$\GL_2(\Z/56\Z)$-generators: |
$\begin{bmatrix}21&36\\48&17\end{bmatrix}$, $\begin{bmatrix}27&4\\4&45\end{bmatrix}$, $\begin{bmatrix}41&22\\2&13\end{bmatrix}$, $\begin{bmatrix}43&0\\0&11\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
56.96.1-56.bq.1.1, 56.96.1-56.bq.1.2, 56.96.1-56.bq.1.3, 56.96.1-56.bq.1.4, 168.96.1-56.bq.1.1, 168.96.1-56.bq.1.2, 168.96.1-56.bq.1.3, 168.96.1-56.bq.1.4, 280.96.1-56.bq.1.1, 280.96.1-56.bq.1.2, 280.96.1-56.bq.1.3, 280.96.1-56.bq.1.4 |
Cyclic 56-isogeny field degree: |
$16$ |
Cyclic 56-torsion field degree: |
$384$ |
Full 56-torsion field degree: |
$64512$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ x^{2} + 2 x y + 2 x z - 2 x w + z^{2} - 2 z w + w^{2} $ |
| $=$ | $3 x^{2} - x y - x z + x w + 4 y^{2} + y z + y w + 2 z w$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 8 x^{4} + 11 x^{3} y - 24 x^{3} z + 34 x^{2} y^{2} + 5 x^{2} y z + 34 x^{2} z^{2} + 67 x y^{3} + \cdots + 8 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{3^3}{2^4}\cdot\frac{1412166645150457xz^{9}w^{2}-12538149835888719xz^{8}w^{3}+49544575357360356xz^{7}w^{4}-114458006641169772xz^{6}w^{5}+171061619127704046xz^{5}w^{6}-172569056341019058xz^{4}w^{7}+118248707346282324xz^{3}w^{8}-53451665368595676xz^{2}w^{9}+14581544209890273xzw^{10}-1861819503014583xw^{11}+696413784615049y^{2}z^{10}-6765268029624126y^{2}z^{9}w+28696786287029469y^{2}z^{8}w^{2}-69591220433383464y^{2}z^{7}w^{3}+105968503795490946y^{2}z^{6}w^{4}-104070361959307764y^{2}z^{5}w^{5}+63764494318148562y^{2}z^{4}w^{6}-20234286619843752y^{2}z^{3}w^{7}-409873417930827y^{2}z^{2}w^{8}+2654524310622210y^{2}zw^{9}-693875677523823y^{2}w^{10}-96721061579295yz^{11}+1461747120076473yz^{10}w-9664316551745977yz^{9}w^{2}+36599110119288639yz^{8}w^{3}-88469407881782262yz^{7}w^{4}+144216006914828538yz^{6}w^{5}-162979516158147090yz^{5}w^{6}+128460807684552798yz^{4}w^{7}-69393569763993723yz^{3}w^{8}+24384700095586749yz^{2}w^{9}-4941901722608037yzw^{10}+410151659180787yw^{11}+154758082710872z^{12}-1658151366352600z^{11}w+8723926166573235z^{10}w^{2}-30418757052625058z^{9}w^{3}+77698332648717039z^{8}w^{4}-148857761560039848z^{7}w^{5}+212830168391257302z^{6}w^{6}-224378149286572524z^{5}w^{7}+171595635451179102z^{4}w^{8}-92551033660250736z^{3}w^{9}+33363731278306935z^{2}w^{10}-7192540878669954zw^{11}+687114682996059w^{12}}{145581667xz^{9}w^{2}-317447109xz^{8}w^{3}+387255708xz^{7}w^{4}-583312859892xz^{6}w^{5}+5079119378634xz^{5}w^{6}-16114286436822xz^{4}w^{7}+25396538488524xz^{3}w^{8}-21385624455972xz^{2}w^{9}+9131344150635xzw^{10}-1531109687373xw^{11}-1040044397y^{2}z^{10}+2962956582y^{2}z^{9}w-4038641937y^{2}z^{8}w^{2}+2188016712y^{2}z^{7}w^{3}+1502273422422y^{2}z^{6}w^{4}-7714842627996y^{2}z^{5}w^{5}+15429498070662y^{2}z^{4}w^{6}-14554291231800y^{2}z^{3}w^{7}+5624231068503y^{2}z^{2}w^{8}+159282433638y^{2}zw^{9}-449508209589y^{2}w^{10}-1061017125yz^{11}+3454565523yz^{10}w-4647457555yz^{9}w^{2}+416178117yz^{8}w^{3}-201097388178yz^{7}w^{4}+2515826537118yz^{6}w^{5}-10383834818790yz^{5}w^{6}+20899284754122yz^{4}w^{7}-22902269342841yz^{3}w^{8}+13634533556127yz^{2}w^{9}-3961917632727yzw^{10}+399669672609yw^{11}-20972728z^{12}-527462728z^{11}w+2756994753z^{10}w^{2}-7659997958z^{9}w^{3}+344122183701z^{8}w^{4}-2304253009080z^{7}w^{5}+8244115279218z^{6}w^{6}-19417082381316z^{5}w^{7}+30132947260170z^{4}w^{8}-29497539328848z^{3}w^{9}+17052316011981z^{2}w^{10}-5149726000038zw^{11}+598909029273w^{12}}$ |
Hi
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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.