$\GL_2(\Z/40\Z)$-generators: |
$\begin{bmatrix}21&12\\10&37\end{bmatrix}$, $\begin{bmatrix}35&16\\2&15\end{bmatrix}$, $\begin{bmatrix}35&36\\18&35\end{bmatrix}$, $\begin{bmatrix}39&4\\10&1\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
40.192.1-40.ba.2.1, 40.192.1-40.ba.2.2, 40.192.1-40.ba.2.3, 40.192.1-40.ba.2.4, 40.192.1-40.ba.2.5, 40.192.1-40.ba.2.6, 40.192.1-40.ba.2.7, 40.192.1-40.ba.2.8, 120.192.1-40.ba.2.1, 120.192.1-40.ba.2.2, 120.192.1-40.ba.2.3, 120.192.1-40.ba.2.4, 120.192.1-40.ba.2.5, 120.192.1-40.ba.2.6, 120.192.1-40.ba.2.7, 120.192.1-40.ba.2.8, 280.192.1-40.ba.2.1, 280.192.1-40.ba.2.2, 280.192.1-40.ba.2.3, 280.192.1-40.ba.2.4, 280.192.1-40.ba.2.5, 280.192.1-40.ba.2.6, 280.192.1-40.ba.2.7, 280.192.1-40.ba.2.8 |
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 $ | $=$ | $ 2 x^{2} + 2 x z - x w - 2 z^{2} + 2 z w - w^{2} $ |
| $=$ | $5 x^{2} + 10 y^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 2 x^{4} - 20 x^{3} z + 12 x^{2} y^{2} + 130 x^{2} z^{2} - 60 x y^{2} z - 400 x z^{3} + 8 y^{4} + \cdots + 675 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{5}w$ |
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}{5^2}\cdot\frac{94961664000000000000xz^{23}-1092059136000000000000xz^{22}w+6154309632000000000000xz^{21}w^{2}-22575974400000000000000xz^{20}w^{3}+60450743193600000000000xz^{19}w^{4}-125656623667200000000000xz^{18}w^{5}+210605847372800000000000xz^{17}w^{6}-291882609651200000000000xz^{16}w^{7}+340402461021465600000000xz^{15}w^{8}-338200977315392000000000xz^{14}w^{9}+288722876164256000000000xz^{13}w^{10}-212994061907952000000000xz^{12}w^{11}+136211886798093504000000xz^{11}w^{12}-75582138556608672000000xz^{10}w^{13}+36340536118161360000000xz^{9}w^{14}-15082494144641080000000xz^{8}w^{15}+5367144671549897680000xz^{7}w^{16}-1620560871115505880000xz^{6}w^{17}+408761977767803580000xz^{5}w^{18}-84147707197904250000xz^{4}w^{19}+13637973479828800200xz^{3}w^{20}-1639429754279890300xz^{2}w^{21}+130598382998114350xzw^{22}-5199247775077125xw^{23}-58689536000000000000z^{24}+704274432000000000000z^{23}w-4151654400000000000000z^{22}w^{2}+15971293184000000000000z^{21}w^{3}-44966121891840000000000z^{20}w^{4}+98544293990400000000000z^{19}w^{5}-174619597516800000000000z^{18}w^{6}+256613391820800000000000z^{17}w^{7}-318321514163008000000000z^{16}w^{8}+337534350252544000000000z^{15}w^{9}-308686917679584000000000z^{14}w^{10}+244981982567968000000000z^{13}w^{11}-169367607930407424000000z^{12}w^{12}+102183461764172544000000z^{11}w^{13}-53790259245003792000000z^{10}w^{14}+24650743878114640000000z^{9}w^{15}-9790233412430825040000z^{8}w^{16}+3345173034997444160000z^{7}w^{17}-972660973418405940000z^{6}w^{18}+236853512494867260000z^{5}w^{19}-47175257792281246800z^{4}w^{20}+7411714640513913600z^{3}w^{21}-865179272263947750z^{2}w^{22}+67033424982840950zw^{23}-2599759999429883w^{24}}{w^{4}(1385472000000000xz^{19}-13161984000000000xz^{18}w+61005056000000000xz^{17}w^{2}-182912384000000000xz^{16}w^{3}+396977040384000000xz^{15}w^{4}-661602490880000000xz^{14}w^{5}+876984195840000000xz^{13}w^{6}-945180529280000000xz^{12}w^{7}+839785881227520000xz^{11}w^{8}-620081616767360000xz^{10}w^{9}+381835171156800000xz^{9}w^{10}-195982555650400000xz^{8}w^{11}+83460615987315200xz^{7}w^{12}-29220032419923200xz^{6}w^{13}+8284835821411200xz^{5}w^{14}-1858411904520000xz^{4}w^{15}+317934468626264xz^{3}w^{16}-39032661820996xz^{2}w^{17}+3066606581142xzw^{18}-115964114805xw^{19}-856268800000000z^{20}+8562688000000000z^{19}w-41925376000000000z^{18}w^{2}+133291776000000000z^{17}w^{3}-307984120320000000z^{16}w^{4}+548891996160000000z^{15}w^{5}-781900117760000000z^{14}w^{6}+910675339520000000z^{13}w^{7}-879986773285120000z^{12}w^{8}+711919870430720000z^{11}w^{9}-484536908512960000z^{10}w^{10}+277776078443200000z^{9}w^{11}-133837962185065600z^{8}w^{12}+53882204978982400z^{7}w^{13}-17942739967401600z^{6}w^{14}+4864677807286400z^{5}w^{15}-1048421177367376z^{4}w^{16}+173066193963552z^{3}w^{17}-20581484339230z^{2}w^{18}+1571958101454zw^{19}-57982060683w^{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.