$\GL_2(\Z/60\Z)$-generators: |
$\begin{bmatrix}7&28\\30&13\end{bmatrix}$, $\begin{bmatrix}11&42\\12&11\end{bmatrix}$, $\begin{bmatrix}37&8\\51&23\end{bmatrix}$, $\begin{bmatrix}41&0\\3&43\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
60.96.1-60.y.1.1, 60.96.1-60.y.1.2, 60.96.1-60.y.1.3, 60.96.1-60.y.1.4, 60.96.1-60.y.1.5, 60.96.1-60.y.1.6, 60.96.1-60.y.1.7, 60.96.1-60.y.1.8, 120.96.1-60.y.1.1, 120.96.1-60.y.1.2, 120.96.1-60.y.1.3, 120.96.1-60.y.1.4, 120.96.1-60.y.1.5, 120.96.1-60.y.1.6, 120.96.1-60.y.1.7, 120.96.1-60.y.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 $ | $=$ | $ x^{2} + 4 x y + 3 x z + y^{2} + 3 y z $ |
| $=$ | $5 x^{2} - 18 x y + 10 x z - x w + 5 y^{2} + 10 y z - y w + 34 z^{2} + z w + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 49 x^{4} - 4 x^{3} y + 20 x^{3} z + x^{2} y^{2} - 14 x^{2} y z + 78 x^{2} z^{2} + 2 x y^{2} z + \cdots + 49 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:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 3w$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle y$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -2^4\,\frac{491397646304577480xz^{11}-287837456254279350xz^{10}w-202103947940629518xz^{9}w^{2}-32920630716222603xz^{8}w^{3}-5262928022646900xz^{7}w^{4}+893091862705176xz^{6}w^{5}+577173663722016xz^{5}w^{6}+114281237439666xz^{4}w^{7}+24016342749084xz^{3}w^{8}+2446128598980xz^{2}w^{9}+241302421932xzw^{10}+14197319136xw^{11}+491397646304577480yz^{11}-287837456254279350yz^{10}w-202103947940629518yz^{9}w^{2}-32920630716222603yz^{8}w^{3}-5262928022646900yz^{7}w^{4}+893091862705176yz^{6}w^{5}+577173663722016yz^{5}w^{6}+114281237439666yz^{4}w^{7}+24016342749084yz^{3}w^{8}+2446128598980yz^{2}w^{9}+241302421932yzw^{10}+14197319136yw^{11}+814173155530714795z^{12}-595506126595195530z^{11}w-347127219760611567z^{10}w^{2}-13924975914959053z^{9}w^{3}+10333677758045067z^{8}w^{4}+7369482403343484z^{7}w^{5}+2449732032804534z^{6}w^{6}+435547562646618z^{5}w^{7}+76540133442528z^{4}w^{8}+8003590895504z^{3}w^{9}+644900637744z^{2}w^{10}+42906435924zw^{11}-1000128041w^{12}}{557083101900xz^{11}+2162165417442xz^{10}w+3644266053906xz^{9}w^{2}+3472527745827xz^{8}w^{3}+1976528983320xz^{7}w^{4}+568912523130xz^{6}w^{5}-58848813258xz^{5}w^{6}-137329504719xz^{4}w^{7}-66750733098xz^{3}w^{8}-18475014720xz^{2}w^{9}-3091692210xzw^{10}-266666400xw^{11}+557083101900yz^{11}+2162165417442yz^{10}w+3644266053906yz^{9}w^{2}+3472527745827yz^{8}w^{3}+1976528983320yz^{7}w^{4}+568912523130yz^{6}w^{5}-58848813258yz^{5}w^{6}-137329504719yz^{4}w^{7}-66750733098yz^{3}w^{8}-18475014720yz^{2}w^{9}-3091692210yzw^{10}-266666400yw^{11}+601394856612z^{12}+2436785738868z^{11}w+4414472540487z^{10}w^{2}+4775085822935z^{9}w^{3}+3469947309852z^{8}w^{4}+1803643956288z^{7}w^{5}+701539218593z^{6}w^{6}+214962210807z^{5}w^{7}+54591363195z^{4}w^{8}+11442687472z^{3}w^{9}+1803374706z^{2}w^{10}+194632350zw^{11}+18785275w^{12}}$ |
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.