| $\GL_2(\Z/50\Z)$-generators: |
$\begin{bmatrix}9&14\\0&49\end{bmatrix}$, $\begin{bmatrix}21&2\\0&17\end{bmatrix}$, $\begin{bmatrix}39&11\\0&33\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
50.360.4-50.a.2.1, 50.360.4-50.a.2.2, 50.360.4-50.a.2.3, 50.360.4-50.a.2.4, 100.360.4-50.a.2.1, 100.360.4-50.a.2.2, 100.360.4-50.a.2.3, 100.360.4-50.a.2.4, 100.360.4-50.a.2.5, 100.360.4-50.a.2.6, 100.360.4-50.a.2.7, 100.360.4-50.a.2.8, 100.360.4-50.a.2.9, 100.360.4-50.a.2.10, 100.360.4-50.a.2.11, 100.360.4-50.a.2.12, 150.360.4-50.a.2.1, 150.360.4-50.a.2.2, 150.360.4-50.a.2.3, 150.360.4-50.a.2.4, 200.360.4-50.a.2.1, 200.360.4-50.a.2.2, 200.360.4-50.a.2.3, 200.360.4-50.a.2.4, 200.360.4-50.a.2.5, 200.360.4-50.a.2.6, 200.360.4-50.a.2.7, 200.360.4-50.a.2.8, 200.360.4-50.a.2.9, 200.360.4-50.a.2.10, 200.360.4-50.a.2.11, 200.360.4-50.a.2.12, 200.360.4-50.a.2.13, 200.360.4-50.a.2.14, 200.360.4-50.a.2.15, 200.360.4-50.a.2.16, 300.360.4-50.a.2.1, 300.360.4-50.a.2.2, 300.360.4-50.a.2.3, 300.360.4-50.a.2.4, 300.360.4-50.a.2.5, 300.360.4-50.a.2.6, 300.360.4-50.a.2.7, 300.360.4-50.a.2.8, 300.360.4-50.a.2.9, 300.360.4-50.a.2.10, 300.360.4-50.a.2.11, 300.360.4-50.a.2.12 |
| Cyclic 50-isogeny field degree: |
$1$ |
| Cyclic 50-torsion field degree: |
$10$ |
| Full 50-torsion field degree: |
$10000$ |
Canonical model in $\mathbb{P}^{ 3 }$
| $ 0 $ | $=$ | $ x z - y w $ |
| $=$ | $ - x^{2} w + x y^{2} - y z^{2} - z w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{4} y - x^{3} z^{2} + x y^{3} z - y^{2} z^{3} $ |
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This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 180 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{x^{30}+18x^{27}yw^{2}+129x^{25}w^{5}+714x^{22}yw^{7}+1902x^{20}w^{10}+5622x^{17}yw^{12}-3642x^{15}w^{15}-88302x^{12}yw^{17}-456399x^{10}w^{20}-2103612x^{7}yw^{22}-5366796x^{5}w^{25}-14448411x^{2}yw^{27}+64y^{30}-1152y^{27}z^{2}w+5952y^{26}zw^{3}-14976y^{25}w^{5}+2688y^{22}z^{2}w^{6}+61056y^{21}zw^{8}+8319y^{20}w^{10}-14700y^{17}z^{2}w^{11}+42354y^{16}zw^{13}+62772y^{15}w^{15}+133191y^{12}z^{2}w^{16}+302880y^{11}zw^{18}+169245y^{10}w^{20}-177123y^{7}z^{2}w^{21}-1446966y^{6}zw^{23}-5006646y^{5}w^{25}-9066123y^{2}z^{2}w^{26}-9080463yzw^{28}+z^{30}-18z^{25}w^{5}+57z^{20}w^{10}+144z^{15}w^{15}+420z^{10}w^{20}+2472z^{5}w^{25}+64w^{30}}{w^{12}(x^{17}y+17x^{15}w^{3}+158x^{12}yw^{5}+805x^{10}w^{8}+4398x^{7}yw^{10}+13905x^{5}w^{13}+38116x^{2}yw^{15}-64y^{12}z^{2}w^{4}+512y^{10}w^{8}+1408y^{7}z^{2}w^{9}+6015y^{6}zw^{11}+14917y^{5}w^{13}+24204y^{2}z^{2}w^{14}+24211yzw^{16}-z^{5}w^{13})}$ |
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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.
