| $\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}7&29\\40&37\end{bmatrix}$, $\begin{bmatrix}9&13\\4&25\end{bmatrix}$, $\begin{bmatrix}11&5\\24&37\end{bmatrix}$, $\begin{bmatrix}39&10\\8&11\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
48.192.1-48.cf.2.1, 48.192.1-48.cf.2.2, 48.192.1-48.cf.2.3, 48.192.1-48.cf.2.4, 48.192.1-48.cf.2.5, 48.192.1-48.cf.2.6, 48.192.1-48.cf.2.7, 48.192.1-48.cf.2.8, 240.192.1-48.cf.2.1, 240.192.1-48.cf.2.2, 240.192.1-48.cf.2.3, 240.192.1-48.cf.2.4, 240.192.1-48.cf.2.5, 240.192.1-48.cf.2.6, 240.192.1-48.cf.2.7, 240.192.1-48.cf.2.8 |
| Cyclic 48-isogeny field degree: |
$8$ |
| Cyclic 48-torsion field degree: |
$64$ |
| Full 48-torsion field degree: |
$12288$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - x z - 2 y^{2} + z^{2} $ |
| $=$ | $ - 6 x z + 4 y^{2} + 3 w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{4} - 24 x^{2} y^{2} - 2 x^{2} z^{2} + 9 y^{4} + 6 y^{2} z^{2} + z^{4} $ |
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This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{2}{3}y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -2^{11}\,\frac{1079070720xz^{23}-2967444480xz^{21}w^{2}+3322805760xz^{19}w^{4}-1912354560xz^{17}w^{6}+573298176xz^{15}w^{8}-70448448xz^{13}w^{10}-3533088xz^{11}w^{12}+1393200xz^{9}w^{14}-10848xz^{7}w^{16}-8688xz^{5}w^{18}-24xz^{3}w^{20}+12xzw^{22}-289136128z^{24}+327873024z^{22}w^{2}+336191616z^{20}w^{4}-780387200z^{18}w^{6}+530198304z^{16}w^{8}-159871392z^{14}w^{10}+16040440z^{12}w^{12}+1792536z^{10}w^{14}-342540z^{8}w^{16}-11272z^{6}w^{18}+1758z^{4}w^{20}+54z^{2}w^{22}-w^{24}}{w^{4}(619790336xz^{19}-1394528256xz^{17}w^{2}+1321226240xz^{15}w^{4}-685196288xz^{13}w^{6}+211640832xz^{11}w^{8}-39672064xz^{9}w^{10}+4398976xz^{7}w^{12}-267264xz^{5}w^{14}+7520xz^{3}w^{16}-64xzw^{18}-166072320z^{20}+105285632z^{18}w^{2}+216279808z^{16}w^{4}-321415680z^{14}w^{6}+184725632z^{12}w^{8}-56824448z^{10}w^{10}+9949200z^{8}w^{12}-968896z^{6}w^{14}+47464z^{4}w^{16}-912z^{2}w^{18}+3w^{20})}$ |
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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.
