YZ^{2}-45X^{2}Z^{3}+9YZ^{4} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
60.72.3.b.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 4z^{4}-6z^{2}w^{2}-3z^{2}wt$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
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.
