$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}17&10\\20&23\end{bmatrix}$, $\begin{bmatrix}33&37\\8&43\end{bmatrix}$, $\begin{bmatrix}35&3\\16&1\end{bmatrix}$, $\begin{bmatrix}35&16\\4&45\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.ca.1.1, 48.192.1-48.ca.1.2, 48.192.1-48.ca.1.3, 48.192.1-48.ca.1.4, 48.192.1-48.ca.1.5, 48.192.1-48.ca.1.6, 48.192.1-48.ca.1.7, 48.192.1-48.ca.1.8, 240.192.1-48.ca.1.1, 240.192.1-48.ca.1.2, 240.192.1-48.ca.1.3, 240.192.1-48.ca.1.4, 240.192.1-48.ca.1.5, 240.192.1-48.ca.1.6, 240.192.1-48.ca.1.7, 240.192.1-48.ca.1.8 |
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 $ | $=$ | $ 4 x^{2} - y^{2} - 2 y w - 2 z^{2} - 2 w^{2} $ |
| $=$ | $2 x^{2} + 2 y^{2} + 3 y w + 2 z^{2} + 3 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 10 x^{2} y^{2} + 2 x^{2} z^{2} + 25 y^{4} - 4 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{2^8}{5^2}\cdot\frac{10546875000yz^{22}w-22664765625000yz^{20}w^{3}-97605281250000yz^{18}w^{5}-254222132812500yz^{16}w^{7}-408424631250000yz^{14}w^{9}-378251069250000yz^{12}w^{11}-178767559800000yz^{10}w^{13}-22162485345000yz^{8}w^{15}+8848213875000yz^{6}w^{17}+678349167000yz^{4}w^{19}-6736289520yz^{2}w^{21}-33058452yw^{23}+48828125z^{24}-908437500000z^{22}w^{2}+13716562500000z^{20}w^{4}+30917010937500z^{18}w^{6}-76354334765625z^{16}w^{8}-348478503000000z^{14}w^{10}-521009548850000z^{12}w^{12}-377824701825000z^{10}w^{14}-123988414127625z^{8}w^{16}-10196749908000z^{6}w^{18}+934542318480z^{4}w^{20}+2828947836z^{2}w^{22}-62984947w^{24}}{z^{2}(10000000000yz^{20}w+90000000000yz^{18}w^{3}+285300000000yz^{16}w^{5}+355610000000yz^{14}w^{7}+18635000000yz^{12}w^{9}-327465900000yz^{10}w^{11}-235202300000yz^{8}w^{13}-37573397000yz^{6}w^{15}+9828660600yz^{4}w^{17}+2669866200yz^{2}w^{19}+95764784yw^{21}+7500000000z^{22}+55000000000z^{20}w^{2}+144750000000z^{18}w^{4}+85160000000z^{16}w^{6}-326917750000z^{14}w^{8}-719134800000z^{12}w^{10}-543553925000z^{10}w^{12}-104086145000z^{8}w^{14}+56252546475z^{6}w^{16}+24943721360z^{4}w^{18}+2239281948z^{2}w^{20}-48328176w^{22})}$ |
Hi
|
Cover information
Click on a modular curve in the diagram to see information about it.
|
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.