Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x z - x w - z t $ |
| $=$ | $x w + y^{2} - z t$ |
| $=$ | $x t + y^{2} - z^{2} - z w + w^{2} - t^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{5} y - 5 x^{4} y^{2} + x^{4} z^{2} + 8 x^{3} y^{3} - 4 x^{3} y z^{2} - 4 x^{2} y^{4} + \cdots + y^{2} z^{4} $ |
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This modular curve has 8 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(1:0:0:0:0)$, $(1:0:0:0:1)$, $(0:0:0:1:1)$, $(0:0:0:-1:1)$, $(-1:-1:1:1:0)$, $(-1:1:1:1:0)$, $(1:1:1:0:1)$, $(1:-1:1:0:1)$ |
Maps to other modular curves
$j$-invariant map
of degree 144 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{x^{18}-6x^{17}t-3x^{16}t^{2}+82x^{15}t^{3}-60x^{14}t^{4}-606x^{13}t^{5}+751x^{12}t^{6}+3498x^{11}t^{7}-5967x^{10}t^{8}-17060x^{9}t^{9}+41142x^{8}t^{10}+65688x^{7}t^{11}-252171x^{6}t^{12}-147906x^{5}t^{13}+1352235x^{4}t^{14}-413318x^{3}t^{15}-6148737x^{2}t^{16}-123375xw^{17}+769345xw^{16}t-2099545xw^{15}t^{2}+3304885xw^{14}t^{3}-3405600xw^{13}t^{4}+2610750xw^{12}t^{5}-1805190xw^{11}t^{6}+1254030xw^{10}t^{7}-639305xw^{9}t^{8}+222465xw^{8}t^{9}-319260xw^{7}t^{10}-450370xw^{6}t^{11}-872105xw^{5}t^{12}-1092525xw^{4}t^{13}-885410xw^{3}t^{14}+58560xw^{2}t^{15}+1506630xwt^{16}+9765444xt^{17}-323000zw^{17}+1814625zw^{16}t-4440280zw^{15}t^{2}+6259615zw^{14}t^{3}-5898140zw^{13}t^{4}+4409470zw^{12}t^{5}-3033950zw^{11}t^{6}+1797650zw^{10}t^{7}-861915zw^{9}t^{8}+443705zw^{8}t^{9}-117155zw^{7}t^{10}+179280zw^{6}t^{11}+271535zw^{5}t^{12}+709545zw^{4}t^{13}+1470475zw^{3}t^{14}+2426220zw^{2}t^{15}+2691360zwt^{16}+199625w^{18}-998125w^{17}t+1940720w^{16}t^{2}-1601915w^{15}t^{3}-13710w^{14}t^{4}+1131140w^{13}t^{5}-1150975w^{12}t^{6}+893000w^{11}t^{7}-726485w^{10}t^{8}+500500w^{9}t^{9}-105635w^{8}t^{10}+345485w^{7}t^{11}+572595w^{6}t^{12}+1283870w^{5}t^{13}+2135490w^{4}t^{14}+2644065w^{3}t^{15}+1391445w^{2}t^{16}-4198020wt^{17}-4243006t^{18}}{t^{4}w^{5}(w-t)^{2}(105xw^{6}-284xw^{5}t+296xw^{4}t^{2}-159xw^{3}t^{3}+51xw^{2}t^{4}-10xwt^{5}+xt^{6}+275zw^{6}-575zw^{5}t+473zw^{4}t^{2}-217zw^{3}t^{3}+62zw^{2}t^{4}-11zwt^{5}+zt^{6}-170w^{7}+250w^{6}t+21w^{5}t^{2}-200w^{4}t^{3}+139w^{3}t^{4}-49w^{2}t^{5}+10wt^{6}-t^{7})}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
40.144.5.gn.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{5}Y-5X^{4}Y^{2}+8X^{3}Y^{3}-4X^{2}Y^{4}+X^{4}Z^{2}-4X^{3}YZ^{2}+4X^{2}Y^{2}Z^{2}-X^{2}Z^{4}-XYZ^{4}+Y^{2}Z^{4} $ |
The following modular covers realize this modular curve as a fiber product over $X(1)$.
This modular curve minimally covers the modular curves listed below.
