$\GL_2(\Z/56\Z)$-generators: |
$\begin{bmatrix}2&9\\47&2\end{bmatrix}$, $\begin{bmatrix}14&1\\53&20\end{bmatrix}$, $\begin{bmatrix}48&19\\31&26\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
112.96.1-56.df.1.1, 112.96.1-56.df.1.2, 112.96.1-56.df.1.3, 112.96.1-56.df.1.4, 112.96.1-56.df.1.5, 112.96.1-56.df.1.6, 112.96.1-56.df.1.7, 112.96.1-56.df.1.8, 112.96.1-56.df.1.9, 112.96.1-56.df.1.10, 112.96.1-56.df.1.11, 112.96.1-56.df.1.12, 112.96.1-56.df.1.13, 112.96.1-56.df.1.14, 112.96.1-56.df.1.15, 112.96.1-56.df.1.16 |
Cyclic 56-isogeny field degree: |
$32$ |
Cyclic 56-torsion field degree: |
$768$ |
Full 56-torsion field degree: |
$64512$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 14 x^{2} + 7 x y + 2 y^{2} - y z + z^{2} $ |
| $=$ | $28 x^{2} - 42 x y + 3 y^{2} + 2 y z - 2 z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 4 x^{3} y + 34 x^{2} y^{2} - 4 x^{2} z^{2} - 60 x y^{3} + 8 x y z^{2} + 421 y^{4} + 52 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^8}{7^2}\cdot\frac{429038482162282527590068608xz^{11}-4790457137130177169492540992xz^{9}w^{2}-15354728609689070850614377536xz^{7}w^{4}+5708760876900730426929695488xz^{5}w^{6}+1673811755403589529728983360xz^{3}w^{8}-58758902212778771164024192xzw^{10}+482818390058089927513851732y^{2}z^{10}+2670159372751429522085339343y^{2}z^{8}w^{2}-4923080610772173466902015276y^{2}z^{6}w^{4}-7367846065980755012969276826y^{2}z^{4}w^{6}-490001310091538455097928348y^{2}z^{2}w^{8}-5559843894573893101473189y^{2}w^{10}-351705969042395024485808352yz^{11}+205655897229201224739452988yz^{9}w^{2}+11498176475241357899838338064yz^{7}w^{4}+8746035138618922564829112536yz^{5}w^{6}-53963318960508813324558656yz^{3}w^{8}-41783957394632001629978164yzw^{10}+105691949965302467356863000z^{12}-129284972212985334745440084z^{10}w^{2}-1523850457016800747925830782z^{8}w^{4}+294671973053872159406508960z^{6}w^{6}+244443656219671686278582844z^{4}w^{8}-42502069606946691729618980z^{2}w^{10}-928571755237278020254174w^{12}}{1297168502380294867997184xz^{11}-713119500974435585026304xz^{9}w^{2}-1780691895965965743427072xz^{7}w^{4}+2117978541845240100596864xz^{5}w^{6}-831455653496559008091328xz^{3}w^{8}+112853131394233206264064xzw^{10}+1459768375081148684849136y^{2}z^{10}-4198788229616122289673344y^{2}z^{8}w^{2}+4634967396915947614323808y^{2}z^{6}w^{4}-2464010625101686749424542y^{2}z^{4}w^{6}+633489452050855277052188y^{2}z^{2}w^{8}-63377246540316276640171y^{2}w^{10}-1063358938903688660576896yz^{11}+3138396035462532975146912yz^{9}w^{2}-3647995075565777004833184yz^{7}w^{4}+2074244892304864850502664yz^{5}w^{6}-574344746396329099648576yz^{3}w^{8}+61792870862985741068116yzw^{10}+319552380847475335924000z^{12}-1049426917307371124923072z^{10}w^{2}+1233491535827478292913240z^{8}w^{4}-640715411117187243237832z^{6}w^{6}+122977220239155882146100z^{4}w^{8}+8836602323248366973764z^{2}w^{10}-4364815605868596395722w^{12}}$ |
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.