Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x^{2} y + 2 x^{2} z + x^{2} w - x y^{2} - x y z - x y w + 3 x z w - x w^{2} + 2 y^{3} + y^{2} z + \cdots + y z w $ |
| $=$ | $2 x^{2} y + 2 x^{2} z - x y^{2} + 4 x y z - 2 x y w - 2 x z^{2} + x z w - 3 y^{3} + 3 y^{2} z - y z^{2}$ |
| $=$ | $2 x^{2} y + 2 x^{2} z + 3 x^{2} w + 9 x y^{2} - 5 x y z - x y w + x z w - 3 y^{3} + 3 y^{2} z - y z^{2}$ |
| $=$ | $2 x^{2} y + 2 x^{2} z - x y^{2} + 4 x y z + x y w - 2 x z^{2} + x z w + 7 y^{3} - 6 y^{2} z + \cdots + y z^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 200 x^{4} + 70 x^{3} y - 30 x^{3} z + 38 x^{2} y^{2} - 54 x^{2} y z - 37 x^{2} z^{2} + 6 x y^{3} + \cdots - z^{4} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{6} - 2x^{5} - 4x^{4} - 4x^{3} - 4x^{2} - 2x + 1 $ |
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This modular curve has rational points, including 2 rational_cusps and 2 known non-cuspidal non-CM points. The following are the known rational points on this modular curve (one row per $j$-invariant).
Maps to other modular curves
$j$-invariant map
of degree 32 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{5}\cdot\frac{130614400000000000x^{7}-99881600000000000x^{6}z-55977600000000000x^{6}w-122931200000000000x^{5}zw-477024800000000000x^{5}w^{2}-1112064800000000000x^{4}zw^{2}+197387926310400000x^{4}w^{3}+394110356677800000x^{3}zw^{3}+277763466735600000x^{3}w^{4}+1301844259231725200x^{2}zw^{4}-207339576407639620x^{2}w^{5}+115528051941719310xyw^{5}+1389879324467200000xz^{6}-1695807243571700000xz^{5}w+1251162780491750000xz^{4}w^{2}-880869154427673600xz^{3}w^{3}+502320398906296910xz^{2}w^{4}-374656451590002785xzw^{5}+96371895233205741xw^{6}+1223471152367100000yz^{6}-2759228863259750000yz^{5}w+2038438633988750000yz^{4}w^{2}-682108298581565450yz^{3}w^{3}-190611837352195895yz^{2}w^{4}+340220211204822902yzw^{5}-74743987053512293yw^{6}-960670392456600000z^{7}+2382474277455300000z^{6}w-2197883256738400000z^{5}w^{2}+991193000528510500z^{4}w^{3}-64080431766090230z^{3}w^{4}-257052774107024586z^{2}w^{5}+114287600000000000zw^{6}-15366400000000000w^{7}}{133280000000000x^{5}w^{2}+297920000000000x^{4}zw^{2}-56255270400000x^{4}w^{3}-102093447400000x^{3}zw^{3}+79252400000000x^{3}w^{4}+315695033018800x^{2}zw^{4}-50957507310780x^{2}w^{5}-5305822059110xyw^{5}+8482252800000xz^{6}-37149760620000xz^{5}w+91090806450000xz^{4}w^{2}-164844693518400xz^{3}w^{3}+167630755828290xz^{2}w^{4}-61374963333915xzw^{5}+8818499575479xw^{6}+2084692260000yz^{6}+18129734910000yz^{5}w-76396073950000yz^{4}w^{2}+134395684607450yz^{3}w^{3}-72650245298005yz^{2}w^{4}+29690063656938yzw^{5}-3779481440767yw^{6}+213496920000z^{7}-11664298100000z^{6}w+40076845200000z^{5}w^{2}-67994028658500z^{4}w^{3}+31663349175630z^{3}w^{4}-7558962881534z^{2}w^{5}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
28.32.2.a.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 2z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 200X^{4}+70X^{3}Y+38X^{2}Y^{2}+6XY^{3}-30X^{3}Z-54X^{2}YZ-24XY^{2}Z-37X^{2}Z^{2}+23XYZ^{2}+3Y^{2}Z^{2}+12XZ^{3}-3YZ^{3}-Z^{4} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
28.32.2.a.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{1}{5}yw$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{3}{5}y^{6}-\frac{86}{75}y^{5}z-\frac{13}{75}y^{5}w+\frac{286}{375}y^{4}z^{2}+\frac{176}{375}y^{4}zw-\frac{41}{375}y^{4}w^{2}-\frac{16}{75}y^{3}z^{3}-\frac{6}{25}y^{3}z^{2}w+\frac{2}{25}y^{3}zw^{2}+\frac{8}{375}y^{2}z^{4}+\frac{4}{125}y^{2}z^{3}w-\frac{2}{125}y^{2}z^{2}w^{2}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -y^{2}+\frac{2}{5}yz$ |
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.
