$\GL_2(\Z/60\Z)$-generators: |
$\begin{bmatrix}11&22\\51&25\end{bmatrix}$, $\begin{bmatrix}29&0\\9&1\end{bmatrix}$, $\begin{bmatrix}41&54\\21&25\end{bmatrix}$, $\begin{bmatrix}43&10\\45&41\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
60.96.1-60.q.1.1, 60.96.1-60.q.1.2, 60.96.1-60.q.1.3, 60.96.1-60.q.1.4, 60.96.1-60.q.1.5, 60.96.1-60.q.1.6, 60.96.1-60.q.1.7, 60.96.1-60.q.1.8, 120.96.1-60.q.1.1, 120.96.1-60.q.1.2, 120.96.1-60.q.1.3, 120.96.1-60.q.1.4, 120.96.1-60.q.1.5, 120.96.1-60.q.1.6, 120.96.1-60.q.1.7, 120.96.1-60.q.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 $ | $=$ | $ x^{2} - 4 x y + 3 x z + y^{2} - 3 y z $ |
| $=$ | $6 x^{2} + 26 x y + 13 x z + 6 y^{2} - 13 y z + 45 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 5 x^{4} + 3 x^{2} y^{2} + 5 x^{2} z^{2} - 6 x y^{2} z + 3 y^{2} z^{2} + 5 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:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle y$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{2^4}{5^2}\cdot\frac{1249996267520xz^{11}-218752332800xz^{9}w^{2}-3519406080xz^{7}w^{4}+398391680xz^{5}w^{6}+12143200xz^{3}w^{8}+94380xzw^{10}-1249996267520yz^{11}+218752332800yz^{9}w^{2}+3519406080yz^{7}w^{4}-398391680yz^{5}w^{6}-12143200yz^{3}w^{8}-94380yzw^{10}+2249996267520z^{12}-456252519424z^{10}w^{2}+699215616z^{8}w^{4}+1089600128z^{6}w^{6}+23041040z^{4}w^{8}+44220z^{2}w^{10}-1331w^{12}}{w^{4}(2880xz^{7}+1720xz^{5}w^{2}+350xz^{3}w^{4}+30xzw^{6}-2880yz^{7}-1720yz^{5}w^{2}-350yz^{3}w^{4}-30yzw^{6}+2880z^{8}+1864z^{6}w^{2}+445z^{4}w^{4}+54z^{2}w^{6}+w^{8})}$ |
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.