$\GL_2(\Z/48\Z)$-generators: |
$\begin{bmatrix}31&9\\40&41\end{bmatrix}$, $\begin{bmatrix}37&46\\24&37\end{bmatrix}$, $\begin{bmatrix}43&25\\0&1\end{bmatrix}$, $\begin{bmatrix}45&37\\16&3\end{bmatrix}$, $\begin{bmatrix}45&41\\32&7\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
48.192.1-48.bp.2.1, 48.192.1-48.bp.2.2, 48.192.1-48.bp.2.3, 48.192.1-48.bp.2.4, 48.192.1-48.bp.2.5, 48.192.1-48.bp.2.6, 48.192.1-48.bp.2.7, 48.192.1-48.bp.2.8, 48.192.1-48.bp.2.9, 48.192.1-48.bp.2.10, 48.192.1-48.bp.2.11, 48.192.1-48.bp.2.12, 240.192.1-48.bp.2.1, 240.192.1-48.bp.2.2, 240.192.1-48.bp.2.3, 240.192.1-48.bp.2.4, 240.192.1-48.bp.2.5, 240.192.1-48.bp.2.6, 240.192.1-48.bp.2.7, 240.192.1-48.bp.2.8, 240.192.1-48.bp.2.9, 240.192.1-48.bp.2.10, 240.192.1-48.bp.2.11, 240.192.1-48.bp.2.12 |
Cyclic 48-isogeny field degree: |
$8$ |
Cyclic 48-torsion field degree: |
$128$ |
Full 48-torsion field degree: |
$12288$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + 2 x z - y^{2} + 2 z^{2} - w^{2} $ |
| $=$ | $x^{2} + 4 y^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 4 x^{2} y^{2} - x^{2} z^{2} + 100 y^{4} + 20 y^{2} z^{2} + 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{5^2}\cdot\frac{135407419392xz^{23}-13795920936960xz^{21}w^{2}-694629547008000xz^{19}w^{4}+4530285504000000xz^{17}w^{6}+5673596248320000xz^{15}w^{8}-22882247654400000xz^{13}w^{10}+24208068432000000xz^{11}w^{12}-13069588200000000xz^{9}w^{14}+4067554125000000xz^{7}w^{16}-780842250000000xz^{5}w^{18}+90659062500000xz^{3}w^{20}+21093750000xzw^{22}+257986342912z^{24}+5793685168128z^{22}w^{2}-956971334123520z^{20}w^{4}-5220735952896000z^{18}w^{6}+31741034016672000z^{16}w^{8}-48361561833600000z^{14}w^{10}+33344611126400000z^{12}w^{12}-11151312096000000z^{10}w^{14}+1221669356250000z^{8}w^{16}+247336087500000z^{6}w^{18}-54866250000000z^{4}w^{20}-1816875000000z^{2}w^{22}-48828125w^{24}}{w^{2}(766118272xz^{21}-10679464800xz^{19}w^{2}+19657321200xz^{17}w^{4}+37573397000xz^{15}w^{6}-117601150000xz^{13}w^{8}+81866475000xz^{11}w^{10}+2329375000xz^{9}w^{12}-22225625000xz^{7}w^{14}+8915625000xz^{5}w^{16}-1406250000xz^{3}w^{18}+78125000xzw^{20}-386625408z^{22}-8957127792z^{20}w^{2}+49887442720z^{18}w^{4}-56252546475z^{16}w^{6}-52043072500z^{14}w^{8}+135888481250z^{12}w^{10}-89891850000z^{10}w^{12}+20432359375z^{8}w^{14}+2661250000z^{6}w^{16}-2261718750z^{4}w^{18}+429687500z^{2}w^{20}-29296875w^{22})}$ |
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.