| $\GL_2(\Z/60\Z)$-generators: |
$\begin{bmatrix}1&40\\32&9\end{bmatrix}$, $\begin{bmatrix}29&6\\43&49\end{bmatrix}$, $\begin{bmatrix}33&44\\19&19\end{bmatrix}$, $\begin{bmatrix}43&16\\42&41\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
60.96.1-60.bn.1.1, 60.96.1-60.bn.1.2, 60.96.1-60.bn.1.3, 60.96.1-60.bn.1.4, 60.96.1-60.bn.1.5, 60.96.1-60.bn.1.6, 60.96.1-60.bn.1.7, 60.96.1-60.bn.1.8, 120.96.1-60.bn.1.1, 120.96.1-60.bn.1.2, 120.96.1-60.bn.1.3, 120.96.1-60.bn.1.4, 120.96.1-60.bn.1.5, 120.96.1-60.bn.1.6, 120.96.1-60.bn.1.7, 120.96.1-60.bn.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 $ | $=$ | $ 3 y^{2} - 3 y z - 3 z^{2} - w^{2} $ |
| $=$ | $5 x^{2} - 2 y^{2} - 3 y z - 3 z^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 3 x^{2} y^{2} + 15 x^{2} z^{2} + y^{4} - 60 y^{2} z^{2} + 900 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:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{15}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -2^4\cdot5^2\,\frac{14760427500xyz^{10}+6695743500xyz^{8}w^{2}+821137500xyz^{6}w^{4}-68388300xyz^{4}w^{6}-22651632xyz^{2}w^{8}-919212xyw^{10}+9121612500xz^{11}+6338776500xz^{9}w^{2}+1359301500xz^{7}w^{4}+34599420xz^{5}w^{6}-25642116xz^{3}w^{8}-2766252xzw^{10}-23947650000yz^{11}-12456787500yz^{9}w^{2}-2433827250yz^{7}w^{4}-366951600yz^{5}w^{6}-51805125yz^{3}w^{8}-2842650yzw^{10}-14800978125z^{12}-11268517500z^{10}w^{2}-3123400500z^{8}w^{4}-498480750z^{6}w^{6}-74390625z^{4}w^{8}-7470675z^{2}w^{10}-195725w^{12}}{w^{2}(51637500xyz^{8}+9618750xyz^{6}w^{2}-1113750xyz^{4}w^{4}-150300xyz^{2}w^{6}-1050xyw^{8}+31893750xz^{9}+13668750xz^{7}w^{2}+243000xz^{5}w^{4}-288900xz^{3}w^{6}-10800xzw^{8}+83531250yz^{9}+20098125yz^{7}w^{2}+3685500yz^{5}w^{4}+399150yz^{3}w^{6}+8130yzw^{8}+51637500z^{10}+24856875z^{8}w^{2}+4471875z^{6}w^{4}+695475z^{4}w^{6}+39480z^{2}w^{8}+223w^{10})}$ |
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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.
