| $\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}21&6\\10&27\end{bmatrix}$, $\begin{bmatrix}29&13\\5&22\end{bmatrix}$, $\begin{bmatrix}32&17\\15&14\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
80.240.5-40.df.1.1, 80.240.5-40.df.1.2, 80.240.5-40.df.1.3, 80.240.5-40.df.1.4, 80.240.5-40.df.1.5, 80.240.5-40.df.1.6, 80.240.5-40.df.1.7, 80.240.5-40.df.1.8, 240.240.5-40.df.1.1, 240.240.5-40.df.1.2, 240.240.5-40.df.1.3, 240.240.5-40.df.1.4, 240.240.5-40.df.1.5, 240.240.5-40.df.1.6, 240.240.5-40.df.1.7, 240.240.5-40.df.1.8 |
| Cyclic 40-isogeny field degree: |
$12$ |
| Cyclic 40-torsion field degree: |
$192$ |
| Full 40-torsion field degree: |
$6144$ |
Embedded model Embedded model in $\mathbb{P}^{7}$
| $ 0 $ | $=$ | $ x u + w v $ |
| $=$ | $y u - z v$ |
| $=$ | $x z + y w$ |
| $=$ | $x z - x r + y t + z v$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 16 x^{8} y^{4} + 1760 x^{8} y^{2} z^{2} + 48400 x^{8} z^{4} - 6408 x^{6} y^{4} z^{2} + 5520 x^{6} y^{2} z^{4} + \cdots + 75625 z^{12} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 355x^{12} + 1540x^{11} - 3630x^{10} + 1100x^{9} - 9075x^{8} - 14520x^{7} - 5060x^{6} + 14520x^{5} - 9075x^{4} - 1100x^{3} - 3630x^{2} - 1540x + 355 $ |
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This modular curve has no $\Q_p$ points for $p=19$, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle r$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 5v$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle u$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ |
$=$ |
$\displaystyle \frac{1}{32}wu^{2}+\frac{1}{128}wur-\frac{1}{128}u^{3}+\frac{1}{128}u^{2}r$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{35}{2097152}w^{12}u^{5}v+\frac{5}{262144}w^{11}u^{6}v+\frac{25}{8388608}w^{11}u^{5}vr-\frac{235}{33554432}w^{10}u^{7}v+\frac{235}{33554432}w^{10}u^{6}vr-\frac{375}{67108864}w^{9}u^{8}v+\frac{235}{67108864}w^{9}u^{7}vr+\frac{1185}{536870912}w^{8}u^{9}v-\frac{535}{268435456}w^{8}u^{8}vr-\frac{675}{536870912}w^{7}u^{10}v-\frac{3035}{2147483648}w^{7}u^{9}vr+\frac{6435}{8589934592}w^{6}u^{11}v+\frac{1575}{8589934592}w^{6}u^{10}vr+\frac{1905}{17179869184}w^{5}u^{12}v+\frac{1835}{8589934592}w^{5}u^{11}vr-\frac{7575}{34359738368}w^{4}u^{13}v+\frac{995}{34359738368}w^{4}u^{12}vr+\frac{4955}{68719476736}w^{3}u^{14}v-\frac{1655}{34359738368}w^{3}u^{13}vr-\frac{85}{8589934592}w^{2}u^{15}v+\frac{365}{34359738368}w^{2}u^{14}vr+\frac{35}{68719476736}wu^{16}v-\frac{25}{34359738368}wu^{15}vr$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{16}w^{2}u-\frac{1}{64}wu^{2}+\frac{1}{128}wur+\frac{1}{128}u^{2}r$ |
Maps to other modular curves
$j$-invariant map
of degree 120 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{2^{12}}{11^8}\cdot\frac{42858057231616000000xv^{9}-23379421633410700000xv^{7}r^{2}+3633482405422100000xv^{5}r^{4}-266025748693606000xv^{3}r^{6}-88992174688700xvr^{8}+2699900380567280000zv^{8}r-1162290149358426000zv^{6}r^{3}+273717898235874400zv^{4}r^{5}+18815157988363360zv^{2}r^{7}+49299502402698zr^{9}-1523917245442350000wuv^{8}+962989963284085000wuv^{6}r^{2}-213007852789479000wuv^{4}r^{4}-31543545953150550wuv^{2}r^{6}-344680196894135wur^{8}+3458291787586350000tuv^{8}-1431712117216645000tuv^{6}r^{2}+274820884602143000tuv^{4}r^{4}-2707895898527050tuv^{2}r^{6}-1615747323105865tur^{8}-1747282110807200000u^{2}v^{8}+934336998336280000u^{2}v^{6}r^{2}-46898582702422000u^{2}v^{4}r^{4}+9391482360469000u^{2}v^{2}r^{6}-1064275678225220u^{2}r^{8}+13718968384000000v^{10}-4059612652060020000v^{8}r^{2}+1969632903117484000v^{6}r^{4}-160801208739683600v^{4}r^{6}+5947012186129660v^{2}r^{8}-28289789160352r^{10}}{2552400000xv^{7}r^{2}-949760000xv^{5}r^{4}+88564000xv^{3}r^{6}+1064800xvr^{8}-2011100000zv^{8}r+1488696000zv^{6}r^{3}-61023600zv^{4}r^{5}+2628560zv^{2}r^{7}+263538zr^{9}-2545350000wuv^{8}+2139000000wuv^{6}r^{2}-233859000wuv^{4}r^{4}+10815200wuv^{2}r^{6}-512435wur^{8}+87750000tuv^{8}-1504280000tuv^{6}r^{2}+198883000tuv^{4}r^{4}-7136800tuv^{2}r^{6}+512435tur^{8}+358400000u^{2}v^{8}+314180000u^{2}v^{6}r^{2}-58342000u^{2}v^{4}r^{4}+1067000u^{2}v^{2}r^{6}-292820u^{2}r^{8}-2552400000v^{8}r^{2}+1122336000v^{6}r^{4}-136061600v^{4}r^{6}+3275360v^{2}r^{8}-468512r^{10}}$ |
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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.
