Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x^{2} w + x y w + y z w $ |
| $=$ | $x^{2} y + x y^{2} + y^{2} z$ |
| $=$ | $x^{3} + x^{2} y + x y z$ |
| $=$ | $x^{2} z + x y z + y z^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} y - x^{6} z + 4 x^{5} y z - 4 x^{5} z^{2} + 39 x^{4} y^{2} z - 78 x^{4} y z^{2} + 39 x^{4} z^{3} + \cdots + 4 z^{7} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{4} + x^{3} + x^{2} + x + 1\right) y $ | $=$ | $ -2x^{7} + 2x^{5} - 7x^{4} + 2x^{3} - 2x $ |
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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 |
|---|
| $(-1:1:0:0:0)$, $(0:0:0:3:1)$, $(0:0:0:0:1)$, $(0:0:0:-1/4:1)$ |
Maps to other modular curves
$j$-invariant map
of degree 56 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -3^2\,\frac{3238717xw^{8}-3458758xw^{7}t-9281869xw^{6}t^{2}+3037648xw^{5}t^{3}+4369880xw^{4}t^{4}-2854216xw^{3}t^{5}-365898xw^{2}t^{6}+1946175xwt^{7}-459459xt^{8}-72y^{9}+486y^{8}t-342y^{7}t^{2}-1728y^{6}t^{3}-234y^{5}t^{4}+1548y^{4}t^{5}-414y^{3}t^{6}-11502y^{2}t^{7}+121981yzw^{7}-4978699yzw^{6}t+309068yzw^{5}t^{2}+6186556yzw^{4}t^{3}+1183892yzw^{3}t^{4}-2189164yzw^{2}t^{5}-414810yzwt^{6}+407115yzt^{7}+3299842yw^{8}-5384324yw^{7}t-4121646yw^{6}t^{2}+5369288yw^{5}t^{3}-1117208yw^{4}t^{4}-6126438yw^{3}t^{5}+808018yw^{2}t^{6}+3493932ywt^{7}-927702yt^{8}+1015847z^{2}w^{7}-59837z^{2}w^{6}t-3179174z^{2}w^{5}t^{2}-1004764z^{2}w^{4}t^{3}+1341424z^{2}w^{3}t^{4}+270628z^{2}w^{2}t^{5}-97974z^{2}wt^{6}+151209z^{2}t^{7}+1759834zw^{8}+5830560zw^{7}t-3986216zw^{6}t^{2}-9600858zw^{5}t^{3}-2296860zw^{4}t^{4}+1946134zw^{3}t^{5}+978038zw^{2}t^{6}+123672zwt^{7}+1098zt^{8}+1351568w^{9}-2374104w^{8}t-8333788w^{7}t^{2}+5303814w^{6}t^{3}+12326790w^{5}t^{4}+4652384w^{4}t^{5}-281720w^{3}t^{6}-3251538w^{2}t^{7}-764550wt^{8}-216t^{9}}{w^{3}(4655xw^{5}+4479xw^{4}t-1297xw^{3}t^{2}-2295xw^{2}t^{3}+275xwt^{4}+63xt^{5}+955yzw^{4}-2376yzw^{3}t-3541yzw^{2}t^{2}-660yzwt^{3}+351yzt^{4}+4868yw^{5}+2912yw^{4}t+266yw^{3}t^{2}+1158yw^{2}t^{3}+998ywt^{4}-288yt^{5}+1611z^{2}w^{4}+1842z^{2}w^{3}t+19z^{2}w^{2}t^{2}-700z^{2}wt^{3}-87z^{2}t^{4}+1368zw^{5}+5022zw^{4}t+7514zw^{3}t^{2}+3682zw^{2}t^{3}+620zwt^{4}+24zt^{5}+1736w^{6}-1558w^{5}t-7946w^{4}t^{2}-8334w^{3}t^{3}-2002w^{2}t^{4}-96wt^{5})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
$X_0(39)$
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{6}Y-X^{6}Z+4X^{5}YZ+39X^{4}Y^{2}Z-4X^{5}Z^{2}-78X^{4}YZ^{2}+78X^{3}Y^{2}Z^{2}+39X^{4}Z^{3}-156X^{3}YZ^{3}+39X^{2}Y^{2}Z^{3}+77X^{3}Z^{4}-100X^{2}YZ^{4}+62X^{2}Z^{5}-25XYZ^{5}+24XZ^{6}+4Z^{7} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
$X_0(39)$
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -x^{4}-x^{3}z+40x^{2}z^{2}-39x^{2}zt+38xz^{3}-39xz^{2}t+12z^{4}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -z$ |
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.
This modular curve is minimally covered by the modular curves in the database listed below.
