Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ x^{2} + x y + 2 y^{2} + z w + w^{2} $ |
| $=$ | $x^{3} - x^{2} y - x z^{2} - x z w + x w^{2} - y z w$ |
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This modular curve has 2 rational cusps but no known non-cuspidal rational 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 60 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{36016xyz^{8}+54304y^{2}z^{8}+16384z^{10}+69168xyz^{7}w+192224y^{2}z^{7}w+109072z^{9}w-223040xyz^{6}w^{2}-170176y^{2}z^{6}w^{2}+303152z^{8}w^{2}-603104xyz^{5}w^{3}-1979456y^{2}z^{5}w^{3}+295424z^{7}w^{3}+229840xyz^{4}w^{4}-3316480y^{2}z^{4}w^{4}-680864z^{6}w^{4}+1914848xyz^{3}w^{5}-695184y^{2}z^{3}w^{5}-2403008z^{5}w^{5}+1822912xyz^{2}w^{6}+2791136y^{2}z^{2}w^{6}-2362432z^{4}w^{6}+520832xyzw^{7}+2343744y^{2}zw^{7}+234496z^{3}w^{7}+520832y^{2}w^{8}+2024704z^{2}w^{8}+1310720zw^{9}+262144w^{10}}{7xyz^{8}+10y^{2}z^{8}+25xyz^{7}w+50y^{2}z^{7}w+5z^{9}w+41xyz^{6}w^{2}+74y^{2}z^{6}w^{2}+29z^{8}w^{2}+53xyz^{5}w^{3}+58y^{2}z^{5}w^{3}+59z^{7}w^{3}+35xyz^{4}w^{4}+30y^{2}z^{4}w^{4}+53z^{6}w^{4}+7xyz^{3}w^{5}-6y^{2}z^{3}w^{5}+17z^{5}w^{5}-7xyz^{2}w^{6}-16y^{2}z^{2}w^{6}-13z^{4}w^{6}-2xyzw^{7}-9y^{2}zw^{7}-16z^{3}w^{7}-2y^{2}w^{8}-4z^{2}w^{8}}$ |
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.
