$\GL_2(\Z/60\Z)$-generators: |
$\begin{bmatrix}11&24\\10&43\end{bmatrix}$, $\begin{bmatrix}39&46\\43&3\end{bmatrix}$, $\begin{bmatrix}41&56\\30&13\end{bmatrix}$, $\begin{bmatrix}57&28\\8&35\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
60.96.1-60.bs.1.1, 60.96.1-60.bs.1.2, 60.96.1-60.bs.1.3, 60.96.1-60.bs.1.4, 60.96.1-60.bs.1.5, 60.96.1-60.bs.1.6, 60.96.1-60.bs.1.7, 60.96.1-60.bs.1.8, 120.96.1-60.bs.1.1, 120.96.1-60.bs.1.2, 120.96.1-60.bs.1.3, 120.96.1-60.bs.1.4, 120.96.1-60.bs.1.5, 120.96.1-60.bs.1.6, 120.96.1-60.bs.1.7, 120.96.1-60.bs.1.8, 120.96.1-60.bs.1.9, 120.96.1-60.bs.1.10, 120.96.1-60.bs.1.11, 120.96.1-60.bs.1.12, 120.96.1-60.bs.1.13, 120.96.1-60.bs.1.14, 120.96.1-60.bs.1.15, 120.96.1-60.bs.1.16 |
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 $ | $=$ | $ 2 x^{2} - x y + 2 x w - 3 y^{2} + 2 y w - 2 w^{2} $ |
| $=$ | $5 x^{2} - y^{2} - y z + z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 4 x^{3} y + 61 x^{2} y^{2} - 30 x^{2} y z + 15 x^{2} z^{2} - 114 x y^{3} + 60 x y^{2} z + \cdots + 25 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
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{3^3}{2^{12}}\cdot\frac{8312788037453525654450xz^{11}-55936195683871388545540xz^{10}w+104234733024785145886810xz^{9}w^{2}-687343423574804117376800xz^{8}w^{3}+1921125604632918136439680xz^{7}w^{4}-4840775577177858398700800xz^{6}w^{5}+15012011486628098852308480xz^{5}w^{6}-25327999054073512354365440xz^{4}w^{7}+34736416618291147010211840xz^{3}w^{8}-32674662558783465903226880xz^{2}w^{9}+21074143215917210966425600xzw^{10}-13835820868250299216691200xw^{11}+81199145737249332712201y^{2}z^{10}+7628302069970201759560y^{2}z^{9}w+472687704703427767042544y^{2}z^{8}w^{2}-1219155715700234148545024y^{2}z^{7}w^{3}+1812522811255685020944640y^{2}z^{6}w^{4}-12214779543834816467605504y^{2}z^{5}w^{5}+22867268617964228801228800y^{2}z^{4}w^{6}-41920913479364990482251776y^{2}z^{3}w^{7}+37777440549019136251592704y^{2}z^{2}w^{8}-29622915300496920176230400y^{2}zw^{9}+15124827386125077247426560y^{2}w^{10}-14812451403321750243599yz^{11}-57843271201363938985430yz^{10}w+38126938484837913276774yz^{9}w^{2}-265016995452996746796224yz^{8}w^{3}+1604425258388201155933760yz^{7}w^{4}-1327137839549179919446784yz^{6}w^{5}+8025869745114704659386880yz^{5}w^{6}-12277384304849219895402496yz^{4}w^{7}+25580781437991938110685184yz^{3}w^{8}-25356513828680762882129920yz^{2}w^{9}+20260306540054452906229760yzw^{10}-13949453681369064878899200yw^{11}+8202933519196097896324z^{12}+1907075517492550439890z^{11}w+100647184861760309403476z^{10}w^{2}-305719348999911660077536z^{9}w^{3}+808074954546456930343920z^{8}w^{4}-4219275379536994101655296z^{7}w^{5}+8033462381131923630048000z^{6}w^{6}-23801659329106571489562624z^{5}w^{7}+32356900798162926039556096z^{4}w^{8}-47514067906364011714969600z^{3}w^{9}+38555313548430492661841920z^{2}w^{10}-27524400143533214924800000zw^{11}+13995636443808275444531200w^{12}}{9604122542080000xz^{11}-53086826056908800xz^{10}w+232860517526405120xz^{9}w^{2}-456242020528947200xz^{8}w^{3}-472852965752176640xz^{7}w^{4}+3943703283381043200xz^{6}w^{5}-2897714741398201450xz^{5}w^{6}-6491841895262729980xz^{4}w^{7}-2655268851072216050xz^{3}w^{8}+2589296398728726960xz^{2}w^{9}+1182190646229866400xzw^{10}-168762102915020800xw^{11}+7555036636774400y^{2}z^{10}-272468182407577600y^{2}z^{9}w+1151933367140220928y^{2}z^{8}w^{2}-1199136611875422208y^{2}z^{7}w^{3}-3844920610451619840y^{2}z^{6}w^{4}+9253327618921988096y^{2}z^{5}w^{5}+6640595728622843995y^{2}z^{4}w^{6}+138288270823702368y^{2}z^{3}w^{7}-5789489318444293408y^{2}z^{2}w^{8}-817941118550890880y^{2}zw^{9}+325752696508242176y^{2}w^{10}+7715363382886400yz^{11}+15030219544985600yz^{10}w-70489420325978112yz^{9}w^{2}-89753642914807808yz^{8}w^{3}+641147374306918400yz^{7}w^{4}+357697911608180736yz^{6}w^{5}-6669337435139489005yz^{5}w^{6}-2637972533461464122yz^{4}w^{7}+1100597986823449042yz^{3}w^{8}+4369338580197906960yz^{2}w^{9}+229409403749306016yzw^{10}-233947871663449600yw^{11}+1888759159193600z^{12}-68117045601894400z^{11}w+299380027438661632z^{10}w^{2}-595075029964685312z^{9}w^{3}+180442207032770560z^{8}w^{4}+1543612764160917504z^{7}w^{5}-2745626609206652020z^{6}w^{6}+4806459439823576342z^{5}w^{7}+4150328112399200588z^{4}w^{8}+616623423076663200z^{3}w^{9}-3913332096281352176z^{2}w^{10}-546918996694089600zw^{11}+213804375805363200w^{12}}$ |
Hi
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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.