$\GL_2(\Z/60\Z)$-generators: |
$\begin{bmatrix}5&32\\39&19\end{bmatrix}$, $\begin{bmatrix}7&46\\36&25\end{bmatrix}$, $\begin{bmatrix}11&12\\42&43\end{bmatrix}$, $\begin{bmatrix}53&24\\36&5\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
60.192.3-60.bv.1.1, 60.192.3-60.bv.1.2, 60.192.3-60.bv.1.3, 60.192.3-60.bv.1.4, 60.192.3-60.bv.1.5, 60.192.3-60.bv.1.6, 60.192.3-60.bv.1.7, 60.192.3-60.bv.1.8, 120.192.3-60.bv.1.1, 120.192.3-60.bv.1.2, 120.192.3-60.bv.1.3, 120.192.3-60.bv.1.4, 120.192.3-60.bv.1.5, 120.192.3-60.bv.1.6, 120.192.3-60.bv.1.7, 120.192.3-60.bv.1.8, 120.192.3-60.bv.1.9, 120.192.3-60.bv.1.10, 120.192.3-60.bv.1.11, 120.192.3-60.bv.1.12, 120.192.3-60.bv.1.13, 120.192.3-60.bv.1.14, 120.192.3-60.bv.1.15, 120.192.3-60.bv.1.16, 120.192.3-60.bv.1.17, 120.192.3-60.bv.1.18, 120.192.3-60.bv.1.19, 120.192.3-60.bv.1.20, 120.192.3-60.bv.1.21, 120.192.3-60.bv.1.22, 120.192.3-60.bv.1.23, 120.192.3-60.bv.1.24 |
Cyclic 60-isogeny field degree: |
$12$ |
Cyclic 60-torsion field degree: |
$192$ |
Full 60-torsion field degree: |
$23040$ |
Embedded model Embedded model in $\mathbb{P}^{5}$
$ 0 $ | $=$ | $ 2 x u + y z - 2 y w - y t $ |
| $=$ | $2 x y + y z - 2 y w + z u - 2 w u + t u$ |
| $=$ | $2 x^{2} + 2 x z - 4 x w + y^{2} + z^{2} + z w - w^{2} + t^{2} + u^{2}$ |
| $=$ | $2 x^{2} - y^{2} - 2 z^{2} + 3 z w - 3 w^{2} - u^{2}$ |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 4500 x^{6} y^{2} - 6000 x^{5} y^{3} + 3000 x^{5} y z^{2} + 1625 x^{4} y^{4} + 1950 x^{4} y^{2} z^{2} + \cdots + 81 z^{8} $ |
Geometric Weierstrass model Geometric Weierstrass model
$ 45 w^{2} $ | $=$ | $ -56 x^{4} + 192 x^{3} y + 20 x^{3} z + 110 x^{2} y z + 49 x^{2} z^{2} + 236 x y z^{2} - 160 x z^{3} - 60 y z^{3} + 133 z^{4} $ |
$0$ | $=$ | $x^{2} + y^{2} + z^{2}$ |
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:
$\displaystyle X$ |
$=$ |
$\displaystyle t$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 10w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{10}{3}u$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{2^7}{3^2}\cdot\frac{67402581840xwt^{10}-6328999735200xwt^{8}u^{2}+69053407200000xwt^{6}u^{4}-176001508800000xwt^{4}u^{6}+101697523200000xwt^{2}u^{8}-6220032000000xwu^{10}-2433353886xt^{11}+790985664720xt^{9}u^{2}-16154461022400xt^{7}u^{4}+70241107392000xt^{5}u^{6}-75568512960000xt^{3}u^{8}+14681356800000xtu^{10}+883394890560ywt^{9}u-20567680185600ywt^{7}u^{3}+94669164288000ywt^{5}u^{5}-107595371520000ywt^{3}u^{7}+22503936000000ywtu^{9}-127985004180yt^{10}u+3540880785600yt^{8}u^{3}-20850833808000yt^{6}u^{5}+34398083520000yt^{4}u^{7}-14092444800000yt^{2}u^{9}+649856000000yu^{11}+2433353886zt^{11}-557219616120zt^{9}u^{2}+9394244438400zt^{7}u^{4}-34368576912000zt^{5}u^{6}+31021479360000zt^{3}u^{8}-5002636800000ztu^{10}+67399928280w^{2}t^{10}-4981489599600w^{2}t^{8}u^{2}+43709286744000w^{2}t^{6}u^{4}-89423796960000w^{2}t^{4}u^{6}+40813718400000w^{2}t^{2}u^{8}-1921920000000w^{2}u^{10}-4866707772wt^{11}+1458830068560wt^{9}u^{2}-28675096416000wt^{7}u^{4}+120756756960000wt^{5}u^{6}-126233688960000wt^{3}u^{8}+23912985600000wtu^{10}+729t^{12}-56946112020t^{10}u^{2}+2400945948000t^{8}u^{4}-18788972760000t^{6}u^{6}+37367124480000t^{4}u^{8}-16952947200000t^{2}u^{10}+797760000000u^{12}}{t^{4}(1477440xwt^{6}-49874400xwt^{4}u^{2}+130320000xwt^{2}u^{4}-24960000xwu^{6}+1822743xt^{7}-75486060xt^{5}u^{2}+295106400xt^{3}u^{4}-157056000xtu^{6}+11400480ywt^{5}u-76032000ywt^{3}u^{3}+54144000ywtu^{5}+15059250yt^{6}u-147605400yt^{4}u^{3}+203304000yt^{2}u^{5}-24320000yu^{7}-1822743zt^{7}+54309960zt^{5}u^{2}-156434400zt^{3}u^{4}+59136000ztu^{6}+1477440w^{2}t^{6}-33609600w^{2}t^{4}u^{2}+61128000w^{2}t^{2}u^{4}-7680000w^{2}u^{6}+3645486wt^{7}-103725360wt^{5}u^{2}+270532800wt^{3}u^{4}-84864000wtu^{6}+6096330t^{6}u^{2}-72903600t^{4}u^{4}+110256000t^{2}u^{6}-13440000u^{8})}$ |
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.