$\GL_2(\Z/56\Z)$-generators: |
$\begin{bmatrix}7&34\\34&11\end{bmatrix}$, $\begin{bmatrix}9&0\\44&53\end{bmatrix}$, $\begin{bmatrix}11&20\\38&33\end{bmatrix}$, $\begin{bmatrix}39&16\\48&7\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
56.192.1-56.t.1.1, 56.192.1-56.t.1.2, 56.192.1-56.t.1.3, 56.192.1-56.t.1.4, 56.192.1-56.t.1.5, 56.192.1-56.t.1.6, 56.192.1-56.t.1.7, 56.192.1-56.t.1.8, 112.192.1-56.t.1.1, 112.192.1-56.t.1.2, 112.192.1-56.t.1.3, 112.192.1-56.t.1.4, 168.192.1-56.t.1.1, 168.192.1-56.t.1.2, 168.192.1-56.t.1.3, 168.192.1-56.t.1.4, 168.192.1-56.t.1.5, 168.192.1-56.t.1.6, 168.192.1-56.t.1.7, 168.192.1-56.t.1.8, 280.192.1-56.t.1.1, 280.192.1-56.t.1.2, 280.192.1-56.t.1.3, 280.192.1-56.t.1.4, 280.192.1-56.t.1.5, 280.192.1-56.t.1.6, 280.192.1-56.t.1.7, 280.192.1-56.t.1.8 |
Cyclic 56-isogeny field degree: |
$16$ |
Cyclic 56-torsion field degree: |
$192$ |
Full 56-torsion field degree: |
$32256$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 x^{2} + x y + 2 x w - 4 y^{2} - 2 y w - 2 w^{2} $ |
| $=$ | $8 x^{2} - 2 x y - 4 x w + y^{2} + 4 y w + z^{2} + 4 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 441 x^{4} + 224 x^{2} y^{2} + 112 x^{2} y z + 56 x^{2} z^{2} + 23 y^{4} + 16 y^{3} z + 12 y^{2} z^{2} + \cdots + 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle 7y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle 14w$ |
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}{7^2\cdot23^4}\cdot\frac{5865266761939745090417538459072xz^{22}w+326343717148743235511973706133376xz^{20}w^{3}+5211221861444877905580411731857920xz^{18}w^{5}+52283668750126149223468257289740288xz^{16}w^{7}+300755369899172844823961247155968000xz^{14}w^{9}+982029399405418585505038951125536768xz^{12}w^{11}+1856701932612479279551329732450066432xz^{10}w^{13}+880662711464358060737459745341046784xz^{8}w^{15}-4105015603322746558551542904941723648xz^{6}w^{17}-14397222841173910285536056805140103168xz^{4}w^{19}-17996581851244677161667177101353549824xz^{2}w^{21}-15364227368222170230451745743983083520xw^{23}-10359260882437342364537724202464y^{2}z^{22}-448974346289668904732518550503920y^{2}z^{20}w^{2}-6741677834470254989541256519040256y^{2}z^{18}w^{4}-57360396288438551149739840346033952y^{2}z^{16}w^{6}-250553684829089953825508740397460480y^{2}z^{14}w^{8}-435791444154831024137986225053615616y^{2}z^{12}w^{10}+671030088912203661432704249216495616y^{2}z^{10}w^{12}+6555051345464784722587930999022201856y^{2}z^{8}w^{14}+18432611840778900388372699063587905536y^{2}z^{6}w^{16}+33693471040491292794787533674329804800y^{2}z^{4}w^{18}+33129115262729054559411576760463523840y^{2}z^{2}w^{20}+21294050420222898841796829754455613440y^{2}w^{22}-5865266761939745090417538459072yz^{22}w-251138236474796323611034904933280yz^{20}w^{3}-2522616943748856364146657736517184yz^{18}w^{5}-14279383343409864311714636692756800yz^{16}w^{7}-12533567348499048134949552180152320yz^{14}w^{9}+350832303185858050555961101357689856yz^{12}w^{11}+2324391392041788052908057673886459904yz^{10}w^{13}+8125813698242404078742996749149083648yz^{8}w^{15}+17243679873028620506332841515259281408yz^{6}w^{17}+25893324326588779507884209277630504960yz^{4}w^{19}+22336680310874043188341997242200145920yz^{2}w^{21}+11788823853959485078642695531387863040yw^{23}-572047977000767019527220039023z^{24}-32401718752061846958879359359584z^{22}w^{2}-587122370847300678341910426696480z^{20}w^{4}-6172801893095288864567429050469568z^{18}w^{6}-40939253442084937115906550991974608z^{16}w^{8}-141134141380265002064072842481949696z^{14}w^{10}-99598445464649194948558607188383744z^{12}w^{12}+1109893631980073929196120411607435264z^{10}w^{14}+5610648748892936948627346171401532160z^{8}w^{16}+13746166369148606379463598556666241024z^{6}w^{18}+22919193601780956252543584044655026176z^{4}w^{20}+21549312736342303256047714356019150848z^{2}w^{22}+13275508805635614348093044670209265664w^{24}}{z^{4}(1015451654772022272xz^{18}w+57072782838927841536xz^{16}w^{3}+1345570526479589464768xz^{14}w^{5}+17545688429571474172784xz^{12}w^{7}-534451808829412381684000xz^{10}w^{9}-13926501743564455674769392xz^{8}w^{11}-89627437473267852322912608xz^{6}w^{13}-116359904277484970136131328xz^{4}w^{15}+442435835561364825084787904xz^{2}w^{17}+879670311652152905484852224xw^{19}-915043188908370816y^{2}z^{18}-56711838219403893120y^{2}z^{16}w^{2}-1432595868729767833168y^{2}z^{14}w^{4}-18861608162782333006072y^{2}z^{12}w^{6}+611134430691215245946880y^{2}z^{10}w^{8}+11796619035179809605265204y^{2}z^{8}w^{10}+27393297196290415810995120y^{2}z^{6}w^{12}-317067736695020428447197252y^{2}z^{4}w^{14}-1535495945785677616270434016y^{2}z^{2}w^{16}-1720773010116855146169043760y^{2}w^{18}-1015451654772022272yz^{18}w-57133406818317216000yz^{16}w^{3}-1359986769665386528960yz^{14}w^{5}-17987330017791506332912yz^{12}w^{7}+680965213190587890040yz^{10}w^{9}-1221897190531687400715880yz^{8}w^{11}-55648951615464029560830904yz^{6}w^{13}-462691897205105020420409288yz^{4}w^{15}-1344646292502133343723205184yz^{2}w^{17}-1213323002629497887807342880yw^{19}-130720455558338688z^{20}-9147454818667265664z^{18}w^{2}-269303366125964788528z^{16}w^{4}-4319523168226685850312z^{14}w^{6}-178481133480428265249z^{12}w^{8}+683683562895666387831168z^{10}w^{10}+4521442138808900444531322z^{8}w^{12}-22150867972413251445195416z^{6}w^{14}-320506590131091111794300961z^{4}w^{16}-1064757621566240650198613792z^{2}w^{18}-1073386226151223934056195984w^{20})}$ |
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.