| $\GL_2(\Z/24\Z)$-generators: |
$\begin{bmatrix}1&1\\12&11\end{bmatrix}$, $\begin{bmatrix}13&20\\0&5\end{bmatrix}$, $\begin{bmatrix}17&15\\6&13\end{bmatrix}$, $\begin{bmatrix}19&10\\6&19\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: |
Group 768.136644 |
| Contains $-I$: |
yes |
| Quadratic refinements: |
24.192.3-24.dl.1.1, 24.192.3-24.dl.1.2, 24.192.3-24.dl.1.3, 24.192.3-24.dl.1.4, 24.192.3-24.dl.1.5, 24.192.3-24.dl.1.6, 24.192.3-24.dl.1.7, 24.192.3-24.dl.1.8, 48.192.3-24.dl.1.1, 48.192.3-24.dl.1.2, 48.192.3-24.dl.1.3, 48.192.3-24.dl.1.4, 48.192.3-24.dl.1.5, 48.192.3-24.dl.1.6, 48.192.3-24.dl.1.7, 48.192.3-24.dl.1.8, 120.192.3-24.dl.1.1, 120.192.3-24.dl.1.2, 120.192.3-24.dl.1.3, 120.192.3-24.dl.1.4, 120.192.3-24.dl.1.5, 120.192.3-24.dl.1.6, 120.192.3-24.dl.1.7, 120.192.3-24.dl.1.8, 168.192.3-24.dl.1.1, 168.192.3-24.dl.1.2, 168.192.3-24.dl.1.3, 168.192.3-24.dl.1.4, 168.192.3-24.dl.1.5, 168.192.3-24.dl.1.6, 168.192.3-24.dl.1.7, 168.192.3-24.dl.1.8, 240.192.3-24.dl.1.1, 240.192.3-24.dl.1.2, 240.192.3-24.dl.1.3, 240.192.3-24.dl.1.4, 240.192.3-24.dl.1.5, 240.192.3-24.dl.1.6, 240.192.3-24.dl.1.7, 240.192.3-24.dl.1.8, 264.192.3-24.dl.1.1, 264.192.3-24.dl.1.2, 264.192.3-24.dl.1.3, 264.192.3-24.dl.1.4, 264.192.3-24.dl.1.5, 264.192.3-24.dl.1.6, 264.192.3-24.dl.1.7, 264.192.3-24.dl.1.8, 312.192.3-24.dl.1.1, 312.192.3-24.dl.1.2, 312.192.3-24.dl.1.3, 312.192.3-24.dl.1.4, 312.192.3-24.dl.1.5, 312.192.3-24.dl.1.6, 312.192.3-24.dl.1.7, 312.192.3-24.dl.1.8 |
| Cyclic 24-isogeny field degree: |
$4$ |
| Cyclic 24-torsion field degree: |
$32$ |
| Full 24-torsion field degree: |
$768$ |
Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ x^{2} - x z + y^{2} - y z $ |
| $=$ | $2 x w + x t + y u$ |
| $=$ | $x w - x t - y w + y t + z w - z t + z u$ |
| $=$ | $x w - x t + y w - y t - 3 z w$ |
| $=$ | $\cdots$ |
![Copy content]()
![Toggle raw display]()
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 10 x^{8} + 8 x^{7} z - 37 x^{6} y^{2} + 12 x^{6} z^{2} - 78 x^{5} y^{2} z + 8 x^{5} z^{3} + \cdots - y^{2} z^{6} $ |
![Copy content]()
![Toggle raw display]()
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 2x^{8} + 80x^{4} + 288 $ |
![Copy content]()
![Toggle raw display]()
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:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{3}u$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ |
$=$ |
$\displaystyle x^{3}+x^{2}y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{37}{3}x^{11}u-\frac{41}{3}x^{10}yu-\frac{10}{3}x^{9}y^{2}u-\frac{10}{3}x^{8}y^{3}u-\frac{5}{3}x^{7}y^{4}u-\frac{1}{3}x^{6}y^{5}u$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle x^{3}$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{2^{15}\cdot3^3}\cdot\frac{5283615080448xy^{9}u^{2}-9001714581504xy^{7}u^{4}+3261490790400xy^{5}u^{6}-391378894848xy^{3}u^{8}-11542724608xyu^{10}+2641807540224y^{12}-7044820107264y^{10}u^{2}+3815944224768y^{8}u^{4}+579820584960y^{6}u^{6}-752558800896y^{4}u^{8}+6979321856y^{2}u^{10}+6484997376yz^{11}+33175775232yz^{9}u^{2}-1033365454848yz^{7}u^{4}+4730520600576yz^{5}u^{6}-58232332615680yz^{3}u^{8}-297578151280640yzu^{10}-5093015616z^{12}+43391199744z^{10}u^{2}+102873120768z^{8}u^{4}-918327656448z^{6}u^{6}+31700954382336z^{4}u^{8}+107983597993984z^{2}u^{10}+660935299536wt^{11}+16161632996652wt^{10}u+81431219833164wt^{9}u^{2}+156388858673814wt^{8}u^{3}+133540373072664wt^{7}u^{4}+21242077059816wt^{6}u^{5}-50901422379408wt^{5}u^{6}-41680590331644wt^{4}u^{7}-48881439962856wt^{3}u^{8}-54272321583472wt^{2}u^{9}-292738149503528wtu^{10}-229350634012408wu^{11}+289157579541t^{12}+4610632463964t^{11}u+8617344948027t^{10}u^{2}-4329070416342t^{9}u^{3}-20820372041421t^{8}u^{4}-8635223807244t^{7}u^{5}+15053246025714t^{6}u^{6}+10266080623632t^{5}u^{7}-29382511662324t^{4}u^{8}-40531052653556t^{3}u^{9}-57354053326456t^{2}u^{10}-177769805357148tu^{11}-38147356589891u^{12}}{u^{6}(12wt^{5}+150wt^{4}u+352wt^{3}u^{2}+348wt^{2}u^{3}+136wtu^{4}+16wu^{5}+5t^{6}+30t^{5}u-12t^{4}u^{2}-40t^{3}u^{3}-14t^{2}u^{4}+16tu^{5}+15u^{6})}$ |
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.
