| $\GL_2(\Z/60\Z)$-generators: |
$\begin{bmatrix}19&18\\45&23\end{bmatrix}$, $\begin{bmatrix}27&32\\19&25\end{bmatrix}$, $\begin{bmatrix}31&52\\51&35\end{bmatrix}$, $\begin{bmatrix}35&26\\47&39\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
60.288.5-60.os.2.1, 60.288.5-60.os.2.2, 60.288.5-60.os.2.3, 60.288.5-60.os.2.4, 60.288.5-60.os.2.5, 60.288.5-60.os.2.6, 60.288.5-60.os.2.7, 60.288.5-60.os.2.8, 120.288.5-60.os.2.1, 120.288.5-60.os.2.2, 120.288.5-60.os.2.3, 120.288.5-60.os.2.4, 120.288.5-60.os.2.5, 120.288.5-60.os.2.6, 120.288.5-60.os.2.7, 120.288.5-60.os.2.8 |
| Cyclic 60-isogeny field degree: |
$8$ |
| Cyclic 60-torsion field degree: |
$128$ |
| Full 60-torsion field degree: |
$15360$ |
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ y w - y t + z w - z t + w^{2} $ |
| $=$ | $y^{2} - 3 y z + z^{2} - t^{2}$ |
| $=$ | $3 x^{2} + y t + z t - t^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 81 x^{8} + 270 x^{6} y^{2} - 135 x^{6} y z - 54 x^{6} z^{2} - 225 x^{4} y^{4} + 450 x^{4} y^{3} z + \cdots + 25 y^{2} z^{6} $ |
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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 canonical model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 144 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{5046875000yz^{17}+16739062500yz^{16}t+27716406250yz^{15}t^{2}+35298437500yz^{14}t^{3}+35469140625yz^{13}t^{4}+27221187500yz^{12}t^{5}+17678875000yz^{11}t^{6}+9342675000yz^{10}t^{7}+3747359375yz^{9}t^{8}+1361087500yz^{8}t^{9}+348802500yz^{7}t^{10}+60749000yz^{6}t^{11}+17774250yz^{5}t^{12}+1287000yz^{4}t^{13}-384000yz^{3}t^{14}+219840yz^{2}t^{15}+7245yzt^{16}-12132yt^{17}-1927734375z^{18}-6393750000z^{17}t-8329687500z^{16}t^{2}-5996875000z^{15}t^{3}-1604296875z^{14}t^{4}+3891187500z^{13}t^{5}+6811171875z^{12}t^{6}+6046800000z^{11}t^{7}+4203637500z^{10}t^{8}+2186962500z^{9}t^{9}+785214375z^{8}t^{10}+260034000z^{7}t^{11}+59250750z^{6}t^{12}+5205000z^{5}t^{13}+2631750z^{4}t^{14}+411040z^{3}t^{15}-187500z^{2}t^{16}+12828zt^{17}+12257t^{18}}{t^{10}(13125yz^{7}+7250yz^{6}t+750yz^{5}t^{2}+4500yz^{4}t^{3}-1500yz^{3}t^{4}+530yz^{2}t^{5}-85yzt^{6}+6yt^{7}-5000z^{8}-2750z^{7}t+5500z^{6}t^{2}+1500z^{5}t^{3}-125z^{4}t^{4}+1180z^{3}t^{5}-375z^{2}t^{6}+76zt^{7}-6t^{8})}$ |
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.
