Canonical model in $\mathbb{P}^{ 10 }$ defined by 36 equations
| $ 0 $ | $=$ | $ x y + x w - x t + x u - x a - y z - y t - y s - y a + v a + r a + s a + a^{2} $ |
| $=$ | $x y - 2 x w + x t - x u - x v + x s - x b - y w - y u - y s - v a - a^{2}$ |
| $=$ | $x^{2} - x y - x w - x v - x r - x s - x b - y z - y u + y r - z^{2} + z w + z r + z s + v b + r b + \cdots + a b$ |
| $=$ | $x^{2} + x y + x z - x w + x t + x s + x a - y z - y u + y r + z^{2} - 2 z r - z s - t r - v r + r s + \cdots - s b$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{14} y^{4} + 24 x^{14} y^{2} z^{2} + 36 x^{14} z^{4} + 20 x^{13} y^{5} + 49 x^{13} y^{4} z + \cdots + z^{18} $ |
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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.
| Canonical model |
|---|
| $(0:-1:0:-1:-1:0:0:0:1:0:1)$, $(0:-1:0:1:0:-1:-1:-1:0:1:0)$, $(0:0:1:0:-1:-1:0:1:0:0:0)$, $(0:0:0:0:0:1:-1:0:0:1:1)$ |
Maps to other modular curves
Map
of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve
39.84.5.b.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -z+w+r+s$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z-r-s$ |
| $\displaystyle W$ |
$=$ |
$\displaystyle z+w+u-r$ |
| $\displaystyle T$ |
$=$ |
$\displaystyle x-z+w-t-s$ |
Equation of the image curve:
| $0$ |
$=$ |
$ XY+XZ-XW+YW $ |
|
$=$ |
$ X^{2}+Y^{2}+XZ+YZ+Z^{2}-XW-ZW $ |
|
$=$ |
$ 2X^{2}-XY-Y^{2}-XZ-YZ+XT+2YT+2ZT-WT-T^{2} $ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
39.168.11.a.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle b$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle a$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 4X^{14}Y^{4}+20X^{13}Y^{5}+41X^{12}Y^{6}+44X^{11}Y^{7}+26X^{10}Y^{8}+8X^{9}Y^{9}+X^{8}Y^{10}+49X^{13}Y^{4}Z+217X^{12}Y^{5}Z+382X^{11}Y^{6}Z+338X^{10}Y^{7}Z+155X^{9}Y^{8}Z+33X^{8}Y^{9}Z+2X^{7}Y^{10}Z+24X^{14}Y^{2}Z^{2}+108X^{13}Y^{3}Z^{2}+530X^{12}Y^{4}Z^{2}+1359X^{11}Y^{5}Z^{2}+1747X^{10}Y^{6}Z^{2}+1180X^{9}Y^{7}Z^{2}+398X^{8}Y^{8}Z^{2}+53X^{7}Y^{9}Z^{2}+2X^{6}Y^{10}Z^{2}+154X^{13}Y^{2}Z^{3}+891X^{12}Y^{3}Z^{3}+2803X^{11}Y^{4}Z^{3}+4843X^{10}Y^{5}Z^{3}+4572X^{9}Y^{6}Z^{3}+2275X^{8}Y^{7}Z^{3}+528X^{7}Y^{8}Z^{3}+44X^{6}Y^{9}Z^{3}+36X^{14}Z^{4}+144X^{13}YZ^{4}+1034X^{12}Y^{2}Z^{4}+3861X^{11}Y^{3}Z^{4}+8186X^{10}Y^{4}Z^{4}+10060X^{9}Y^{5}Z^{4}+6848X^{8}Y^{6}Z^{4}+2361X^{7}Y^{7}Z^{4}+342X^{6}Y^{8}Z^{4}+5X^{5}Y^{9}Z^{4}-X^{4}Y^{10}Z^{4}+69X^{13}Z^{5}+732X^{12}YZ^{5}+3574X^{11}Y^{2}Z^{5}+8974X^{10}Y^{3}Z^{5}+13161X^{9}Y^{4}Z^{5}+11095X^{8}Y^{5}Z^{5}+4894X^{7}Y^{6}Z^{5}+901X^{6}Y^{7}Z^{5}-9X^{5}Y^{8}Z^{5}-14X^{4}Y^{9}Z^{5}+235X^{12}Z^{6}+1915X^{11}YZ^{6}+6215X^{10}Y^{2}Z^{6}+10111X^{9}Y^{3}Z^{6}+8580X^{8}Y^{4}Z^{6}+2964X^{7}Y^{5}Z^{6}-524X^{6}Y^{6}Z^{6}-636X^{5}Y^{7}Z^{6}-120X^{4}Y^{8}Z^{6}+3X^{3}Y^{9}Z^{6}+2X^{2}Y^{10}Z^{6}+441X^{11}Z^{7}+2455X^{10}YZ^{7}+4081X^{9}Y^{2}Z^{7}+324X^{8}Y^{3}Z^{7}-6229X^{7}Y^{4}Z^{7}-7493X^{6}Y^{5}Z^{7}-3695X^{5}Y^{6}Z^{7}-732X^{4}Y^{7}Z^{7}+5X^{3}Y^{8}Z^{7}+23X^{2}Y^{9}Z^{7}+2XY^{10}Z^{7}+436X^{10}Z^{8}+621X^{9}YZ^{8}-4333X^{8}Y^{2}Z^{8}-13819X^{7}Y^{3}Z^{8}-16553X^{6}Y^{4}Z^{8}-9550X^{5}Y^{5}Z^{8}-2502X^{4}Y^{6}Z^{8}-114X^{3}Y^{7}Z^{8}+88X^{2}Y^{8}Z^{8}+16XY^{9}Z^{8}+Y^{10}Z^{8}+11X^{9}Z^{9}-2717X^{8}YZ^{9}-11226X^{7}Y^{2}Z^{9}-17653X^{6}Y^{3}Z^{9}-13333X^{5}Y^{4}Z^{9}-4848X^{4}Y^{5}Z^{9}-630X^{3}Y^{6}Z^{9}+108X^{2}Y^{7}Z^{9}+42XY^{8}Z^{9}+6Y^{9}Z^{9}-456X^{8}Z^{10}-4111X^{7}YZ^{10}-9868X^{6}Y^{2}Z^{10}-10452X^{5}Y^{3}Z^{10}-5516X^{4}Y^{4}Z^{10}-1492X^{3}Y^{5}Z^{10}-168X^{2}Y^{6}Z^{10}+16XY^{7}Z^{10}+13Y^{8}Z^{10}-521X^{7}Z^{11}-2615X^{6}YZ^{11}-4357X^{5}Y^{2}Z^{11}-3668X^{4}Y^{3}Z^{11}-1972X^{3}Y^{4}Z^{11}-742X^{2}Y^{5}Z^{11}-140XY^{6}Z^{11}+8Y^{7}Z^{11}-215X^{6}Z^{12}-775X^{5}YZ^{12}-1329X^{4}Y^{2}Z^{12}-1550X^{3}Y^{3}Z^{12}-1092X^{2}Y^{4}Z^{12}-336XY^{5}Z^{12}-14Y^{6}Z^{12}-14X^{5}Z^{13}-210X^{4}YZ^{13}-714X^{3}Y^{2}Z^{13}-852X^{2}Y^{3}Z^{13}-364XY^{4}Z^{13}-28Y^{5}Z^{13}-4X^{4}Z^{14}-175X^{3}YZ^{14}-362X^{2}Y^{2}Z^{14}-208XY^{3}Z^{14}-14Y^{4}Z^{14}-17X^{3}Z^{15}-73X^{2}YZ^{15}-54XY^{2}Z^{15}+8Y^{3}Z^{15}-4X^{2}Z^{16}+13Y^{2}Z^{16}+2XZ^{17}+6YZ^{17}+Z^{18} $ |
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.
