$\GL_2(\Z/60\Z)$-generators: |
$\begin{bmatrix}13&35\\58&9\end{bmatrix}$, $\begin{bmatrix}17&40\\4&49\end{bmatrix}$, $\begin{bmatrix}29&25\\28&1\end{bmatrix}$, $\begin{bmatrix}39&40\\40&47\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
120.144.1-60.k.2.1, 120.144.1-60.k.2.2, 120.144.1-60.k.2.3, 120.144.1-60.k.2.4, 120.144.1-60.k.2.5, 120.144.1-60.k.2.6, 120.144.1-60.k.2.7, 120.144.1-60.k.2.8, 120.144.1-60.k.2.9, 120.144.1-60.k.2.10, 120.144.1-60.k.2.11, 120.144.1-60.k.2.12, 120.144.1-60.k.2.13, 120.144.1-60.k.2.14, 120.144.1-60.k.2.15, 120.144.1-60.k.2.16 |
Cyclic 60-isogeny field degree: |
$8$ |
Cyclic 60-torsion field degree: |
$128$ |
Full 60-torsion field degree: |
$30720$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 x^{2} - y^{2} + y z + y w $ |
| $=$ | $y^{2} - 2 y z - 2 y w + 2 z^{2} + z w + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 2 x^{4} + 3 x^{2} y z - 6 x^{2} z^{2} + 3 y^{2} z^{2} - 9 y z^{3} + 9 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:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}y$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -3^3\,\frac{236yz^{17}-2420yz^{16}w+11024yz^{15}w^{2}-32880yz^{14}w^{3}+70800yz^{13}w^{4}-109488yz^{12}w^{5}+133232yz^{11}w^{6}-109328yz^{10}w^{7}+42920yz^{9}w^{8}+42920yz^{8}w^{9}-109328yz^{7}w^{10}+133232yz^{6}w^{11}-109488yz^{5}w^{12}+70800yz^{4}w^{13}-32880yz^{3}w^{14}+11024yz^{2}w^{15}-2420yzw^{16}+236yw^{17}-121z^{18}+990z^{17}w-5265z^{16}w^{2}+23296z^{15}w^{3}-64980z^{14}w^{4}+133416z^{13}w^{5}-246116z^{12}w^{6}+342144z^{11}w^{7}-445374z^{10}w^{8}+462580z^{9}w^{9}-445374z^{8}w^{10}+342144z^{7}w^{11}-246116z^{6}w^{12}+133416z^{5}w^{13}-64980z^{4}w^{14}+23296z^{3}w^{15}-5265z^{2}w^{16}+990zw^{17}-121w^{18}}{(z+w)^{5}(z^{2}-zw+w^{2})^{5}(2yz^{2}-8yzw+2yw^{2}+6z^{3}+9z^{2}w+9zw^{2}+6w^{3})}$ |
Hi
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Cover information
Click on a modular curve in the diagram to see information about it.
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The following modular covers realize this modular curve as a fiber product over $X(1)$.
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.