$\GL_2(\Z/60\Z)$-generators: |
$\begin{bmatrix}13&25\\48&1\end{bmatrix}$, $\begin{bmatrix}23&25\\0&59\end{bmatrix}$, $\begin{bmatrix}41&35\\24&23\end{bmatrix}$, $\begin{bmatrix}53&25\\51&16\end{bmatrix}$, $\begin{bmatrix}53&40\\15&47\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
60.96.1-60.bw.2.1, 60.96.1-60.bw.2.2, 60.96.1-60.bw.2.3, 60.96.1-60.bw.2.4, 60.96.1-60.bw.2.5, 60.96.1-60.bw.2.6, 60.96.1-60.bw.2.7, 60.96.1-60.bw.2.8, 60.96.1-60.bw.2.9, 60.96.1-60.bw.2.10, 60.96.1-60.bw.2.11, 60.96.1-60.bw.2.12, 60.96.1-60.bw.2.13, 60.96.1-60.bw.2.14, 60.96.1-60.bw.2.15, 60.96.1-60.bw.2.16, 120.96.1-60.bw.2.1, 120.96.1-60.bw.2.2, 120.96.1-60.bw.2.3, 120.96.1-60.bw.2.4, 120.96.1-60.bw.2.5, 120.96.1-60.bw.2.6, 120.96.1-60.bw.2.7, 120.96.1-60.bw.2.8, 120.96.1-60.bw.2.9, 120.96.1-60.bw.2.10, 120.96.1-60.bw.2.11, 120.96.1-60.bw.2.12, 120.96.1-60.bw.2.13, 120.96.1-60.bw.2.14, 120.96.1-60.bw.2.15, 120.96.1-60.bw.2.16 |
Cyclic 60-isogeny field degree: |
$6$ |
Cyclic 60-torsion field degree: |
$96$ |
Full 60-torsion field degree: |
$46080$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 x^{2} - 2 x z - y^{2} - z^{2} $ |
| $=$ | $17 x^{2} + 22 x z + 5 y^{2} + 6 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 225 x^{4} + 65 x^{2} y^{2} + 410 x^{2} z^{2} + 4 y^{4} + 12 y^{2} z^{2} + 9 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 4y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{5}{2^3}\cdot\frac{2862251226000000xyz^{9}w-25907146704000000xyz^{7}w^{3}+50504229495360000xyz^{5}w^{5}-9666964164096000xyz^{3}w^{7}-552282791270400xyzw^{9}+637762306562500xz^{11}-10147300124750000xz^{9}w^{2}+41920976925000000xz^{7}w^{4}-38089377182800000xz^{5}w^{6}-278465368456000xz^{3}w^{8}-65822725228800xzw^{10}+715563267300000yz^{10}w-7192320912900000yz^{8}w^{3}+17671391840232000yz^{6}w^{5}-7842991057545600yz^{4}w^{7}-296306055475200yz^{2}w^{9}-9471147969024yw^{11}+159440544640625z^{12}-2696264955028125z^{10}w^{2}+12698300313956250z^{8}w^{4}-16201138796356000z^{6}w^{6}+2408948618538600z^{4}w^{8}+191799673223760z^{2}w^{10}+1138274232736w^{12}}{210600000xyz^{9}w-10663520000xyz^{7}w^{3}+93406316800xyz^{5}w^{5}-89236561920xyz^{3}w^{7}-3643952128xyzw^{9}-9112500xz^{11}+1599750000xz^{9}w^{2}-33438000000xz^{7}w^{4}+123476208000xz^{5}w^{6}-26214590400xz^{3}w^{8}-471614208xzw^{10}+52650000yz^{10}w-2692930000yz^{8}w^{3}+25630559200yz^{6}w^{5}-39316559680yz^{4}w^{7}+1018498048yz^{2}w^{9}-8279808yw^{11}-2278125z^{12}+402215625z^{10}w^{2}-8658881250z^{8}w^{4}+38536502000z^{6}w^{6}-22367466600z^{4}w^{8}-1260580752z^{2}w^{10}-12713888w^{12}}$ |
Hi
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Cover information
Click on a modular curve in the diagram to see information about it.
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The following modular covers realize this modular curve as a fiber product over $X(1)$.
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.