Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ y^{2} - 3 y z + y w + w^{2} $ |
| $=$ | $y^{2} + y w + 3 z^{2} + w^{2} + t^{2}$ |
| $=$ | $5 x^{2} - y w + z^{2} - w^{2} - t^{2}$ |
![Copy content]()
![Toggle raw display]()
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 21 x^{8} + 60 x^{7} y + 85 x^{6} y^{2} + 612 x^{6} z^{2} + 50 x^{5} y^{3} + 990 x^{5} y z^{2} + \cdots + 20250 z^{8} $ |
![Copy content]()
![Toggle raw display]()
This modular curve has no real points and no $\Q_p$ points for $p=11$, and therefore no rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle y+z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle x+w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}t$ |
Maps to other modular curves
$j$-invariant map
of degree 144 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{2^8\cdot3^2}\cdot\frac{140874638488155yw^{15}t^{2}+563584079868120yw^{13}t^{4}+845694626827680yw^{11}t^{6}+629647796044800yw^{9}t^{8}+248100712238592yw^{7}t^{10}+79423819149312yw^{5}t^{12}+34123679907840yw^{3}t^{14}+5821365288960ywt^{16}-338083943492547z^{2}w^{16}-1689858641245860z^{2}w^{14}t^{2}-3371490608616720z^{2}w^{12}t^{4}-3511543210676352z^{2}w^{10}t^{6}-2159468414772480z^{2}w^{8}t^{8}-836885443774464z^{2}w^{6}t^{10}-451830142033920z^{2}w^{4}t^{12}-294443547033600z^{2}w^{2}t^{14}-70744615354368z^{2}t^{16}+84603132258543zw^{17}+450915890089896zw^{15}t^{2}+844208691547680zw^{13}t^{4}+737163580412160zw^{11}t^{6}+288955725166080zw^{9}t^{8}-103983987437568zw^{7}t^{10}-213929312993280zw^{5}t^{12}-76150212526080zw^{3}t^{14}-84537841287168w^{18}-535428091809189w^{16}t^{2}-1408950804318300w^{14}t^{4}-2034783065271600w^{12}t^{6}-1764670155960960w^{10}t^{8}-954235991326464w^{8}t^{10}-283251360872448w^{6}t^{12}-37582369136640w^{4}t^{14}-16176271196160w^{2}t^{16}-5989352407040t^{18}}{t^{4}(93555yw^{11}t^{2}+1990440yw^{9}t^{4}-62784yw^{7}t^{6}-11520yw^{5}t^{8}-132864yw^{3}t^{10}+129024ywt^{12}+5087205z^{2}w^{12}+11994156z^{2}w^{10}t^{2}-7173360z^{2}w^{8}t^{4}+1780992z^{2}w^{6}t^{6}-1227264z^{2}w^{4}t^{8}+857088z^{2}w^{2}t^{10}-110592z^{2}t^{12}-280665zw^{13}-7947720zw^{11}t^{2}-7110720zw^{9}t^{4}+2115072zw^{7}t^{6}-719616zw^{5}t^{8}+165888zw^{3}t^{10}+93555w^{12}t^{2}+2524500w^{10}t^{4}-64080w^{8}t^{6}-267264w^{6}t^{8}-55808w^{4}t^{10}+248832w^{2}t^{12}-36864t^{14})}$ |
Hi
|
|
Cover information
Click on a modular curve in the diagram to see information about it.
|
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.
