$\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}5&16\\6&3\end{bmatrix}$, $\begin{bmatrix}5&32\\24&1\end{bmatrix}$, $\begin{bmatrix}7&24\\16&15\end{bmatrix}$, $\begin{bmatrix}9&20\\4&21\end{bmatrix}$, $\begin{bmatrix}15&32\\12&39\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
40.192.1-40.a.1.1, 40.192.1-40.a.1.2, 40.192.1-40.a.1.3, 40.192.1-40.a.1.4, 40.192.1-40.a.1.5, 40.192.1-40.a.1.6, 40.192.1-40.a.1.7, 40.192.1-40.a.1.8, 40.192.1-40.a.1.9, 40.192.1-40.a.1.10, 40.192.1-40.a.1.11, 40.192.1-40.a.1.12, 120.192.1-40.a.1.1, 120.192.1-40.a.1.2, 120.192.1-40.a.1.3, 120.192.1-40.a.1.4, 120.192.1-40.a.1.5, 120.192.1-40.a.1.6, 120.192.1-40.a.1.7, 120.192.1-40.a.1.8, 120.192.1-40.a.1.9, 120.192.1-40.a.1.10, 120.192.1-40.a.1.11, 120.192.1-40.a.1.12, 280.192.1-40.a.1.1, 280.192.1-40.a.1.2, 280.192.1-40.a.1.3, 280.192.1-40.a.1.4, 280.192.1-40.a.1.5, 280.192.1-40.a.1.6, 280.192.1-40.a.1.7, 280.192.1-40.a.1.8, 280.192.1-40.a.1.9, 280.192.1-40.a.1.10, 280.192.1-40.a.1.11, 280.192.1-40.a.1.12 |
Cyclic 40-isogeny field degree: |
$12$ |
Cyclic 40-torsion field degree: |
$192$ |
Full 40-torsion field degree: |
$7680$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + 2 y^{2} + z^{2} $ |
| $=$ | $3 x^{2} + x z - 2 x w - 2 z^{2} - 2 z w + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 13 x^{4} + 8 x^{3} z + 31 x^{2} y^{2} - 6 x^{2} z^{2} + 8 x y^{2} z - 4 x z^{3} + 18 y^{4} - 8 y^{2} z^{2} + 2 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
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}{3^4}\cdot\frac{228255878482099xz^{23}-119199109414702xz^{22}w-1063354766461328xz^{21}w^{2}+539001154514128xz^{20}w^{3}+1228815074753720xz^{19}w^{4}-1694320898086256xz^{18}w^{5}-354248696980032xz^{17}w^{6}+1852987320115584xz^{16}w^{7}-482585374318848xz^{15}w^{8}-794729866831360xz^{14}w^{9}+451435281919744xz^{13}w^{10}+81415674278912xz^{12}w^{11}-150730458701056xz^{11}w^{12}+42762903591424xz^{10}w^{13}+19876847836160xz^{9}w^{14}-15192956608512xz^{8}w^{15}-809776844544xz^{7}w^{16}+3272432936448xz^{6}w^{17}-927240507392xz^{5}w^{18}-48055029760xz^{4}w^{19}+97096050688xz^{3}w^{20}-34674987008xz^{2}w^{21}+6934839296xzw^{22}-603029504xw^{23}-157354581993293z^{24}-119199109414702z^{23}w+696273092427714z^{22}w^{2}-117170345275608z^{21}w^{3}-1918203216260812z^{20}w^{4}+725336315570352z^{19}w^{5}+2203943518803728z^{18}w^{6}-1317906107817024z^{17}w^{7}-948873757134960z^{16}w^{8}+1100156286597632z^{15}w^{9}-109025564384768z^{14}w^{10}-420486513427968z^{13}w^{11}+230653947197952z^{12}w^{12}+64867038141952z^{11}w^{13}-92512416907776z^{10}w^{14}+16263651198976z^{9}w^{15}+10255988668416z^{8}w^{16}-4866194631168z^{7}w^{17}+123222398464z^{6}w^{18}+525975861248z^{5}w^{19}-132688892928z^{4}w^{20}-55104073728z^{3}w^{21}+40458235904z^{2}w^{22}-9247506432zw^{23}+770625536w^{24}}{z^{4}(3570125xz^{19}-1208350xz^{18}w-33081750xz^{17}w^{2}+17421924xz^{16}w^{3}+125255775xz^{15}w^{4}-81516510xz^{14}w^{5}-255244110xz^{13}w^{6}+181791768xz^{12}w^{7}+39865365xz^{11}w^{8}+332030350xz^{10}w^{9}-238125800xz^{9}w^{10}-731967408xz^{8}w^{11}+928078320xz^{7}w^{12}-47274720xz^{6}w^{13}-485956800xz^{5}w^{14}+282026496xz^{4}w^{15}-11942160xz^{3}w^{16}-39489120xz^{2}w^{17}+13631360xzw^{18}-1434880xw^{19}+3570125z^{20}-1208350z^{19}w-34509800z^{18}w^{2}+18476484z^{17}w^{3}+137723919z^{16}w^{4}-93082350z^{15}w^{5}-297966060z^{14}w^{6}+227736288z^{13}w^{7}+591485181z^{12}w^{8}-519415610z^{11}w^{9}-892244690z^{10}w^{10}+1202249192z^{9}w^{11}+217176468z^{8}w^{12}-1038168480z^{7}w^{13}+492931680z^{6}w^{14}+115664256z^{5}w^{15}-185869440z^{4}w^{16}+71175840z^{3}w^{17}-13840480z^{2}w^{18}+1873280zw^{19}-187328w^{20})}$ |
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.