Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{3} - x^{2} y + 2 x^{2} z - x^{2} w - x y^{2} + x y z + x y w - x z^{2} + y^{3} - y^{2} z $ |
| $=$ | $x^{3} - x^{2} y - x^{2} w - x y^{2} + x y z + x y w + 5 x z^{2} + y^{3} + y^{2} z + 2 y z^{2} - z w^{2}$ |
| $=$ | $3 x^{3} - 5 x^{2} y + x y^{2} + x y z - x y w + x z^{2} + x z w + y^{3} + y^{2} z + y^{2} w$ |
| $=$ | $x^{3} + x^{2} y + 2 x^{2} z - x^{2} w - x y^{2} - 5 x y z + x y w - x z^{2} - y^{3} - 3 y^{2} z + y w^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 7 x^{4} - 2 x^{3} y - 6 x^{3} z + 2 x^{2} y^{2} - x^{2} y z - 4 x^{2} z^{2} + 2 x y^{3} + 2 x y^{2} z + \cdots + z^{4} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{3} + x\right) y $ | $=$ | $ -2x^{4} + 4x^{2} - 2 $ |
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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 40 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{92643732xyw^{7}+1830391792xz^{8}+704600656xz^{7}w+765751308xz^{6}w^{2}+1097628608xz^{5}w^{3}+932961020xz^{4}w^{4}+977869364xz^{3}w^{5}+974080990xz^{2}w^{6}-324610178xzw^{7}-46291798xw^{8}-2048y^{8}w-3072y^{7}w^{2}-2816y^{6}w^{3}+4864y^{5}w^{4}+8832y^{4}w^{5}+7056y^{3}w^{6}+459346432y^{2}z^{7}+94920256y^{2}z^{6}w+152880232y^{2}z^{5}w^{2}+221339156y^{2}z^{4}w^{3}+169133640y^{2}z^{3}w^{4}+188878224y^{2}z^{2}w^{5}+184921992y^{2}zw^{6}-92632984y^{2}w^{7}+645060848yz^{8}+262237264yz^{7}w+347178564yz^{6}w^{2}+451762414yz^{5}w^{3}+394746876yz^{4}w^{4}+418863858yz^{3}w^{5}+416515334yz^{2}w^{6}-46343278yzw^{7}+46315134yw^{8}-368z^{9}+720z^{8}w-322532064z^{7}w^{2}-131117248z^{6}w^{3}-153360434z^{5}w^{4}-204580447z^{4}w^{5}-177283027z^{3}w^{6}-186746369z^{2}w^{7}-185187568zw^{8}+46325707w^{9}}{13126xyw^{7}+636xz^{8}+2464xz^{7}w+7196xz^{6}w^{2}-9812xz^{5}w^{3}-55074xz^{4}w^{4}-23814xz^{3}w^{5}+97160xz^{2}w^{6}-30392xzw^{7}-2831xw^{8}+168y^{2}z^{7}+468y^{2}z^{6}w+1236y^{2}z^{5}w^{2}-1564y^{2}z^{4}w^{3}-9124y^{2}z^{3}w^{4}-4516y^{2}z^{2}w^{5}+15324y^{2}zw^{6}-13126y^{2}w^{7}+228yz^{8}+942yz^{7}w+2548yz^{6}w^{2}-4158yz^{5}w^{3}-19594yz^{4}w^{4}-6644yz^{3}w^{5}+33752yz^{2}w^{6}-13184yzw^{7}+2831yw^{8}-114z^{7}w^{2}-471z^{6}w^{3}-1271z^{5}w^{4}+2107z^{4}w^{5}+10019z^{3}w^{6}+3389z^{2}w^{7}-18170zw^{8}+5197w^{9}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
20.40.2.b.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 7X^{4}-2X^{3}Y+2X^{2}Y^{2}+2XY^{3}-6X^{3}Z-X^{2}YZ+2XY^{2}Z-4X^{2}Z^{2}+2XYZ^{2}+2XZ^{3}+YZ^{3}+Z^{4} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
20.40.2.b.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle \frac{2}{3}x-\frac{1}{6}w$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{26}{27}x^{3}-\frac{4}{27}x^{2}y+\frac{1}{27}x^{2}w+\frac{10}{27}xy^{2}+\frac{1}{27}xyw-\frac{2}{27}xw^{2}+\frac{4}{27}y^{3}+\frac{2}{27}y^{2}w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -\frac{1}{3}x-\frac{1}{3}y-\frac{1}{6}w$ |
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.
