$\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}15&8\\34&17\end{bmatrix}$, $\begin{bmatrix}25&2\\7&17\end{bmatrix}$, $\begin{bmatrix}31&26\\31&39\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
80.96.1-40.de.1.1, 80.96.1-40.de.1.2, 80.96.1-40.de.1.3, 80.96.1-40.de.1.4, 80.96.1-40.de.1.5, 80.96.1-40.de.1.6, 80.96.1-40.de.1.7, 80.96.1-40.de.1.8, 80.96.1-40.de.1.9, 80.96.1-40.de.1.10, 80.96.1-40.de.1.11, 80.96.1-40.de.1.12, 80.96.1-40.de.1.13, 80.96.1-40.de.1.14, 80.96.1-40.de.1.15, 80.96.1-40.de.1.16, 240.96.1-40.de.1.1, 240.96.1-40.de.1.2, 240.96.1-40.de.1.3, 240.96.1-40.de.1.4, 240.96.1-40.de.1.5, 240.96.1-40.de.1.6, 240.96.1-40.de.1.7, 240.96.1-40.de.1.8, 240.96.1-40.de.1.9, 240.96.1-40.de.1.10, 240.96.1-40.de.1.11, 240.96.1-40.de.1.12, 240.96.1-40.de.1.13, 240.96.1-40.de.1.14, 240.96.1-40.de.1.15, 240.96.1-40.de.1.16 |
Cyclic 40-isogeny field degree: |
$24$ |
Cyclic 40-torsion field degree: |
$384$ |
Full 40-torsion field degree: |
$15360$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 5 x^{2} - 5 x y + 3 y^{2} - 2 y z + 2 z^{2} $ |
| $=$ | $10 x^{2} + 10 x y + 5 y^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 4 x^{3} y + 26 x^{2} y^{2} - 4 x^{2} z^{2} - 44 x y^{3} + 8 x y z^{2} + 221 y^{4} + 36 y^{2} z^{2} + 4 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^7}{5^4}\cdot\frac{795429993595271400000000xz^{11}+24397788632132350792800000xz^{9}w^{2}-163364451756048127334880000xz^{7}w^{4}-16843319465188316468368000xz^{5}w^{6}+46558914963193434960340000xz^{3}w^{8}-197790122601746602018000xzw^{10}-6065165529805719964500000y^{2}z^{10}+37062461602346639889690000y^{2}z^{8}w^{2}-23654899201324703291226000y^{2}z^{6}w^{4}-188385164658935894280631000y^{2}z^{4}w^{6}+4515122525427580229398950y^{2}z^{2}w^{8}+389416500321001420967869y^{2}w^{10}+3464413426731346020000000yz^{11}-8125426434168560381760000yz^{9}w^{2}-87059267136741894682896000yz^{7}w^{4}+140414609305031396044944000yz^{5}w^{6}-8404087169982922109884400yz^{3}w^{8}-1293927541596419257476416yzw^{10}-1334129297275747755000000z^{12}+1918267698668289476760000z^{10}w^{2}+27012982378820007187566000z^{8}w^{4}-123306000682746692060172000z^{6}w^{6}-23663234608696200315149300z^{4}w^{8}+3600960813758247306061916z^{2}w^{10}+14166600316902224405253w^{12}}{94273184426106240000xz^{11}-932874893544318394880xz^{9}w^{2}+1366971351729385825280xz^{7}w^{4}-655518340355307287040xz^{5}w^{6}+80091808062753997120xz^{3}w^{8}+8320905924480674720xzw^{10}-718834433162159403200y^{2}z^{10}+819079070365792287376y^{2}z^{8}w^{2}+836935326416663910176y^{2}z^{6}w^{4}-840959164911482152952y^{2}z^{4}w^{6}+91075613537472189652y^{2}z^{2}w^{8}+15389812431944372045y^{2}w^{10}+410597146871863232000yz^{11}-1446652004492304064704yz^{9}w^{2}+560505341242787261568yz^{7}w^{4}+608811230191203501280yz^{5}w^{6}-290089121833215952832yz^{3}w^{8}+14470570506515561204yzw^{10}-158119027825273808000z^{12}+540266133442027472704z^{10}w^{2}-540960442652018379744z^{8}w^{4}+155934105283147228128z^{6}w^{6}+34911174538823126440z^{4}w^{8}-21849871645117612044z^{2}w^{10}+2102699832917314358w^{12}}$ |
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.