Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x y + y z - t^{2} $ |
| $=$ | $x y + x z + w^{2} + w t + t^{2}$ |
| $=$ | $x w - y w - z w - z t$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} y^{2} + x^{4} y z + 4 x^{2} y^{4} + 8 x^{2} y^{3} z + 11 x^{2} y^{2} z^{2} + 7 x^{2} y z^{3} + \cdots + y z^{5} $ |
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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 |
|---|
| $(-1:-1:1:0:0)$, $(0:1:0:0:0)$, $(1:0:0:0:0)$, $(0:0:1:0:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^2\,\frac{x^{12}-6x^{10}t^{2}+717x^{8}t^{4}-4250x^{6}t^{6}+185538x^{4}t^{8}-1082166x^{2}t^{10}+117649y^{12}+2117682y^{10}t^{2}+10941357y^{8}t^{4}+23294502y^{6}t^{6}+21882714y^{4}t^{8}+78354234y^{2}t^{10}+2yz^{11}+12yz^{9}t^{2}+1462yz^{7}t^{4}+2948yz^{5}t^{6}+365694yz^{3}t^{8}-244709886yzw^{10}-1345904378yzw^{9}t-4098884826yzw^{8}t^{2}-8350698873yzw^{7}t^{3}-12525660686yzw^{6}t^{4}-14291217106yzw^{5}t^{5}-12702214652yzw^{4}t^{6}-8782935907yzw^{3}t^{7}-4687469216yzw^{2}t^{8}-1845780342yzwt^{9}-358310904yzt^{10}+z^{12}+4z^{10}t^{2}+721z^{8}t^{4}+14z^{6}t^{6}+181351z^{4}t^{8}-726812z^{2}t^{10}+7529534w^{12}+45177204w^{11}t+280475114w^{10}t^{2}+988251180w^{9}t^{3}+2451799393w^{8}t^{4}+4374639632w^{7}t^{5}+5938320602w^{6}t^{6}+6157424198w^{5}t^{7}+4977426935w^{4}t^{8}+3068183890w^{3}t^{9}+1433145274w^{2}t^{10}+454997108wt^{11}+20727973t^{12}}{t^{2}(4x^{8}t^{2}-24x^{6}t^{4}-60x^{4}t^{6}+536x^{2}t^{8}+8yz^{7}t^{2}+16yz^{5}t^{4}-152yz^{3}t^{6}+32yzw^{8}+144yzw^{7}t+184yzw^{6}t^{2}-40yzw^{5}t^{3}-56yzw^{4}t^{4}+296yzw^{3}t^{5}+304yzw^{2}t^{6}+136yzwt^{7}+472yzt^{8}+4z^{8}t^{2}-84z^{4}t^{6}+328z^{2}t^{8}-24w^{8}t^{2}-96w^{7}t^{3}-244w^{6}t^{4}-420w^{5}t^{5}-1083w^{4}t^{6}-1658w^{3}t^{7}-1595w^{2}t^{8}-900wt^{9}-944t^{10})}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
28.96.5.a.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{4}Y^{2}+4X^{2}Y^{4}+X^{4}YZ+8X^{2}Y^{3}Z+11X^{2}Y^{2}Z^{2}+7X^{2}YZ^{3}+2X^{2}Z^{4}+Y^{2}Z^{4}+YZ^{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.
