$\GL_2(\Z/56\Z)$-generators: |
$\begin{bmatrix}13&36\\46&15\end{bmatrix}$, $\begin{bmatrix}13&40\\2&3\end{bmatrix}$, $\begin{bmatrix}43&16\\22&13\end{bmatrix}$, $\begin{bmatrix}43&20\\48&53\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
56.192.1-56.ba.2.1, 56.192.1-56.ba.2.2, 56.192.1-56.ba.2.3, 56.192.1-56.ba.2.4, 56.192.1-56.ba.2.5, 56.192.1-56.ba.2.6, 56.192.1-56.ba.2.7, 56.192.1-56.ba.2.8, 168.192.1-56.ba.2.1, 168.192.1-56.ba.2.2, 168.192.1-56.ba.2.3, 168.192.1-56.ba.2.4, 168.192.1-56.ba.2.5, 168.192.1-56.ba.2.6, 168.192.1-56.ba.2.7, 168.192.1-56.ba.2.8, 280.192.1-56.ba.2.1, 280.192.1-56.ba.2.2, 280.192.1-56.ba.2.3, 280.192.1-56.ba.2.4, 280.192.1-56.ba.2.5, 280.192.1-56.ba.2.6, 280.192.1-56.ba.2.7, 280.192.1-56.ba.2.8 |
Cyclic 56-isogeny field degree: |
$16$ |
Cyclic 56-torsion field degree: |
$384$ |
Full 56-torsion field degree: |
$32256$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 4 x^{2} - 2 x z - x w + 2 z^{2} + 2 z w + w^{2} $ |
| $=$ | $3 x^{2} + 2 x z + x w + 14 y^{2} - 2 z^{2} - 2 z w - 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 2 x^{4} + 28 x^{3} z - 12 x^{2} y^{2} + 238 x^{2} z^{2} - 84 x y^{2} z + 980 x z^{3} + 32 y^{4} + \cdots + 2793 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:
$\displaystyle X$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{7}w$ |
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}{2^4\cdot7^2}\cdot\frac{245570815331842129920xz^{23}+2824064376316184494080xz^{22}w+16531705373550440349696xz^{21}w^{2}+64856427934106520649728xz^{20}w^{3}+190921981832422400458752xz^{19}w^{4}+448572973253069302923264xz^{18}w^{5}+872975439703702654025728xz^{17}w^{6}+1439043166360706665873408xz^{16}w^{7}+2033657450616098554576896xz^{15}w^{8}+2474495232184153348538368xz^{14}w^{9}+2586730544975565011795968xz^{13}w^{10}+2302535975785014778798080xz^{12}w^{11}+1712785247155960397844480xz^{11}w^{12}+1023613041009738184857600xz^{10}w^{13}+443751726056125736481792xz^{9}w^{14}+84429214391598667490816xz^{8}w^{15}-63762849930466914424960xz^{7}w^{16}-81249569545337670429120xz^{6}w^{17}-51345296389127797236768xz^{5}w^{18}-22121909649583499127792xz^{4}w^{19}-6701587816991843610744xz^{3}w^{20}-1350605517488535433636xz^{2}w^{21}-152862778285781720590xzw^{22}-5470484204623982313xw^{23}+112277531170634203136z^{24}+1347330374047610437632z^{23}w+7834144418332806217728z^{22}w^{2}+29363157829319961608192z^{21}w^{3}+78563561990574577287168z^{20}w^{4}+155612111285933464092672z^{19}w^{5}+224275019813007188164608z^{18}w^{6}+203171439593861620826112z^{17}w^{7}-56293753565906468864z^{16}w^{8}-417286617468842023583744z^{15}w^{9}-963824477299428041342976z^{14}w^{10}-1456481398043766486941696z^{13}w^{11}-1708787111659908931424256z^{12}w^{12}-1638088394668048176193536z^{11}w^{13}-1305627702351765007574016z^{10}w^{14}-865946526964141727777792z^{9}w^{15}-471781905619352765910144z^{8}w^{16}-204486080908439008311808z^{7}w^{17}-65406424780116287690592z^{6}w^{18}-11995946536954803347232z^{5}w^{19}+1069821377567471693976z^{4}w^{20}+1634829363142617297936z^{3}w^{21}+579490940020532231286z^{2}w^{22}+106496750423221280318zw^{23}+8477760777047822573w^{24}}{w^{4}(1895054823653376xz^{19}+18003020824707072xz^{18}w+84184108081217536xz^{17}w^{2}+256487887660318720xz^{16}w^{3}+568305541058002944xz^{15}w^{4}+968841928848769024xz^{14}w^{5}+1312527292604645376xz^{13}w^{6}+1440254794596007936xz^{12}w^{7}+1293854459018956800xz^{11}w^{8}+955938070985734144xz^{10}w^{9}+580560401040485376xz^{9}w^{10}+288282242809111040xz^{8}w^{11}+115819909564710400xz^{7}w^{12}+37033047790611712xz^{6}w^{13}+9205096639337088xz^{5}w^{14}+1725411532791744xz^{4}w^{15}+236719814961880xz^{3}w^{16}+23882639113988xz^{2}w^{17}+1929535865466xzw^{18}+112247335431xw^{19}-1381244105916416z^{20}-13812441059164160z^{19}w-69854833888329728z^{18}w^{2}-235038934808788992z^{17}w^{3}-585447659071930368z^{16}w^{4}-1140599468420235264z^{15}w^{5}-1794978887010254848z^{14}w^{6}-2327946748994879488z^{13}w^{7}-2519833495001919488z^{12}w^{8}-2293893593667149824z^{11}w^{9}-1762975681900633088z^{10}w^{10}-1144631492859051008z^{9}w^{11}-626293756583505920z^{8}w^{12}-287146107091859456z^{7}w^{13}-109253881401404544z^{6}w^{14}-33983026553789824z^{5}w^{15}-8445448481203976z^{4}w^{16}-1617717860445072z^{3}w^{17}-224873415587186z^{2}w^{18}-20229545322474zw^{19}-884483118531w^{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.