Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x z w + y z t $ |
| $=$ | $x y w + y^{2} t$ |
| $=$ | $x w t + y t^{2}$ |
| $=$ | $x w^{2} + y w t$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} y + x^{2} y^{2} z + 60 x^{2} y z^{2} + 1125 x^{2} z^{3} + 225 y z^{4} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{4} + 1\right) y $ | $=$ | $ 30x^{6} - 113x^{4} + 6750x^{2} + 12656 $ |
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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 |
|---|
| $(0:0:0:0:1)$, $(1:0:0:0:0)$, $(0:0:1:0:0)$, $(0:1:0:0:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{1483154296875x^{11}-593261718750x^{9}t^{2}-59326171875x^{7}t^{4}+174023437500x^{5}t^{6}-128349609375x^{3}t^{8}+75753906250xt^{10}-949218750000y^{11}+253125000000y^{9}w^{2}+8859375000000y^{9}wt+6530625000000y^{9}t^{2}-25974000000000y^{7}w^{2}t^{2}-16563015000000y^{7}wt^{3}-70982541000000y^{7}t^{4}-14327847000000y^{5}w^{2}t^{4}-31062187260000y^{5}wt^{5}-58978713528000y^{5}t^{6}+15457278240750y^{3}w^{2}t^{6}+22115591266740y^{3}wt^{7}+12684121044240y^{3}t^{8}-1749953720876yw^{2}t^{8}-1708747399134ywt^{9}-622958984375yt^{10}-652587890625z^{11}+213574218750z^{9}w^{2}+6347900390625z^{9}wt+6100312500000z^{9}t^{2}-20131083984375z^{7}w^{2}t^{2}-19776964218750z^{7}wt^{3}-73715892328125z^{7}t^{4}+4514280187500z^{5}w^{2}t^{4}+16559476796250z^{5}wt^{5}+3129494472000z^{5}t^{6}-1337733152250z^{3}w^{2}t^{6}-288260063460z^{3}wt^{7}-293549398290z^{3}t^{8}-91559690562zw^{2}t^{8}-340194959886zwt^{9}-75410156250zt^{10}}{t(12656250000y^{9}w+10125000000y^{9}t+9787500000y^{7}w^{2}t+13635000000y^{7}wt^{2}-384750000y^{7}t^{3}+1236600000y^{5}w^{2}t^{3}+457560000y^{5}wt^{4}-269640000y^{5}t^{5}-23860800y^{3}w^{2}t^{5}-17431350y^{3}wt^{6}-2915370y^{3}t^{7}-476001yw^{2}t^{7}-194358ywt^{8}+8701171875z^{9}w+8542968750z^{9}t+3723046875z^{7}w^{2}t+3636562500z^{7}wt^{2}-1992093750z^{7}t^{3}+117225000z^{5}w^{2}t^{3}-292083750z^{5}wt^{4}-57262500z^{5}t^{5}-32593425z^{3}w^{2}t^{5}-7248450z^{3}wt^{6}+3976815z^{3}t^{7}+637396zw^{2}t^{7}+265121zwt^{8})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
60.72.3.c.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{15}w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{4}Y+X^{2}Y^{2}Z+60X^{2}YZ^{2}+1125X^{2}Z^{3}+225YZ^{4} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
60.72.3.c.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -113z^{4}-30z^{2}w^{2}-15z^{2}wt-w^{4}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
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.
