Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} - x y + 2 y^{2} - y z + z^{2} $ |
| $=$ | $x^{2} + x y - 4 x z + y^{2} + y z + z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 4 x^{3} y + 12 x^{2} y^{2} - 16 x y^{3} + 7 y^{4} + z^{4} $ |
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle 2^2\cdot3^3\,\frac{10982655558144xz^{9}w^{2}-7321764804480xz^{7}w^{4}+1044628945920xz^{5}w^{6}+32597512640xz^{3}w^{8}-5409971616xzw^{10}+7687690611840y^{2}z^{10}-6406881410160y^{2}z^{8}w^{2}-274934937312y^{2}z^{6}w^{4}+727151493600y^{2}z^{4}w^{6}-89611831512y^{2}z^{2}w^{8}+1239432215y^{2}w^{10}-4392646536960yz^{11}-1829001295872yz^{9}w^{2}+2536551448704yz^{7}w^{4}-140009197440yz^{5}w^{6}-99165871504yz^{3}w^{8}+7580360368yzw^{10}+4392861061248z^{12}-3662402688000z^{10}w^{2}-2310705033840z^{8}w^{4}+1825146739360z^{6}w^{6}-271186899480z^{4}w^{8}+5825978480z^{2}w^{10}+107884133w^{12}}{686415972384xz^{9}w^{2}+91522060056xz^{7}w^{4}-41470906176xz^{5}w^{6}-2860685856xz^{3}w^{8}+42353640xzw^{10}+480480663240y^{2}z^{10}+80086017627y^{2}z^{8}w^{2}-141812044542y^{2}z^{6}w^{4}-13342757967y^{2}z^{4}w^{6}+806836842y^{2}z^{2}w^{8}+17294403y^{2}w^{10}-274540408560yz^{11}-388987067232yz^{9}w^{2}-44804993184yz^{7}w^{4}+18342558864yz^{5}w^{6}+3126908736yz^{3}w^{8}+67765824yzw^{10}+274553816328z^{12}+45780033600z^{10}w^{2}-215460640143z^{8}w^{4}-25261433442z^{6}w^{6}+6701620779z^{4}w^{8}+387535806z^{2}w^{10}-823543w^{12}}$ |
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Cover information
Click on a modular curve in the diagram to see information about it.
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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.