| $\GL_2(\Z/60\Z)$-generators: |
$\begin{bmatrix}17&50\\6&11\end{bmatrix}$, $\begin{bmatrix}37&32\\27&23\end{bmatrix}$, $\begin{bmatrix}53&18\\0&47\end{bmatrix}$, $\begin{bmatrix}59&32\\12&47\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
60.192.3-60.bp.1.1, 60.192.3-60.bp.1.2, 60.192.3-60.bp.1.3, 60.192.3-60.bp.1.4, 60.192.3-60.bp.1.5, 60.192.3-60.bp.1.6, 60.192.3-60.bp.1.7, 60.192.3-60.bp.1.8, 120.192.3-60.bp.1.1, 120.192.3-60.bp.1.2, 120.192.3-60.bp.1.3, 120.192.3-60.bp.1.4, 120.192.3-60.bp.1.5, 120.192.3-60.bp.1.6, 120.192.3-60.bp.1.7, 120.192.3-60.bp.1.8, 120.192.3-60.bp.1.9, 120.192.3-60.bp.1.10, 120.192.3-60.bp.1.11, 120.192.3-60.bp.1.12, 120.192.3-60.bp.1.13, 120.192.3-60.bp.1.14, 120.192.3-60.bp.1.15, 120.192.3-60.bp.1.16, 120.192.3-60.bp.1.17, 120.192.3-60.bp.1.18, 120.192.3-60.bp.1.19, 120.192.3-60.bp.1.20, 120.192.3-60.bp.1.21, 120.192.3-60.bp.1.22, 120.192.3-60.bp.1.23, 120.192.3-60.bp.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 t - x u + y t - y u + z u - w u $ |
| $=$ | $x t + x u - y u - z t + z u + w t - w u$ |
| $=$ | $x z + x w - z^{2} - w^{2}$ |
| $=$ | $3 x^{2} + x z + x w - y^{2} - 2 z w$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} z^{2} - 20 x^{3} y^{2} z + 4 x^{3} z^{3} + 25 x^{2} y^{4} + 50 x^{2} y^{2} z^{2} - 15 x^{2} z^{4} + \cdots - 2 z^{6} $ |
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Geometric Weierstrass model Geometric Weierstrass model
| $ 81 w^{2} $ | $=$ | $ -190 x^{3} z + 220 x^{2} y z - 105 x^{2} z^{2} + 420 x y z^{2} + 158 x z^{3} + 156 y z^{3} + 84 z^{4} $ |
| $0$ | $=$ | $2 x^{2} - 2 x y + 3 y^{2} - z^{2}$ |
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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:
| $\displaystyle X$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle u$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{3^2}\cdot\frac{50220000xy^{7}u^{4}-1687392000xy^{5}u^{6}-1239883200xy^{3}u^{8}+1715948893440xyu^{10}+15625y^{12}-29295000y^{8}u^{4}+974268000y^{6}u^{6}+719328600y^{4}u^{8}-990754856640y^{2}u^{10}+39803400000yw^{9}u^{2}-447110280000yw^{7}u^{4}+5012534016000yw^{5}u^{6}-53595577281600yw^{3}u^{8}-11046373237440ywu^{10}+238878720zwt^{9}u-2030469120zwt^{8}u^{2}+11466178560zwt^{7}u^{3}-97462517760zwt^{6}u^{4}+665754992640zwt^{5}u^{5}-3725657026560zwt^{4}u^{6}+18410804720640zwt^{3}u^{7}-77282454067200zwt^{2}u^{8}+197534464542720zwtu^{9}-59441733411840zwu^{10}+8292375000w^{12}-59705100000w^{10}u^{2}+772492140000w^{8}u^{4}-8609212980000w^{6}u^{6}+92027114609400w^{4}u^{8}-22306622477760w^{2}u^{10}-4096t^{12}+98304t^{11}u-192405504t^{10}u^{2}+681377792t^{9}u^{3}+4893603840t^{8}u^{4}-37483425792t^{7}u^{5}+157546836480t^{6}u^{6}-613441405440t^{5}u^{7}+2134669985280t^{4}u^{8}-3649683933184t^{3}u^{9}-30244749095424t^{2}u^{10}+31798375190016tu^{11}-7918620943600u^{12}}{7500xy^{7}u^{4}-252000xy^{5}u^{6}+9817200xy^{3}u^{8}-412464960xyu^{10}-4375y^{8}u^{4}+145500y^{6}u^{6}-5667750y^{4}u^{8}+238129200y^{2}u^{10}-202500yw^{7}u^{4}+4252500yw^{5}u^{6}-66606300yw^{3}u^{8}-13622580ywu^{10}+1920zwt^{9}u-16320zwt^{8}u^{2}+92160zwt^{7}u^{3}-433920zwt^{6}u^{4}+1856640zwt^{5}u^{5}-7439040zwt^{4}u^{6}+28286400zwt^{3}u^{7}-100381440zwt^{2}u^{8}+245499840zwtu^{9}-73532160zwu^{10}+354375w^{8}u^{4}-7330500w^{6}u^{6}+114683850w^{4}u^{8}-27405540w^{2}u^{10}-48t^{12}+576t^{11}u-5568t^{10}u^{2}+41088t^{9}u^{3}-246176t^{8}u^{4}+1264640t^{7}u^{5}-5657600t^{6}u^{6}+22650752t^{5}u^{7}-76615712t^{4}u^{8}+233958656t^{3}u^{9}-352435328t^{2}u^{10}+297446912tu^{11}-104568281u^{12}}$ |
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.
