Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - 2 x w - 2 y^{2} + 2 y z + 2 z^{2} $ |
| $=$ | $3 x^{2} + 4 x y + 8 x z + 2 x w - 3 y^{2} - 2 y z - 2 z^{2} + 2 w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{4} + 24 x^{3} y + 4 x^{3} z + 12 x^{2} y^{2} + 24 x^{2} y z - 2 x^{2} z^{2} - 144 x y^{3} + \cdots + 9 z^{4} $ |
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This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{4}y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
Maps to other modular curves
$j$-invariant map
of degree 32 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2^2\,\frac{3038159005920000xz^{7}+17509032738248000xz^{6}w+42474322985744000xz^{5}w^{2}+56574847658488000xz^{4}w^{3}+44862418190174400xz^{3}w^{4}+21232522205607440xz^{2}w^{5}+5562336372789280xzw^{6}+622842076501888xw^{7}+3667083612314000y^{2}z^{6}+15163826591598000y^{2}z^{5}w+26326311163188100y^{2}z^{4}w^{2}+24550305121658800y^{2}z^{3}w^{3}+12962748654518400y^{2}z^{2}w^{4}+3672232040940460y^{2}zw^{5}+435783243236753y^{2}w^{6}-4824955607064000yz^{7}-20062658030744000yz^{6}w-35306033962781600yz^{5}w^{2}-33817150353499600yz^{4}w^{3}-18796695716150800yz^{3}w^{4}-5921856907574240yz^{2}w^{5}-929914768272088yzw^{6}-47874787301300yw^{7}-5507766164754000z^{8}-22439545351040000z^{7}w-37358975758745600z^{6}w^{2}-31362096957126400z^{5}w^{3}-12149167601901200z^{4}w^{4}+112980560953280z^{3}w^{5}+1875190612767512z^{2}w^{6}+618527592796320zw^{7}+63623336273173w^{8}}{8403506000000xz^{7}+47038145860500xz^{6}w+110806278359000xz^{5}w^{2}+143569767161300xz^{4}w^{3}+111056050916800xz^{3}w^{4}+51454500019600xz^{2}w^{5}+13251502416880xzw^{6}+1465806901224xw^{7}+10143103401250y^{2}z^{6}+40263891950625y^{2}z^{5}w+67493188591300y^{2}z^{4}w^{2}+61102204669900y^{2}z^{3}w^{3}+31489356620140y^{2}z^{2}w^{4}+8754966970370y^{2}zw^{5}+1025577009144y^{2}w^{6}-13345759717500yz^{7}-53283729439000yz^{6}w-90598645801300yz^{5}w^{2}-84329361587200yz^{4}w^{3}-45804733047840yz^{3}w^{4}-14180695399600yz^{2}w^{5}-2200844101064yzw^{6}-112669276800yw^{7}-15234404959375z^{8}-59545410023000z^{7}w-95489948772300z^{6}w^{2}-77379557274250z^{5}w^{3}-28740581821640z^{4}w^{4}+753202141720z^{3}w^{5}+4577911501256z^{2}w^{6}+1472086343980zw^{7}+149733050464w^{8}}$ |
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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.
