Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x^{2} t - 2 x z t + x w t - y^{2} t - y z t + z^{2} t - z w t $ |
| $=$ | $x^{3} + x^{2} y + x^{2} w - x y^{2} + x y w - x z^{2} - y^{3} + z^{2} w - z w^{2}$ |
| $=$ | $2 x^{2} t + 3 x y t + y^{2} t - y w t - z w t$ |
| $=$ | $2 x z t - x w t + y z t - 2 y w t - 2 z w t + w^{2} t$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 52 x^{6} - 138 x^{5} z + 24 x^{4} y^{2} + 141 x^{4} z^{2} - 60 x^{3} y^{2} z - 80 x^{3} z^{3} + \cdots + z^{6} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -x^{7} - 7x^{4} + 8x $ |
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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 |
|---|
| $(0:0:0:0:1)$, $(1:-2:2:1:0)$, $(-1:1:0:0:0)$, $(1:0:2:1:0)$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{2\cdot3\cdot13^2}\cdot\frac{804736250520xw^{10}+3317808735132xw^{8}t^{2}+4256794007295xw^{6}t^{4}+823124733084xw^{4}t^{6}+1144990119780xw^{2}t^{8}+40989660192xt^{10}+82427046y^{11}+1033679712y^{9}t^{2}+7039715280y^{7}t^{4}+22738725906y^{5}t^{6}+35459224086y^{3}t^{8}+1030203236448yzw^{9}-403604016246yzw^{7}t^{2}+3485997811317yzw^{5}t^{4}-706208745294yzw^{3}t^{6}+132521313834yzwt^{8}+1012402513728yw^{10}+2627508053172yw^{8}t^{2}+5959463996430yw^{6}t^{4}+8777742588yw^{4}t^{6}+1256984820702yw^{2}t^{8}+76306307766yt^{10}+681413896066z^{2}w^{9}+3076914831286z^{2}w^{7}t^{2}+230053581926z^{2}w^{5}t^{4}+1689851406282z^{2}w^{3}t^{6}+48662061786z^{2}wt^{8}+508734799472zw^{10}-3150358493866zw^{8}t^{2}+5164437000148zw^{6}t^{4}-1800164416218zw^{4}t^{6}-34650943470zw^{2}t^{8}+46360782702zt^{10}-359533817216w^{11}+2803381331590w^{9}t^{2}-1276132121899w^{7}t^{4}+1013615582208w^{5}t^{6}+586373755500w^{3}t^{8}-7046397540wt^{10}}{t^{6}(200060xw^{4}+196405xw^{2}t^{2}+251784xt^{4}+243986yzw^{3}-128934yzwt^{2}+272190yw^{4}+135812yw^{2}t^{2}+251784yt^{4}+140039z^{2}w^{3}+152308z^{2}wt^{2}+176077zw^{4}-145733zw^{2}t^{2}-75296zt^{4}-103947w^{5}+161228w^{3}t^{2}+163540wt^{4})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
12.72.3.cy.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 2w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 52X^{6}+24X^{4}Y^{2}-72X^{2}Y^{4}-138X^{5}Z-60X^{3}Y^{2}Z-18XY^{4}Z+141X^{4}Z^{2}+54X^{2}Y^{2}Z^{2}+9Y^{4}Z^{2}-80X^{3}Z^{3}-24XY^{2}Z^{3}+30X^{2}Z^{4}+6Y^{2}Z^{4}-6XZ^{5}+Z^{6} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
12.72.3.cy.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{1}{6}z^{4}-\frac{1}{6}z^{3}w+\frac{5}{4}z^{2}w^{2}-\frac{1}{2}z^{2}t^{2}-\frac{2}{3}zw^{3}-\frac{1}{4}zwt^{2}+\frac{1}{3}w^{4}+\frac{1}{4}w^{2}t^{2}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{3}{16}z^{15}t-\frac{165}{64}z^{14}wt+\frac{939}{64}z^{13}w^{2}t-\frac{9}{32}z^{13}t^{3}-\frac{11793}{256}z^{12}w^{3}t+\frac{135}{64}z^{12}wt^{3}+\frac{23007}{256}z^{11}w^{4}t-\frac{837}{128}z^{11}w^{2}t^{3}-\frac{14901}{128}z^{10}w^{5}t+\frac{2637}{256}z^{10}w^{3}t^{3}+\frac{3291}{32}z^{9}w^{6}t-\frac{1431}{256}z^{9}w^{4}t^{3}-\frac{483}{8}z^{8}w^{7}t-\frac{999}{128}z^{8}w^{5}t^{3}+\frac{363}{16}z^{7}w^{8}t+\frac{819}{64}z^{7}w^{6}t^{3}-\frac{45}{8}z^{6}w^{9}t-\frac{135}{32}z^{6}w^{7}t^{3}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -\frac{1}{3}z^{4}+\frac{17}{12}z^{3}w-\frac{7}{4}z^{2}w^{2}+\frac{1}{2}z^{2}t^{2}+\frac{2}{3}zw^{3}+\frac{1}{4}zwt^{2}-\frac{1}{3}w^{4}-\frac{1}{4}w^{2}t^{2}$ |
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.
