$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}17&25\\8&23\end{bmatrix}$, $\begin{bmatrix}19&14\\44&17\end{bmatrix}$, $\begin{bmatrix}25&33\\0&35\end{bmatrix}$, $\begin{bmatrix}33&28\\20&7\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.do.2.1, 48.192.1-48.do.2.2, 48.192.1-48.do.2.3, 48.192.1-48.do.2.4, 48.192.1-48.do.2.5, 48.192.1-48.do.2.6, 48.192.1-48.do.2.7, 48.192.1-48.do.2.8, 96.192.1-48.do.2.1, 96.192.1-48.do.2.2, 96.192.1-48.do.2.3, 96.192.1-48.do.2.4, 240.192.1-48.do.2.1, 240.192.1-48.do.2.2, 240.192.1-48.do.2.3, 240.192.1-48.do.2.4, 240.192.1-48.do.2.5, 240.192.1-48.do.2.6, 240.192.1-48.do.2.7, 240.192.1-48.do.2.8 |
Cyclic 48-isogeny field degree: |
$4$ |
Cyclic 48-torsion field degree: |
$64$ |
Full 48-torsion field degree: |
$12288$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ - x w - y w + z^{2} - w^{2} $ |
| $=$ | $x^{2} - x y + x w + y^{2} + y w + z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 3 x^{2} y z + 3 y^{2} z^{2} + 3 y z^{3} + 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
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{3}\cdot\frac{8503056xy^{22}w+178564176xy^{21}w^{2}+1436071680xy^{20}w^{3}+4777772688xy^{19}w^{4}-1907833824xy^{18}w^{5}-68013110592xy^{17}w^{6}-222723380160xy^{16}w^{7}-282997135152xy^{15}w^{8}+63708744672xy^{14}w^{9}+646420652928xy^{13}w^{10}+784829502720xy^{12}w^{11}+323904147840xy^{11}w^{12}-135813563136xy^{10}w^{13}-219841205760xy^{9}w^{14}-106957283328xy^{8}w^{15}-21744018432xy^{7}w^{16}+1868949504xy^{6}w^{17}+2326413312xy^{5}w^{18}+652492800xy^{4}w^{19}+97984512xy^{3}w^{20}+8110080xy^{2}w^{21}+294912xyw^{22}+531441y^{24}+8503056y^{23}w+8503056y^{22}w^{2}-718035840y^{21}w^{3}-6988567248y^{20}w^{4}-31199916960y^{19}w^{5}-70734088512y^{18}w^{6}-38841959808y^{17}w^{7}+231905106000y^{16}w^{8}+729612243360y^{15}w^{9}+1016872628832y^{14}w^{10}+624335195520y^{13}w^{11}-215940624192y^{12}w^{12}-734878842624y^{11}w^{13}-614909470464y^{10}w^{14}-242082266112y^{9}w^{15}-15733688832y^{8}w^{16}+31372793856y^{7}w^{17}+17043277824y^{6}w^{18}+4556169216y^{5}w^{19}+688656384y^{4}w^{20}+47038464y^{3}w^{21}-1916928y^{2}w^{22}-589824yw^{23}-32768w^{24}}{w^{8}(2187xy^{15}+7290xy^{14}w-177876xy^{13}w^{2}-1065312xy^{12}w^{3}-985608xy^{11}w^{4}+3418200xy^{10}w^{5}+5387796xy^{9}w^{6}+1372680xy^{8}w^{7}-1838619xy^{7}w^{8}-1466298xy^{6}w^{9}-355104xy^{5}w^{10}+18960xy^{4}w^{11}+25080xy^{3}w^{12}+4432xy^{2}w^{13}+256xyw^{14}-2187y^{16}-36450y^{15}w-138510y^{14}w^{2}+200232y^{13}w^{3}+2171934y^{12}w^{4}+4570992y^{11}w^{5}+3373164y^{10}w^{6}-2169936y^{9}w^{7}-5541912y^{8}w^{8}-3671334y^{7}w^{9}-765090y^{6}w^{10}+264696y^{5}w^{11}+187260y^{4}w^{12}+41296y^{3}w^{13}+3344y^{2}w^{14}-64yw^{15}-16w^{16})}$ |
Hi
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Cover information
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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.