$\GL_2(\Z/56\Z)$-generators: |
$\begin{bmatrix}4&45\\5&14\end{bmatrix}$, $\begin{bmatrix}32&55\\43&50\end{bmatrix}$, $\begin{bmatrix}39&16\\32&31\end{bmatrix}$, $\begin{bmatrix}45&16\\0&29\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
56.96.1-56.dd.1.1, 56.96.1-56.dd.1.2, 56.96.1-56.dd.1.3, 56.96.1-56.dd.1.4, 112.96.1-56.dd.1.1, 112.96.1-56.dd.1.2, 112.96.1-56.dd.1.3, 112.96.1-56.dd.1.4, 168.96.1-56.dd.1.1, 168.96.1-56.dd.1.2, 168.96.1-56.dd.1.3, 168.96.1-56.dd.1.4, 280.96.1-56.dd.1.1, 280.96.1-56.dd.1.2, 280.96.1-56.dd.1.3, 280.96.1-56.dd.1.4 |
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} $ |
| $=$ | $28 x^{2} - 42 x y + 3 y^{2} - 2 y z - 2 z^{2} - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 4 x^{3} y + 34 x^{2} y^{2} + 2 x^{2} z^{2} + 60 x y^{3} + 4 x y z^{2} + 421 y^{4} - 26 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^8}{7^2}\cdot\frac{13729231429193040882882195456xz^{11}+76647314194082834711880655872xz^{9}w^{2}-122837828877512566804915020288xz^{7}w^{4}-22835043507602921707718781952xz^{5}w^{6}+3347623510807179059457966720xz^{3}w^{8}+58758902212778771164024192xzw^{10}-15450188481858877680443255424y^{2}z^{10}+42722549964022872353365429488y^{2}z^{8}w^{2}+39384644886177387735216122208y^{2}z^{6}w^{4}-29471384263923020051877107304y^{2}z^{4}w^{6}+980002620183076910195856696y^{2}z^{2}w^{8}-5559843894573893101473189y^{2}w^{10}-11254591009356640783545867264yz^{11}-3290494355667219595831247808yz^{9}w^{2}+91985411801930863198706704512yz^{7}w^{4}-34984140554475690259316450144yz^{5}w^{6}-107926637921017626649117312yz^{3}w^{8}+41783957394632001629978164yzw^{10}-3382142398889678955419616000z^{12}-2068559555407765355927041344z^{10}w^{2}+12190803656134405983406646256z^{8}w^{4}+1178687892215488637626035840z^{6}w^{6}-488887312439343372557165688z^{4}w^{8}-42502069606946691729618980z^{2}w^{10}+464285877618639010127087w^{12}}{41509392076169435775909888xz^{11}+11409912015590969360420864xz^{9}w^{2}-14245535167727725947416576xz^{7}w^{4}-8471914167380960402387456xz^{5}w^{6}-1662911306993118016182656xz^{3}w^{8}-112853131394233206264064xzw^{10}-46712588002596757915172352y^{2}z^{10}-67180611673857956634773504y^{2}z^{8}w^{2}-37079739175327580914590464y^{2}z^{6}w^{4}-9856042500406746997698168y^{2}z^{4}w^{6}-1266978904101710554104376y^{2}z^{2}w^{8}-63377246540316276640171y^{2}w^{10}-34027486044918037138460672yz^{11}-50214336567400527602350592yz^{9}w^{2}-29183960604526216038665472yz^{7}w^{4}-8296979569219459402010656yz^{5}w^{6}-1148689492792658199297152yz^{3}w^{8}-61792870862985741068116yzw^{10}-10225676187119210749568000z^{12}-16790830676917937998769152z^{10}w^{2}-9867932286619826343305920z^{8}w^{4}-2562861644468748972951328z^{6}w^{6}-245954440478311764292200z^{4}w^{8}+8836602323248366973764z^{2}w^{10}+2182407802934298197861w^{12}}$ |
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.