| $\GL_2(\Z/56\Z)$-generators: |
$\begin{bmatrix}31&24\\20&9\end{bmatrix}$, $\begin{bmatrix}41&32\\12&33\end{bmatrix}$, $\begin{bmatrix}45&4\\4&41\end{bmatrix}$, $\begin{bmatrix}45&32\\20&7\end{bmatrix}$, $\begin{bmatrix}51&24\\44&25\end{bmatrix}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
56.192.3-56.w.1.1, 56.192.3-56.w.1.2, 56.192.3-56.w.1.3, 56.192.3-56.w.1.4, 56.192.3-56.w.1.5, 56.192.3-56.w.1.6, 56.192.3-56.w.1.7, 56.192.3-56.w.1.8, 56.192.3-56.w.1.9, 56.192.3-56.w.1.10, 56.192.3-56.w.1.11, 56.192.3-56.w.1.12, 112.192.3-56.w.1.1, 112.192.3-56.w.1.2, 112.192.3-56.w.1.3, 112.192.3-56.w.1.4, 112.192.3-56.w.1.5, 112.192.3-56.w.1.6, 112.192.3-56.w.1.7, 112.192.3-56.w.1.8, 168.192.3-56.w.1.1, 168.192.3-56.w.1.2, 168.192.3-56.w.1.3, 168.192.3-56.w.1.4, 168.192.3-56.w.1.5, 168.192.3-56.w.1.6, 168.192.3-56.w.1.7, 168.192.3-56.w.1.8, 168.192.3-56.w.1.9, 168.192.3-56.w.1.10, 168.192.3-56.w.1.11, 168.192.3-56.w.1.12, 280.192.3-56.w.1.1, 280.192.3-56.w.1.2, 280.192.3-56.w.1.3, 280.192.3-56.w.1.4, 280.192.3-56.w.1.5, 280.192.3-56.w.1.6, 280.192.3-56.w.1.7, 280.192.3-56.w.1.8, 280.192.3-56.w.1.9, 280.192.3-56.w.1.10, 280.192.3-56.w.1.11, 280.192.3-56.w.1.12 |
| Cyclic 56-isogeny field degree: |
$16$ |
| Cyclic 56-torsion field degree: |
$192$ |
| Full 56-torsion field degree: |
$32256$ |
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ 2 x z w - x w t - y w^{2} + y w t $ |
| $=$ | $2 x z^{2} - x z t - y z w + y z t$ |
| $=$ | $2 x z t - x t^{2} - y w t + y t^{2}$ |
| $=$ | $2 x y z - x y t - y^{2} w + y^{2} t$ |
| $=$ | $\cdots$ |
![Copy content]()
![Toggle raw display]()
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{4} z + 7 x^{3} y^{2} + x^{3} z^{2} - 42 x^{2} y^{2} z + 2 x^{2} z^{3} + 70 x y^{2} z^{2} + \cdots - 28 y^{2} z^{3} $ |
![Copy content]()
![Toggle raw display]()
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -7x^{7} + 49x^{5} - 49x^{3} + 7x $ |
![Copy content]()
![Toggle raw display]()
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:0:1/2:1:1)$, $(0:1:0:0:0)$, $(1:1:0:0:0)$, $(1:0:0:0:0)$ |
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{7}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ |
$=$ |
$\displaystyle -x+y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle x^{3}t-6x^{2}yt+10xy^{2}t-4y^{3}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y$ |
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{823543x^{14}-705894x^{12}t^{2}+1176490x^{10}t^{4}-1709512x^{8}t^{6}+2762522x^{6}t^{8}-4495456x^{4}t^{10}+7336308x^{2}t^{12}-1647086xy^{13}+5411854xy^{11}t^{2}-1747928xy^{9}t^{4}+14425208xy^{7}t^{6}-2730966xy^{5}t^{8}+24969910xy^{3}t^{10}-4659424xyt^{12}-2352980y^{12}t^{2}-2958032y^{10}t^{4}-7049336y^{8}t^{6}-5718496y^{6}t^{8}-12071052y^{4}t^{10}-9209312y^{2}t^{12}+640zw^{13}+128zw^{12}t+4352zw^{11}t^{2}+256zw^{10}t^{3}+43904zw^{9}t^{4}+4480zw^{8}t^{5}+232960zw^{7}t^{6}+3592zw^{6}t^{7}+1216704zw^{5}t^{8}+12128zw^{4}t^{9}+4893120zw^{3}t^{10}-214360zw^{2}t^{11}+13187584zwt^{12}+640zt^{13}-224w^{14}-64w^{13}t-1568w^{12}t^{2}-384w^{11}t^{3}-15456w^{10}t^{4}-4032w^{9}t^{5}-83104w^{8}t^{6}-19840w^{7}t^{7}-425024w^{6}t^{8}-120488w^{5}t^{9}-1615232w^{4}t^{10}-821832w^{3}t^{11}-3403488w^{2}t^{12}-3182088wt^{13}-224t^{14}}{t^{4}(343x^{6}t^{4}-1078x^{4}t^{6}+2506x^{2}t^{8}-686xy^{5}t^{4}+3038xy^{3}t^{6}-2240xyt^{8}-980y^{4}t^{6}-1904y^{2}t^{8}+40zw^{9}+8zw^{8}t+272zw^{7}t^{2}+16zw^{6}t^{3}+904zw^{5}t^{4}-88zw^{4}t^{5}+2048zw^{3}t^{6}-504zw^{2}t^{7}+3456zwt^{8}-14w^{10}-4w^{9}t-98w^{8}t^{2}-24w^{7}t^{3}-322w^{6}t^{4}-68w^{5}t^{5}-686w^{4}t^{6}-136w^{3}t^{7}-984w^{2}t^{8}-740wt^{9})}$ |
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.
