| $\GL_2(\Z/38\Z)$-generators: |
$\begin{bmatrix}5&15\\0&21\end{bmatrix}$, $\begin{bmatrix}9&31\\15&36\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
38.240.4-38.d.1.1, 38.240.4-38.d.1.2, 76.240.4-38.d.1.1, 76.240.4-38.d.1.2, 114.240.4-38.d.1.1, 114.240.4-38.d.1.2, 152.240.4-38.d.1.1, 152.240.4-38.d.1.2, 152.240.4-38.d.1.3, 152.240.4-38.d.1.4, 190.240.4-38.d.1.1, 190.240.4-38.d.1.2, 228.240.4-38.d.1.1, 228.240.4-38.d.1.2, 266.240.4-38.d.1.1, 266.240.4-38.d.1.2 |
| Cyclic 38-isogeny field degree: |
$3$ |
| Cyclic 38-torsion field degree: |
$54$ |
| Full 38-torsion field degree: |
$6156$ |
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ 19 x^{2} + y^{2} + 2 y z + y w + z w $ |
| $=$ | $12 y^{3} + 2 y^{2} z + 33 y^{2} w - 2 y z^{2} - 28 y z w + 35 y w^{2} + 7 z^{3} + 14 z^{2} w + \cdots + 7 w^{3}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 48013 x^{6} + 5054 x^{4} y^{2} - 8303 x^{4} y z - 5776 x^{4} z^{2} + 304 x^{2} y^{4} + 1064 x^{2} y^{3} z + \cdots + 7 z^{6} $ |
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This modular curve has 1 rational CM point but no rational cusps or other known rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Maps to other modular curves
$j$-invariant map
of degree 120 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -2^{15}\cdot3^2\,\frac{3369808953868288y^{2}z^{18}+21615575273308160y^{2}z^{17}w+180885327031435264y^{2}z^{16}w^{2}+5705376164352000y^{2}z^{15}w^{3}-1498633078932191232y^{2}z^{14}w^{4}-6035135298994990080y^{2}z^{13}w^{5}+13670142159459732480y^{2}z^{12}w^{6}+75560760625468872960y^{2}z^{11}w^{7}+108400663310568727680y^{2}z^{10}w^{8}-363437347071092256000y^{2}z^{9}w^{9}-770602466770548021792y^{2}z^{8}w^{10}+90111560660514578160y^{2}z^{7}w^{11}+2779291944633252875760y^{2}z^{6}w^{12}-62726189053195326420y^{2}z^{5}w^{13}-2761829501305305477330y^{2}z^{4}w^{14}+413245139576712832755y^{2}z^{3}w^{15}+262766351390697647532y^{2}z^{2}w^{16}-52255117153760551920y^{2}zw^{17}+844185713469033024y^{2}w^{18}-5069534744018944yz^{19}+13006025911697408yz^{18}w+66369167535767552yz^{17}w^{2}+463593499329552384yz^{16}w^{3}+1064535551317719040yz^{15}w^{4}+3141457007371895808yz^{14}w^{5}-13986448503752543232yz^{13}w^{6}-60371156534054849280yz^{12}w^{7}-58519122024649272192yz^{11}w^{8}+384893270734607775360yz^{10}w^{9}+617634425402776584288yz^{9}w^{10}-663206169533664423312yz^{8}w^{11}-2753640937098260442000yz^{7}w^{12}+1158075675858624865740yz^{6}w^{13}+4780581784456329360486yz^{5}w^{14}-3199206485762377189020yz^{4}w^{15}-718238383034052226023yz^{3}w^{16}+572513491146393151812yz^{2}w^{17}-56044824781967740368yzw^{18}-2090889973910949696yw^{19}-10653111171284992z^{20}-37882242636316672z^{19}w-32298532447387648z^{18}w^{2}+112785673343277056z^{17}w^{3}-313895278479972352z^{16}w^{4}-1916311612401303040z^{15}w^{5}+7135485246242622720z^{14}w^{6}+42877551552441544704z^{13}w^{7}+33039055691565905088z^{12}w^{8}-228870735552748450656z^{11}w^{9}-386576529121788923808z^{10}w^{10}+406970192065783013784z^{9}w^{11}+1429708242405960687132z^{8}w^{12}-661288215402237530880z^{7}w^{13}-2169865323443113619895z^{6}w^{14}+1882708354521976186098z^{5}w^{15}-180750928270142188080z^{4}w^{16}-398060767751105235519z^{3}w^{17}+140871569310711814500z^{2}w^{18}-9852050763123295824zw^{19}-590761527685399872w^{20}}{483208743907753984y^{2}z^{18}-24504894376284782592y^{2}z^{17}w-30854147929145278464y^{2}z^{16}w^{2}-17798298632274051072y^{2}z^{15}w^{3}-653124481036754485248y^{2}z^{14}w^{4}+2071279640144352706560y^{2}z^{13}w^{5}-953256210972463153152y^{2}z^{12}w^{6}+378626329036313247744y^{2}z^{11}w^{7}-5957021620046586636288y^{2}z^{10}w^{8}+12933854795506304194560y^{2}z^{9}w^{9}-13650409587539470975488y^{2}z^{8}w^{10}+8897776495558138893312y^{2}z^{7}w^{11}-3915306067443673660416y^{2}z^{6}w^{12}+1206787559748356793600y^{2}z^{5}w^{13}-262836137648487131712y^{2}z^{4}w^{14}+39866466566301765696y^{2}z^{3}w^{15}-4028122018336732596y^{2}z^{2}w^{16}+244948447939137012y^{2}zw^{17}-6815571829761015y^{2}w^{18}+2318970647281664000yz^{19}-777792479112593408yz^{18}w-58652968453530583040yz^{17}w^{2}+368787706290675056640yz^{16}w^{3}-180911885921860190208yz^{15}w^{4}-1573965681308375482368yz^{14}w^{5}+2612217643674133118976yz^{13}w^{6}-4721738899265766309888yz^{12}w^{7}+15831993705121573761024yz^{11}w^{8}-31233881955031259102208yz^{10}w^{9}+35398165482756961320960yz^{9}w^{10}-25417413399842401540608yz^{8}w^{11}+12195759515810826479616yz^{7}w^{12}-3975477385687897458816yz^{6}w^{13}+863679990162009499200yz^{5}w^{14}-115179551974685456064yz^{4}w^{15}+6695849541176112456yz^{3}w^{16}+405083684266561350yz^{2}w^{17}-89730597687574158yzw^{18}+4246782924416355yw^{19}+1183272246739337216z^{20}+11593037176165040128z^{19}w-46167246465946419200z^{18}w^{2}-144015604705836924928z^{17}w^{3}+317408400412451389440z^{16}w^{4}+21128430496768770048z^{15}w^{5}+372444246755363672064z^{14}w^{6}-87831643573253505024z^{13}w^{7}-5381362783291638398976z^{12}w^{8}+14329658780676558752256z^{11}w^{9}-18741420627561218131200z^{10}w^{10}+15297016681064307995136z^{9}w^{11}-8418295608403033596096z^{8}w^{12}+3186762594535533594816z^{7}w^{13}-809009103990818800656z^{6}w^{14}+122406284971799634240z^{5}w^{15}-4958935885728214650z^{4}w^{16}-2059088405452656633z^{3}w^{17}+463018033975823766z^{2}w^{18}-42327927953061066zw^{19}+1534356507522699w^{20}}$ |
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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.
