Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x z + x t + y z - y w $ |
| $=$ | $2 x y - z^{2} + z w - z t - w t$ |
| $=$ | $2 x^{2} + 2 x y + 2 y^{2} + w t$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{5} z + 4 x^{4} y^{2} + 2 x^{4} z^{2} + 10 x^{3} y^{2} z + 3 x^{3} z^{3} + 16 x^{2} y^{2} z^{2} + \cdots + 4 y^{2} z^{4} $ |
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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 |
|---|
| $(0:0:-1:0:1)$, $(0:0:1:1:0)$, $(0:0:0:0:1)$, $(0:0:0:1:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^2\,\frac{235296y^{2}w^{10}-235296y^{2}w^{9}t-4001472y^{2}w^{8}t^{2}-8113824y^{2}w^{7}t^{3}-3755232y^{2}w^{6}t^{4}+3755232y^{2}w^{4}t^{6}+8113824y^{2}w^{3}t^{7}+4001472y^{2}w^{2}t^{8}+235296y^{2}wt^{9}-235296y^{2}t^{10}+117649z^{2}w^{10}-352949z^{2}w^{9}t-2000033z^{2}w^{8}t^{2}-4190446z^{2}w^{7}t^{3}+2100857z^{2}w^{6}t^{4}+6502705z^{2}w^{5}t^{5}+3978473z^{2}w^{4}t^{6}-133534z^{2}w^{3}t^{7}+703z^{2}w^{2}t^{8}-235301z^{2}wt^{9}+z^{2}t^{10}-zw^{11}+235302zw^{10}t-236004zw^{9}t^{2}+424253zw^{8}t^{3}-4287111zw^{7}t^{4}-279272zw^{6}t^{5}-1598344zw^{5}t^{6}+2107815zw^{4}t^{7}+1631923zw^{3}t^{8}+2119092zw^{2}t^{9}-6zwt^{10}-117647zt^{11}+w^{12}-5w^{11}t-352242w^{10}t^{2}-824235w^{9}t^{3}-1549808w^{8}t^{4}-1345973w^{7}t^{5}+6165496w^{6}t^{6}+2409259w^{5}t^{7}+6564016w^{4}t^{8}+3177237w^{3}t^{9}-116946w^{2}t^{10}-235301wt^{11}+t^{12}}{720y^{2}w^{10}-240y^{2}w^{9}t-2168y^{2}w^{8}t^{2}-3048y^{2}w^{7}t^{3}-1800y^{2}w^{6}t^{4}+1800y^{2}w^{4}t^{6}+3048y^{2}w^{3}t^{7}+2168y^{2}w^{2}t^{8}+240y^{2}wt^{9}-720y^{2}t^{10}+323z^{2}w^{10}-203z^{2}w^{9}t-969z^{2}w^{8}t^{2}-2056z^{2}w^{7}t^{3}-1538z^{2}w^{6}t^{4}-1242z^{2}w^{5}t^{5}-638z^{2}w^{4}t^{6}-532z^{2}w^{3}t^{7}+115z^{2}w^{2}t^{8}-83z^{2}wt^{9}-37z^{2}t^{10}-323zw^{11}+886zw^{10}t+1002zw^{9}t^{2}+239zw^{8}t^{3}-2038zw^{7}t^{4}-1092zw^{6}t^{5}+192zw^{5}t^{6}+1414zw^{4}t^{7}+201zw^{3}t^{8}-38zw^{2}t^{9}-406zwt^{10}-37zt^{11}+323w^{11}t+87w^{10}t^{2}-973w^{9}t^{3}-3136w^{8}t^{4}-5090w^{7}t^{5}-5422w^{6}t^{6}-3290w^{5}t^{7}-88w^{4}t^{8}+1195w^{3}t^{9}+327w^{2}t^{10}-397wt^{11}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
56.96.5.a.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 4X^{4}Y^{2}+X^{5}Z+10X^{3}Y^{2}Z+4XY^{4}Z+2X^{4}Z^{2}+16X^{2}Y^{2}Z^{2}+3X^{3}Z^{3}+10XY^{2}Z^{3}+2X^{2}Z^{4}+4Y^{2}Z^{4}+XZ^{5} $ |
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.
