| $\GL_2(\Z/60\Z)$-generators: |
$\begin{bmatrix}7&40\\27&41\end{bmatrix}$, $\begin{bmatrix}11&42\\39&11\end{bmatrix}$, $\begin{bmatrix}35&18\\12&17\end{bmatrix}$, $\begin{bmatrix}41&40\\42&29\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
60.96.1-60.u.1.1, 60.96.1-60.u.1.2, 60.96.1-60.u.1.3, 60.96.1-60.u.1.4, 60.96.1-60.u.1.5, 60.96.1-60.u.1.6, 60.96.1-60.u.1.7, 60.96.1-60.u.1.8, 120.96.1-60.u.1.1, 120.96.1-60.u.1.2, 120.96.1-60.u.1.3, 120.96.1-60.u.1.4, 120.96.1-60.u.1.5, 120.96.1-60.u.1.6, 120.96.1-60.u.1.7, 120.96.1-60.u.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 $ | $=$ | $ 5 x y - 3 y^{2} + 2 y z - 2 z^{2} $ |
| $=$ | $27 x^{2} - 12 x y - 3 y^{2} + 12 y z - 12 z^{2} - w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{4} - 24 x^{3} z + 3 x^{2} y^{2} - x^{2} z^{2} + 6 x z^{3} - 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:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{5}{2}w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 3z$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{2^6}{3}\cdot\frac{1271133388800xz^{11}-58058726400xz^{9}w^{2}+4481775360xz^{7}w^{4}-75193920xz^{5}w^{6}+628560xz^{3}w^{8}-123309195264y^{2}z^{10}+21892239360y^{2}z^{8}w^{2}-2688650496y^{2}z^{6}w^{4}+185526720y^{2}z^{4}w^{6}-7001712y^{2}z^{2}w^{8}+111972y^{2}w^{10}+172840697856yz^{11}-80396375040yz^{9}w^{2}+6987803904yz^{7}w^{4}-369835200yz^{5}w^{6}+8278128yz^{3}w^{8}-55968yzw^{10}-845358944256z^{12}+26075105280z^{10}w^{2}-3345110784z^{8}w^{4}+38119680z^{6}w^{6}-1083888z^{4}w^{8}-2232z^{2}w^{10}-5w^{12}}{w^{4}(14929920xz^{7}-2941920xz^{5}w^{2}+125280xz^{3}w^{4}+5211648y^{2}z^{6}-1212480y^{2}z^{4}w^{2}+73164y^{2}z^{2}w^{4}-729y^{2}w^{6}-9718272yz^{7}+485280yz^{5}w^{2}+196824yz^{3}w^{4}-11664yzw^{6}-5211648z^{8}+1054080z^{6}w^{2}-47124z^{4}w^{4}+64z^{2}w^{6})}$ |
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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.
