Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x^{2} t + x y t - x w t - y z t + z w t - w^{2} t $ |
| $=$ | $x^{2} t - 2 x y t + x z t + y w t - z w t$ |
| $=$ | $2 x^{2} t - x y t + x w t + y z t - z^{2} t$ |
| $=$ | $x^{2} y + x y^{2} - x y w - y^{2} z + y z w - y w^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 52 x^{6} - 138 x^{5} z - 40 x^{4} y^{2} + 141 x^{4} z^{2} + 100 x^{3} y^{2} z - 80 x^{3} z^{3} + \cdots + z^{6} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -15x^{7} + 105x^{4} + 120x $ |
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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 |
|---|
| $(0:0:0:0:1)$, $(0:1:0:0:0)$, $(-1:1:2:1:0)$, $(1:1:2:1:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{13^3}\cdot\frac{123571062950400000xw^{10}-57795246044150625xw^{8}t^{2}-1448314522578000xw^{6}t^{4}-186857627607000xw^{4}t^{6}-2184101698740xw^{2}t^{8}-3201151863xt^{10}-36653580843750y^{11}+60581790286875y^{9}t^{2}-41924177802000y^{7}t^{4}+14261438388600y^{5}t^{6}-2253065330880y^{3}t^{8}-14638701639853125yzw^{9}-32492122360441875yzw^{7}t^{2}-3127221748932750yzw^{5}t^{4}-251293916855475yzw^{3}t^{6}+5070525472545yzwt^{8}-2704076313187500yw^{10}+23653031846293125yw^{8}t^{2}+1153718335932750yw^{6}t^{4}+119764443786975yw^{4}t^{6}-1391551969080yw^{2}t^{8}+128630746920yt^{10}-78388308676743750z^{2}w^{9}+18780466724026875z^{2}w^{7}t^{2}+214513515209250z^{2}w^{5}t^{4}+104418174629925z^{2}w^{3}t^{6}-1643038217470z^{2}wt^{8}+104829633197690625zw^{10}-5340465770934375zw^{8}t^{2}+894808035277875zw^{6}t^{4}-15386475670875zw^{4}t^{6}-2942678939525zw^{2}t^{8}-27983379632zt^{10}-91597238713068750w^{11}-12756423448338750w^{9}t^{2}-3321732230728875w^{7}t^{4}-55307347794450w^{5}t^{6}+6297149481155w^{3}t^{8}-156256707776wt^{10}}{t^{6}(4068675xw^{4}-786660xw^{2}t^{2}-5863xt^{4}-128475yzw^{3}-672360yzwt^{2}-576900yw^{4}+438555yw^{2}t^{2}-2646450z^{2}w^{3}+202280z^{2}wt^{2}+3351825zw^{4}+93445zw^{2}t^{2}-3731zt^{4}-2774925w^{5}-361135w^{3}t^{2}+2132wt^{4})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
60.72.3.ld.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{5}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 2w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 52X^{6}-40X^{4}Y^{2}-200X^{2}Y^{4}-138X^{5}Z+100X^{3}Y^{2}Z-50XY^{4}Z+141X^{4}Z^{2}-90X^{2}Y^{2}Z^{2}+25Y^{4}Z^{2}-80X^{3}Z^{3}+40XY^{2}Z^{3}+30X^{2}Z^{4}-10Y^{2}Z^{4}-6XZ^{5}+Z^{6} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
60.72.3.ld.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{2}{11}z^{4}+\frac{6}{11}z^{3}w-\frac{19}{44}z^{2}w^{2}-\frac{1}{110}z^{2}t^{2}+\frac{2}{11}zw^{3}-\frac{1}{220}zwt^{2}-\frac{1}{11}w^{4}+\frac{1}{220}w^{2}t^{2}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{1431}{234256}z^{15}t+\frac{66057}{937024}z^{14}wt-\frac{328731}{937024}z^{13}w^{2}t-\frac{963}{2342560}z^{13}t^{3}+\frac{3719985}{3748096}z^{12}w^{3}t+\frac{3021}{937024}z^{12}wt^{3}-\frac{6661887}{3748096}z^{11}w^{4}t-\frac{91107}{9370240}z^{11}w^{2}t^{3}+\frac{3981477}{1874048}z^{10}w^{5}t+\frac{241767}{18740480}z^{10}w^{3}t^{3}-\frac{809499}{468512}z^{9}w^{6}t-\frac{53373}{18740480}z^{9}w^{4}t^{3}+\frac{110523}{117128}z^{8}w^{7}t-\frac{114381}{9370240}z^{8}w^{5}t^{3}-\frac{79179}{234256}z^{7}w^{8}t+\frac{61713}{4685120}z^{7}w^{6}t^{3}+\frac{8733}{117128}z^{6}w^{9}t-\frac{8733}{2342560}z^{6}w^{7}t^{3}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -\frac{5}{66}z^{4}+\frac{53}{132}z^{3}w-\frac{13}{22}z^{2}w^{2}-\frac{2}{165}z^{2}t^{2}+\frac{8}{33}zw^{3}-\frac{1}{165}zwt^{2}-\frac{4}{33}w^{4}+\frac{1}{165}w^{2}t^{2}$ |
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.
