Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 4 x^{2} + x y - x w + 2 y^{2} - 2 y w + w^{2} $ |
| $=$ | $x y + 3 y^{2} - 2 y w - 3 z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 35 x^{4} - 45 x^{3} y + 15 x^{2} y^{2} - 69 x^{2} z^{2} + 45 x y z^{2} + 36 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:
$\displaystyle X$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{3}{2^{20}\cdot7^2}\cdot\frac{243298845170666693990357761327104xz^{16}w+401295437874324777775298923069440xz^{14}w^{3}+208709130937842434229590038732800xz^{12}w^{5}+48714954348953805260401213440000xz^{10}w^{7}+2181857424964836024707128800000xz^{8}w^{9}-3860758375303675269833724000000xz^{6}w^{11}-1989330566165946668761083750000xz^{4}w^{13}-398415451087263923840298750000xz^{2}w^{15}-27089179141240131536865234375xw^{17}-170467526803606924425869301055488y^{2}z^{16}-1458492129205153306008993517731840y^{2}z^{14}w^{2}-1528245233684143058675255523532800y^{2}z^{12}w^{4}-494581402068373960724938235136000y^{2}z^{10}w^{6}-60112462881441142909331229600000y^{2}z^{8}w^{8}-12801835251114282084957714000000y^{2}z^{6}w^{10}-6292823204412284979856121250000y^{2}z^{4}w^{12}-1421337657014634262125501562500y^{2}z^{2}w^{14}-113446652302179869958830859375y^{2}w^{16}+96018001929009029293736878669824yz^{16}w+838561832014754533405314237726720yz^{14}w^{3}+929987750557551019074016569753600yz^{12}w^{5}+319485723877480292549361455616000yz^{10}w^{7}+46926487452029389062743539200000yz^{8}w^{9}+10092055316849491568811228000000yz^{6}w^{11}+3744461723366150509763565000000yz^{4}w^{13}+782899161940840662258373125000yz^{2}w^{15}+64592456481646243295329687500yw^{17}+173302917245348945412143382331392z^{18}+1315405601575592263987821722468352z^{16}w^{2}+1413473108806194952701122511544320z^{14}w^{4}+463964828550276888085275982182400z^{12}w^{6}+49691119115141561134589322720000z^{10}w^{8}+6188279678517874299703328400000z^{8}w^{10}+5373614147164715222944601750000z^{6}w^{12}+1816075203250046141326290937500z^{4}w^{14}+217569130515215109296322890625z^{2}w^{16}+5558361794426124285888671875w^{18}}{z^{2}(266698721138576434176xz^{14}w-1207258763082727148640xz^{12}w^{3}-269144887044525773400xz^{10}w^{5}+433059906587220467625xz^{8}w^{7}+2679448038113280000xz^{6}w^{9}-100936559001600000xz^{4}w^{11}+1348568064000000xz^{2}w^{13}+325158996940725975104y^{2}z^{14}-1128864175160051351840y^{2}z^{12}w^{2}-2429268755775453965100y^{2}z^{10}w^{4}+1706242436330848586625y^{2}z^{8}w^{6}+63135110124298240000y^{2}z^{6}w^{8}-4265791250739200000y^{2}z^{4}w^{10}+153586694912000000y^{2}z^{2}w^{12}-1593472000000000y^{2}w^{14}+86073818824871988096yz^{14}w+780592524568603750080yz^{12}w^{3}+852907717631550709800yz^{10}w^{5}-1204575612152277088500yz^{8}w^{7}-40450847894446080000yz^{6}w^{9}+2957287366656000000yz^{4}w^{11}-119488286208000000yz^{2}w^{13}+1685710080000000yw^{15}-368047389874400031936z^{16}+514657472762685281568z^{14}w^{2}+2993038992447359842980z^{12}w^{4}-1036239908036307493575z^{10}w^{6}-3927522272797822125z^{8}w^{8}+4140956801802240000z^{6}w^{10}-168326198476800000z^{4}w^{12}+2359994112000000z^{2}w^{14})}$ |
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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.