| $\GL_2(\Z/60\Z)$-generators: |
$\begin{bmatrix}3&8\\35&9\end{bmatrix}$, $\begin{bmatrix}25&58\\42&17\end{bmatrix}$, $\begin{bmatrix}37&52\\38&21\end{bmatrix}$, $\begin{bmatrix}51&2\\31&49\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
60.96.1-60.bg.1.1, 60.96.1-60.bg.1.2, 60.96.1-60.bg.1.3, 60.96.1-60.bg.1.4, 60.96.1-60.bg.1.5, 60.96.1-60.bg.1.6, 60.96.1-60.bg.1.7, 60.96.1-60.bg.1.8, 120.96.1-60.bg.1.1, 120.96.1-60.bg.1.2, 120.96.1-60.bg.1.3, 120.96.1-60.bg.1.4, 120.96.1-60.bg.1.5, 120.96.1-60.bg.1.6, 120.96.1-60.bg.1.7, 120.96.1-60.bg.1.8 |
| Cyclic 60-isogeny field degree: |
$12$ |
| Cyclic 60-torsion field degree: |
$192$ |
| Full 60-torsion field degree: |
$46080$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 5 x^{2} - y^{2} - y w + w^{2} $ |
| $=$ | $x^{2} + 3 x y - 6 x z + 6 y z - y w - 6 z^{2} + w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 225 x^{4} - 900 x^{3} y + 660 x^{2} y^{2} + 30 x^{2} y z - 15 x^{2} z^{2} + 480 x y^{3} - 60 x y^{2} z + \cdots + 4 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:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
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 \frac{1}{2^4}\cdot\frac{137724009833596263963033600xz^{11}-362757645214298893438156800xz^{10}w+404952990441794019759882240xz^{9}w^{2}-333067194912341425725112320xz^{8}w^{3}+194722531175747857677189120xz^{7}w^{4}-91753938190226438699005440xz^{6}w^{5}+33267954654462242757254400xz^{5}w^{6}-9856482238780078589389440xz^{4}w^{7}+2219163622889511462760800xz^{3}w^{8}-395815676197407592087530xz^{2}w^{9}+46643860305617281503720xzw^{10}-3788749997051769758100xw^{11}-86146931807990202362757120y^{2}z^{10}+143655472749752957848780800y^{2}z^{9}w-175245319033842723145187328y^{2}z^{8}w^{2}+128563477064953361145004032y^{2}z^{7}w^{3}-74284767724151375903232000y^{2}z^{6}w^{4}+31237360860426103515506688y^{2}z^{5}w^{5}-10610262223676821559566080y^{2}z^{4}w^{6}+2659031648947679450300928y^{2}z^{3}w^{7}-525508253328689609445648y^{2}z^{2}w^{8}+67117128506834244767880y^{2}zw^{9}-5933756213210357148347y^{2}w^{10}-129635760674424313715097600yz^{11}+518876526168942484499988480yz^{10}w-550116367601927870186127360yz^{9}w^{2}+467457955656397136452288512yz^{8}w^{3}-269177550637481578528432128yz^{7}w^{4}+128108162562414793435722240yz^{6}w^{5}-45966896993184446531549952yz^{5}w^{6}+13670290567891240551478080yz^{4}w^{7}-3044189370732589149708672yz^{3}w^{8}+545378672724485147744742yz^{2}w^{9}-63423142432325842695690yzw^{10}+5272189072822653105803yw^{11}+197889643941060502329753600z^{12}-587485633635005196533760000z^{11}w+585301684833121053724508160z^{10}w^{2}-424806833366807879201587200z^{9}w^{3}+202887163544661275037769728z^{8}w^{4}-71161978116238541227487232z^{7}w^{5}+14309913787201073333318400z^{6}w^{6}-287248376034898570706688z^{5}w^{7}-1182196607443535159018640z^{4}w^{8}+481650365784070400409792z^{3}w^{9}-118122335444504667601992z^{2}w^{10}+16779282126708561191970zw^{11}-1483438957107165798128w^{12}}{2574322216405784985600xz^{11}-7868829464670899404800xz^{10}w+11495874455135797800960xz^{9}w^{2}-10673267117887681436160xz^{8}w^{3}+7027490507334538506240xz^{7}w^{4}-3454858958378360917920xz^{6}w^{5}+1297548133077309258000xz^{5}w^{6}-373634070349885120530xz^{4}w^{7}+81253078109366325480xz^{3}w^{8}-12814836654360264420xz^{2}w^{9}+1336337888559759360xzw^{10}-72264673080115200xw^{11}-1074356937402225721344y^{2}z^{10}+3310525838899139051520y^{2}z^{9}w-4792309629522018865152y^{2}z^{8}w^{2}+4319146192607845146624y^{2}z^{7}w^{3}-2696738498683290734080y^{2}z^{6}w^{4}+1223581197454061546880y^{2}z^{5}w^{5}-410266483709856551680y^{2}z^{4}w^{6}+100915300054277591376y^{2}z^{3}w^{7}-17569666485130319847y^{2}z^{2}w^{8}+1979679666780241920y^{2}zw^{9}-113177416930689024y^{2}w^{10}-3462665817888188006400yz^{11}+10731408246926086864896yz^{10}w-15844194125516038533120yz^{9}w^{2}+14809947848038786656768yz^{8}w^{3}-9780584176070753565696yz^{7}w^{4}+4807222491556956113440yz^{6}w^{5}-1800391049906874294000yz^{5}w^{6}+516122049920415448990yz^{4}w^{7}-111682830668240694234yz^{3}w^{8}+17554117773241519863yz^{2}w^{9}-1831257805254819840yzw^{10}+100559027312787456yw^{11}+4181351014616997888000z^{12}-12525084219859859865600z^{11}w+17308484280869144408064z^{10}w^{2}-14487674746131359539200z^{9}w^{3}+8017758944752832696832z^{8}w^{4}-2934158679884949281664z^{7}w^{5}+610749326155911976880z^{6}w^{6}+6429122890758393120z^{5}w^{7}-53752747364077857880z^{4}w^{8}+20173780502436504234z^{3}w^{9}-4139605304661356208z^{2}w^{10}+494919916695060480zw^{11}-28294354232672256w^{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.
