↑

Properties

Label 40.0.22971935915...5625.2
Degree $40$
Signature $[0, 20]$
Discriminant $5^{30}\cdot 7^{20}\cdot 11^{36}$
Root discriminant $76.56$
Ramified primes $5, 7, 11$
Class number Not computed
Class group Not computed
Galois group $C_2\times C_{20}$ (as 40T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25937424601, 21221529219, 38584598580, 46574338690, 65388316510, 86168888091, 116259627649, 155444907860, 208525783180, 279391609420, 374504727081, -148285290440, 140732503379, -65624674550, 40886456620, -19553743119, 10626053880, -5055544111, 2618352220, -1182457980, 668412581, -199019700, 102866780, -31939171, 15359600, -4823929, 2172130, -662420, 277429, -73920, 29261, -1540, 4180, -380, 589, -69, 80, -10, 10, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 + 10*x^38 - 10*x^37 + 80*x^36 - 69*x^35 + 589*x^34 - 380*x^33 + 4180*x^32 - 1540*x^31 + 29261*x^30 - 73920*x^29 + 277429*x^28 - 662420*x^27 + 2172130*x^26 - 4823929*x^25 + 15359600*x^24 - 31939171*x^23 + 102866780*x^22 - 199019700*x^21 + 668412581*x^20 - 1182457980*x^19 + 2618352220*x^18 - 5055544111*x^17 + 10626053880*x^16 - 19553743119*x^15 + 40886456620*x^14 - 65624674550*x^13 + 140732503379*x^12 - 148285290440*x^11 + 374504727081*x^10 + 279391609420*x^9 + 208525783180*x^8 + 155444907860*x^7 + 116259627649*x^6 + 86168888091*x^5 + 65388316510*x^4 + 46574338690*x^3 + 38584598580*x^2 + 21221529219*x + 25937424601)
 
gp: K = bnfinit(x^40 - x^39 + 10*x^38 - 10*x^37 + 80*x^36 - 69*x^35 + 589*x^34 - 380*x^33 + 4180*x^32 - 1540*x^31 + 29261*x^30 - 73920*x^29 + 277429*x^28 - 662420*x^27 + 2172130*x^26 - 4823929*x^25 + 15359600*x^24 - 31939171*x^23 + 102866780*x^22 - 199019700*x^21 + 668412581*x^20 - 1182457980*x^19 + 2618352220*x^18 - 5055544111*x^17 + 10626053880*x^16 - 19553743119*x^15 + 40886456620*x^14 - 65624674550*x^13 + 140732503379*x^12 - 148285290440*x^11 + 374504727081*x^10 + 279391609420*x^9 + 208525783180*x^8 + 155444907860*x^7 + 116259627649*x^6 + 86168888091*x^5 + 65388316510*x^4 + 46574338690*x^3 + 38584598580*x^2 + 21221529219*x + 25937424601, 1)
 

Normalized defining polynomial

\( x^{40} - x^{39} + 10 x^{38} - 10 x^{37} + 80 x^{36} - 69 x^{35} + 589 x^{34} - 380 x^{33} + 4180 x^{32} - 1540 x^{31} + 29261 x^{30} - 73920 x^{29} + 277429 x^{28} - 662420 x^{27} + 2172130 x^{26} - 4823929 x^{25} + 15359600 x^{24} - 31939171 x^{23} + 102866780 x^{22} - 199019700 x^{21} + 668412581 x^{20} - 1182457980 x^{19} + 2618352220 x^{18} - 5055544111 x^{17} + 10626053880 x^{16} - 19553743119 x^{15} + 40886456620 x^{14} - 65624674550 x^{13} + 140732503379 x^{12} - 148285290440 x^{11} + 374504727081 x^{10} + 279391609420 x^{9} + 208525783180 x^{8} + 155444907860 x^{7} + 116259627649 x^{6} + 86168888091 x^{5} + 65388316510 x^{4} + 46574338690 x^{3} + 38584598580 x^{2} + 21221529219 x + 25937424601 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 20]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2297193591531201418681029625209696069890454162435033694840967655181884765625=5^{30}\cdot 7^{20}\cdot 11^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.56$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(385=5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{385}(1,·)$, $\chi_{385}(258,·)$, $\chi_{385}(134,·)$, $\chi_{385}(13,·)$, $\chi_{385}(272,·)$, $\chi_{385}(274,·)$, $\chi_{385}(281,·)$, $\chi_{385}(153,·)$, $\chi_{385}(27,·)$, $\chi_{385}(29,·)$, $\chi_{385}(36,·)$, $\chi_{385}(293,·)$, $\chi_{385}(167,·)$, $\chi_{385}(169,·)$, $\chi_{385}(48,·)$, $\chi_{385}(307,·)$, $\chi_{385}(309,·)$, $\chi_{385}(223,·)$, $\chi_{385}(188,·)$, $\chi_{385}(62,·)$, $\chi_{385}(64,·)$, $\chi_{385}(118,·)$, $\chi_{385}(71,·)$, $\chi_{385}(328,·)$, $\chi_{385}(202,·)$, $\chi_{385}(204,·)$, $\chi_{385}(141,·)$, $\chi_{385}(83,·)$, $\chi_{385}(342,·)$, $\chi_{385}(344,·)$, $\chi_{385}(351,·)$, $\chi_{385}(97,·)$, $\chi_{385}(316,·)$, $\chi_{385}(106,·)$, $\chi_{385}(237,·)$, $\chi_{385}(239,·)$, $\chi_{385}(211,·)$, $\chi_{385}(246,·)$, $\chi_{385}(377,·)$, $\chi_{385}(379,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $\frac{1}{70071654809032040280680531} a^{31} - \frac{26201730051702969994725441}{70071654809032040280680531} a^{30} - \frac{22574345016632633827658276}{70071654809032040280680531} a^{29} - \frac{3026261021597975282829091}{70071654809032040280680531} a^{28} - \frac{15528485739230217434541187}{70071654809032040280680531} a^{27} + \frac{3270407154855365101454204}{70071654809032040280680531} a^{26} + \frac{7983012371617832265773799}{70071654809032040280680531} a^{25} - \frac{14945193584381734871756114}{70071654809032040280680531} a^{24} + \frac{613881201640431075821734}{6370150437184730934607321} a^{23} + \frac{3159682434703199923407719}{6370150437184730934607321} a^{22} + \frac{13912818260531829421591535}{70071654809032040280680531} a^{21} + \frac{2002834725302585644744390}{6370150437184730934607321} a^{20} - \frac{21435309116570467876217869}{70071654809032040280680531} a^{19} + \frac{149655631590741573560786}{6370150437184730934607321} a^{18} - \frac{9181047439462432441580680}{70071654809032040280680531} a^{17} + \frac{833281819126456000740879}{6370150437184730934607321} a^{16} + \frac{18665471226141573877177522}{70071654809032040280680531} a^{15} + \frac{3014847754433453463797164}{6370150437184730934607321} a^{14} - \frac{32045936381048123679839260}{70071654809032040280680531} a^{13} - \frac{2068356714841748753033941}{6370150437184730934607321} a^{12} - \frac{32368449746755216006852433}{70071654809032040280680531} a^{11} + \frac{1386453360902326416968042}{6370150437184730934607321} a^{10} + \frac{2421685169474903829973184}{6370150437184730934607321} a^{9} - \frac{17756281011675776076232431}{70071654809032040280680531} a^{8} - \frac{23025954351562915283061914}{70071654809032040280680531} a^{7} + \frac{5277055164346953591862700}{70071654809032040280680531} a^{6} - \frac{34696185658670176726142290}{70071654809032040280680531} a^{5} + \frac{203529293128018063225574}{70071654809032040280680531} a^{4} - \frac{11690205874844958158060830}{70071654809032040280680531} a^{3} - \frac{6433034177690757915276659}{70071654809032040280680531} a^{2} + \frac{4424367951482079926602421}{70071654809032040280680531} a - \frac{2693549184604331156432734}{6370150437184730934607321}$, $\frac{1}{770788202899352443087485841} a^{32} - \frac{1}{770788202899352443087485841} a^{31} + \frac{72537206757948725022156354}{770788202899352443087485841} a^{30} + \frac{72598931596962433543482086}{770788202899352443087485841} a^{29} - \frac{190552273674776351431560761}{770788202899352443087485841} a^{28} - \frac{44798886929728107652664950}{770788202899352443087485841} a^{27} - \frac{273114060309328242408791337}{770788202899352443087485841} a^{26} - \frac{331270023999741329065085485}{770788202899352443087485841} a^{25} + \frac{30628792015466044304239374}{70071654809032040280680531} a^{24} + \frac{10175609597338514107290075}{70071654809032040280680531} a^{23} - \frac{140009660232056336499500821}{770788202899352443087485841} a^{22} - \frac{28748238134089857512415868}{70071654809032040280680531} a^{21} + \frac{211386208148365155387249535}{770788202899352443087485841} a^{20} - \frac{18946487464198307484214387}{70071654809032040280680531} a^{19} - \frac{338363737221143102038722511}{770788202899352443087485841} a^{18} - \frac{933349879190217895305924}{70071654809032040280680531} a^{17} - \frac{298863508812672683909734064}{770788202899352443087485841} a^{16} + \frac{20409220168172803873033787}{70071654809032040280680531} a^{15} - \frac{55461732670564890087802929}{770788202899352443087485841} a^{14} + \frac{24538771662453375159094901}{70071654809032040280680531} a^{13} - \frac{85799906129678689627765701}{770788202899352443087485841} a^{12} + \frac{2621944373784479289572993}{6370150437184730934607321} a^{11} - \frac{16084908454658123216754535}{70071654809032040280680531} a^{10} + \frac{61629092627923143916400773}{770788202899352443087485841} a^{9} - \frac{120547039456499470707265448}{770788202899352443087485841} a^{8} + \frac{311659170873718784749249113}{770788202899352443087485841} a^{7} + \frac{333565027783374740272781706}{770788202899352443087485841} a^{6} - \frac{62264066834907940062823065}{770788202899352443087485841} a^{5} + \frac{258205462676071855253647795}{770788202899352443087485841} a^{4} + \frac{296828712047927600569211413}{770788202899352443087485841} a^{3} + \frac{109793153603482822410069914}{770788202899352443087485841} a^{2} + \frac{15900157618275683552210526}{70071654809032040280680531} a - \frac{1044241613838690604449549}{6370150437184730934607321}$, $\frac{1}{8478670231892876873962344251} a^{33} - \frac{1}{8478670231892876873962344251} a^{32} + \frac{10}{8478670231892876873962344251} a^{31} - \frac{4175034092594407132281329170}{8478670231892876873962344251} a^{30} - \frac{2925750618615504206789052106}{8478670231892876873962344251} a^{29} - \frac{734875287162652487893533319}{8478670231892876873962344251} a^{28} - \frac{3821495490476411446003488518}{8478670231892876873962344251} a^{27} - \frac{156103106774274098969847939}{8478670231892876873962344251} a^{26} - \frac{17209714119443468860130107}{770788202899352443087485841} a^{25} - \frac{189949242315862159871154334}{770788202899352443087485841} a^{24} - \frac{1376148757542745013072161927}{8478670231892876873962344251} a^{23} - \frac{221059451947888592875047848}{770788202899352443087485841} a^{22} - \frac{1240273731596994069305434381}{8478670231892876873962344251} a^{21} - \frac{142492934274225774963794659}{770788202899352443087485841} a^{20} - \frac{2156727719022946077260699025}{8478670231892876873962344251} a^{19} - \frac{379636119644519743334960446}{770788202899352443087485841} a^{18} + \frac{1258998863087539270207069133}{8478670231892876873962344251} a^{17} + \frac{125588366140381325883311862}{770788202899352443087485841} a^{16} - \frac{3910453195311388893146203164}{8478670231892876873962344251} a^{15} - \frac{245215490822909939515744054}{770788202899352443087485841} a^{14} + \frac{2233322755836082376846997}{8478670231892876873962344251} a^{13} + \frac{13547058777052324801892778}{70071654809032040280680531} a^{12} + \frac{101679334089339653720465574}{770788202899352443087485841} a^{11} + \frac{932097468137488571321046180}{8478670231892876873962344251} a^{10} - \frac{1573673631075708671766068832}{8478670231892876873962344251} a^{9} + \frac{1432245261341090331915464720}{8478670231892876873962344251} a^{8} - \frac{3436421660268391790346959357}{8478670231892876873962344251} a^{7} + \frac{389164415037329321122358620}{8478670231892876873962344251} a^{6} - \frac{1116342063550247057579053201}{8478670231892876873962344251} a^{5} + \frac{329680141183075553299255903}{8478670231892876873962344251} a^{4} - \frac{2988321642418532983442331209}{8478670231892876873962344251} a^{3} + \frac{9656909781476707867323178}{770788202899352443087485841} a^{2} - \frac{27070798246031125210713250}{70071654809032040280680531} a - \frac{3103829742186863509916180}{6370150437184730934607321}$, $\frac{1}{93265372550821645613585786761} a^{34} - \frac{1}{93265372550821645613585786761} a^{33} + \frac{10}{93265372550821645613585786761} a^{32} - \frac{10}{93265372550821645613585786761} a^{31} + \frac{2419832083870625678827110926}{93265372550821645613585786761} a^{30} - \frac{2439975451311186939065017677}{93265372550821645613585786761} a^{29} + \frac{24218464206146818048509015811}{93265372550821645613585786761} a^{28} - \frac{24399754513111869390650176460}{93265372550821645613585786761} a^{27} + \frac{659750480219306880125124178}{8478670231892876873962344251} a^{26} + \frac{1631896538120474415369852968}{8478670231892876873962344251} a^{25} + \frac{27690401490872567581797082718}{93265372550821645613585786761} a^{24} + \frac{3788445517652315376758803529}{8478670231892876873962344251} a^{23} + \frac{9472544238603536877544963654}{93265372550821645613585786761} a^{22} + \frac{4025109464633164267604997637}{8478670231892876873962344251} a^{21} + \frac{18173011333626358240017551591}{93265372550821645613585786761} a^{20} + \frac{2632511769030852729188899333}{8478670231892876873962344251} a^{19} - \frac{46363042371896810636182819897}{93265372550821645613585786761} a^{18} - \frac{1499948701093129217651172354}{8478670231892876873962344251} a^{17} + \frac{33009085032404187672368630331}{93265372550821645613585786761} a^{16} + \frac{1395310333435272723501906897}{8478670231892876873962344251} a^{15} - \frac{9625711549256705751758854394}{93265372550821645613585786761} a^{14} - \frac{219178637789227186194573077}{770788202899352443087485841} a^{13} + \frac{2690186659263692728808352613}{8478670231892876873962344251} a^{12} + \frac{35850691211582496589172336887}{93265372550821645613585786761} a^{11} + \frac{4409705994056795452726102520}{93265372550821645613585786761} a^{10} + \frac{36913408117357828304556522788}{93265372550821645613585786761} a^{9} + \frac{21217459144325258182745115440}{93265372550821645613585786761} a^{8} + \frac{37846651892268340902971605222}{93265372550821645613585786761} a^{7} - \frac{29502475225853540565172922695}{93265372550821645613585786761} a^{6} - \frac{31498759837383470240669915537}{93265372550821645613585786761} a^{5} - \frac{33530328201639521115752424799}{93265372550821645613585786761} a^{4} + \frac{78160665979193642394280989}{8478670231892876873962344251} a^{3} + \frac{150427979007238270938193025}{770788202899352443087485841} a^{2} - \frac{34061562451084709009936408}{70071654809032040280680531} a + \frac{1784802240715883746514127}{6370150437184730934607321}$, $\frac{1}{1025919098059038101749443654371} a^{35} - \frac{1}{1025919098059038101749443654371} a^{34} + \frac{10}{1025919098059038101749443654371} a^{33} - \frac{10}{1025919098059038101749443654371} a^{32} + \frac{80}{1025919098059038101749443654371} a^{31} - \frac{149276798040437418035728161682}{1025919098059038101749443654371} a^{30} - \frac{302015401386238314445710745909}{1025919098059038101749443654371} a^{29} - \frac{15556682918660346126399054028}{1025919098059038101749443654371} a^{28} + \frac{5236661846793741890111227483}{93265372550821645613585786761} a^{27} - \frac{22526196595906309617978024443}{93265372550821645613585786761} a^{26} - \frac{155299437867924210313778109047}{1025919098059038101749443654371} a^{25} + \frac{21880923059481740119930123280}{93265372550821645613585786761} a^{24} - \frac{398438248686232048028742665319}{1025919098059038101749443654371} a^{23} - \frac{19186537578898410558601070115}{93265372550821645613585786761} a^{22} + \frac{499612874627652669210569187631}{1025919098059038101749443654371} a^{21} - \frac{38660294439969953389704981709}{93265372550821645613585786761} a^{20} - \frac{458525840325876046847090507372}{1025919098059038101749443654371} a^{19} + \frac{13482863827005162760768841693}{93265372550821645613585786761} a^{18} - \frac{261206560723650636146558571280}{1025919098059038101749443654371} a^{17} + \frac{8666813434778680408570437326}{93265372550821645613585786761} a^{16} - \frac{171282624073425988176965660891}{1025919098059038101749443654371} a^{15} + \frac{1031147757959911028939776305}{8478670231892876873962344251} a^{14} + \frac{29800278261841415771806433318}{93265372550821645613585786761} a^{13} + \frac{256727001387109517494719073924}{1025919098059038101749443654371} a^{12} - \frac{454985268593008986120505435722}{1025919098059038101749443654371} a^{11} + \frac{239621770918500818465197120081}{1025919098059038101749443654371} a^{10} - \frac{500184355159216835738634535886}{1025919098059038101749443654371} a^{9} + \frac{380801974610836829307235070295}{1025919098059038101749443654371} a^{8} + \frac{72254859520030881427422649175}{1025919098059038101749443654371} a^{7} + \frac{321140627677192464570588457099}{1025919098059038101749443654371} a^{6} + \frac{25126903720659209421318023538}{1025919098059038101749443654371} a^{5} + \frac{32048168944925807630058164914}{93265372550821645613585786761} a^{4} - \frac{683997740925080307471222628}{8478670231892876873962344251} a^{3} + \frac{182271755784200946234540591}{770788202899352443087485841} a^{2} - \frac{9763175897401575441019207}{70071654809032040280680531} a - \frac{1518076666599768853337068}{6370150437184730934607321}$, $\frac{1}{11285110078649419119243880198081} a^{36} - \frac{1}{11285110078649419119243880198081} a^{35} + \frac{10}{11285110078649419119243880198081} a^{34} - \frac{10}{11285110078649419119243880198081} a^{33} + \frac{80}{11285110078649419119243880198081} a^{32} - \frac{69}{11285110078649419119243880198081} a^{31} - \frac{1673274688705596825174582847871}{11285110078649419119243880198081} a^{30} - \frac{2561993238479103614791452476416}{11285110078649419119243880198081} a^{29} - \frac{263043801523508069545652627}{2448494267444005016108457409} a^{28} - \frac{277246566135654355402433124158}{1025919098059038101749443654371} a^{27} - \frac{1228415195405388660355233017931}{11285110078649419119243880198081} a^{26} + \frac{360241096612889404846806463284}{1025919098059038101749443654371} a^{25} + \frac{4725589060568530394273835581371}{11285110078649419119243880198081} a^{24} - \frac{272379376852172350014735076811}{1025919098059038101749443654371} a^{23} + \frac{4422029775430642788416967554969}{11285110078649419119243880198081} a^{22} + \frac{128079182464518861805167146360}{1025919098059038101749443654371} a^{21} + \frac{4582799311744857104400426177715}{11285110078649419119243880198081} a^{20} - \frac{27903206965760036404953070448}{1025919098059038101749443654371} a^{19} - \frac{4304619749516690160483382211916}{11285110078649419119243880198081} a^{18} + \frac{429054075464049089007625339207}{1025919098059038101749443654371} a^{17} + \frac{4931608213286348897265561214991}{11285110078649419119243880198081} a^{16} + \frac{18012029379018709711886189946}{93265372550821645613585786761} a^{15} + \frac{299316826680108969840711543}{1025919098059038101749443654371} a^{14} + \frac{795628218435674863799317717012}{11285110078649419119243880198081} a^{13} - \frac{1543255969044549711730559193719}{11285110078649419119243880198081} a^{12} - \frac{3955358724017529233337989530461}{11285110078649419119243880198081} a^{11} - \frac{2809508367490637583393909856605}{11285110078649419119243880198081} a^{10} + \frac{995616120386599172203877442258}{11285110078649419119243880198081} a^{9} + \frac{4691128095580693127156875671147}{11285110078649419119243880198081} a^{8} + \frac{2124752423400469296847560191340}{11285110078649419119243880198081} a^{7} + \frac{964767012156240980025064865393}{11285110078649419119243880198081} a^{6} + \frac{389630910893086678617880476348}{1025919098059038101749443654371} a^{5} - \frac{45557471023986339943644063766}{93265372550821645613585786761} a^{4} + \frac{3807632078598336920818843675}{8478670231892876873962344251} a^{3} - \frac{283368693377483914954992988}{770788202899352443087485841} a^{2} - \frac{11527777780440455887088075}{70071654809032040280680531} a + \frac{1724642229208488782143458}{6370150437184730934607321}$, $\frac{1}{124136210865143610311682682178891} a^{37} - \frac{1}{124136210865143610311682682178891} a^{36} + \frac{10}{124136210865143610311682682178891} a^{35} - \frac{10}{124136210865143610311682682178891} a^{34} + \frac{80}{124136210865143610311682682178891} a^{33} - \frac{69}{124136210865143610311682682178891} a^{32} + \frac{589}{124136210865143610311682682178891} a^{31} + \frac{14760354742941567596909779173131}{124136210865143610311682682178891} a^{30} + \frac{4861969756406215897855907051353}{11285110078649419119243880198081} a^{29} - \frac{4070425954467079710537240652685}{11285110078649419119243880198081} a^{28} + \frac{38271829744109307517419046920727}{124136210865143610311682682178891} a^{27} + \frac{1343123323922201474611072233791}{11285110078649419119243880198081} a^{26} - \frac{40917344557835411989288539220231}{124136210865143610311682682178891} a^{25} + \frac{1614988626589535283100070381440}{11285110078649419119243880198081} a^{24} - \frac{53360839704189981164397581266152}{124136210865143610311682682178891} a^{23} + \frac{4989029374261792617710545663145}{11285110078649419119243880198081} a^{22} - \frac{49288427151179634676996016683753}{124136210865143610311682682178891} a^{21} - \frac{756603189709353235696729821419}{11285110078649419119243880198081} a^{20} + \frac{14651293031144003700519466350217}{124136210865143610311682682178891} a^{19} - \frac{4540455335687845212384127580736}{11285110078649419119243880198081} a^{18} + \frac{28394574528894290195467120846396}{124136210865143610311682682178891} a^{17} - \frac{51376923938644386563230425106}{1025919098059038101749443654371} a^{16} - \frac{2684183315302383273389842763970}{11285110078649419119243880198081} a^{15} + \frac{33705547486558699627517769276510}{124136210865143610311682682178891} a^{14} - \frac{47048431362797521252195054265875}{124136210865143610311682682178891} a^{13} + \frac{28910685424081942038237557758630}{124136210865143610311682682178891} a^{12} + \frac{51653329976213429108155169598564}{124136210865143610311682682178891} a^{11} + \frac{35026988548918713621945310689076}{124136210865143610311682682178891} a^{10} - \frac{33780213606923742647091989338615}{124136210865143610311682682178891} a^{9} + \frac{105357989970142032909763681307}{296267806360724606949123346489} a^{8} + \frac{27629696563226242862804226973548}{124136210865143610311682682178891} a^{7} + \frac{4796829064471878228157849104041}{11285110078649419119243880198081} a^{6} - \frac{233886014113525471216768841985}{1025919098059038101749443654371} a^{5} - \frac{38679839207727545335079788113}{93265372550821645613585786761} a^{4} - \frac{2203373280303025901051123991}{8478670231892876873962344251} a^{3} - \frac{154251033472266946476367512}{770788202899352443087485841} a^{2} - \frac{25474080242984796122173326}{70071654809032040280680531} a - \frac{1365649249621211133523729}{6370150437184730934607321}$, $\frac{1}{1365498319516579713428509503967801} a^{38} - \frac{1}{1365498319516579713428509503967801} a^{37} + \frac{10}{1365498319516579713428509503967801} a^{36} - \frac{10}{1365498319516579713428509503967801} a^{35} + \frac{80}{1365498319516579713428509503967801} a^{34} - \frac{69}{1365498319516579713428509503967801} a^{33} + \frac{589}{1365498319516579713428509503967801} a^{32} - \frac{380}{1365498319516579713428509503967801} a^{31} - \frac{23167299968495341713604399375652}{124136210865143610311682682178891} a^{30} + \frac{48383352463081334496313516966705}{124136210865143610311682682178891} a^{29} - \frac{94782934941774082249265216897600}{1365498319516579713428509503967801} a^{28} - \frac{252477730885070396359155508258}{124136210865143610311682682178891} a^{27} - \frac{84565566986141070544813257168880}{1365498319516579713428509503967801} a^{26} + \frac{16329982427509950377316534240058}{124136210865143610311682682178891} a^{25} + \frac{76897079424013846448559799395500}{1365498319516579713428509503967801} a^{24} - \frac{50569766914116780860599446510751}{124136210865143610311682682178891} a^{23} - \frac{301264310507293069434613687102220}{1365498319516579713428509503967801} a^{22} - \frac{47909408621502614515359631825313}{124136210865143610311682682178891} a^{21} + \frac{156339819208655799194986792050420}{1365498319516579713428509503967801} a^{20} - \frac{24837196002927654530461744275555}{124136210865143610311682682178891} a^{19} - \frac{295753420583890006704393450138345}{1365498319516579713428509503967801} a^{18} - \frac{3479549137954942945553393964431}{11285110078649419119243880198081} a^{17} + \frac{43565357023257064647466718089445}{124136210865143610311682682178891} a^{16} + \frac{171566951160214699302078591751199}{1365498319516579713428509503967801} a^{15} - \frac{491032949217647233447720088543111}{1365498319516579713428509503967801} a^{14} + \frac{55399201447420280369123840074210}{1365498319516579713428509503967801} a^{13} - \frac{9011466678939629452313176217457}{1365498319516579713428509503967801} a^{12} - \frac{336935447949145786863879553394637}{1365498319516579713428509503967801} a^{11} - \frac{671704095323307354102451017759370}{1365498319516579713428509503967801} a^{10} - \frac{320212713217925049946626781630184}{1365498319516579713428509503967801} a^{9} + \frac{499865662440339197429205433959811}{1365498319516579713428509503967801} a^{8} - \frac{59416005169717423546242011605974}{124136210865143610311682682178891} a^{7} + \frac{832309188503651962638550960420}{11285110078649419119243880198081} a^{6} - \frac{386143017427432699790948997467}{1025919098059038101749443654371} a^{5} - \frac{27448773960337968585787670463}{93265372550821645613585786761} a^{4} - \frac{1393989065223823612446516129}{8478670231892876873962344251} a^{3} + \frac{244238464223399112398243966}{770788202899352443087485841} a^{2} - \frac{21377700859353003541346847}{70071654809032040280680531} a - \frac{544078906117657956509072}{6370150437184730934607321}$, $\frac{1}{15020481514682376847713604543645811} a^{39} - \frac{1}{15020481514682376847713604543645811} a^{38} + \frac{10}{15020481514682376847713604543645811} a^{37} - \frac{10}{15020481514682376847713604543645811} a^{36} + \frac{80}{15020481514682376847713604543645811} a^{35} - \frac{69}{15020481514682376847713604543645811} a^{34} + \frac{589}{15020481514682376847713604543645811} a^{33} - \frac{380}{15020481514682376847713604543645811} a^{32} + \frac{380}{1365498319516579713428509503967801} a^{31} - \frac{307606871933658114483141266920269}{1365498319516579713428509503967801} a^{30} - \frac{770294465817006602039054714913575}{15020481514682376847713604543645811} a^{29} - \frac{203252019381446925314740017442409}{1365498319516579713428509503967801} a^{28} - \frac{5108995271224542878293128580216665}{15020481514682376847713604543645811} a^{27} + \frac{23122286308953896229896652082067}{1365498319516579713428509503967801} a^{26} + \frac{2197912673533955643957737916981356}{15020481514682376847713604543645811} a^{25} - \frac{570526173648268273730816357148128}{1365498319516579713428509503967801} a^{24} - \frac{5577832788101301891638823141905306}{15020481514682376847713604543645811} a^{23} - \frac{352716177753665832771149161596877}{1365498319516579713428509503967801} a^{22} - \frac{6815895620214449265443867594875207}{15020481514682376847713604543645811} a^{21} + \frac{522786132346604796863006916200539}{1365498319516579713428509503967801} a^{20} - \frac{6544275648381567551403248176228098}{15020481514682376847713604543645811} a^{19} + \frac{47080188422620432023018733994146}{124136210865143610311682682178891} a^{18} - \frac{427529025580938789865773062705250}{1365498319516579713428509503967801} a^{17} - \frac{6100532317019803830607747262352113}{15020481514682376847713604543645811} a^{16} - \frac{3090372979156503913882281949208431}{15020481514682376847713604543645811} a^{15} - \frac{6598707050218697303800866129530552}{15020481514682376847713604543645811} a^{14} + \frac{5652534509290274872576352900112528}{15020481514682376847713604543645811} a^{13} + \frac{4292563742915510000387743364401490}{15020481514682376847713604543645811} a^{12} + \frac{6165504644767854323583134468483224}{15020481514682376847713604543645811} a^{11} + \frac{5721342030666111397767213900520900}{15020481514682376847713604543645811} a^{10} - \frac{3817569173073609775796404393087904}{15020481514682376847713604543645811} a^{9} + \frac{440408503008097341687875780880109}{1365498319516579713428509503967801} a^{8} + \frac{47059279814549719142066015593364}{124136210865143610311682682178891} a^{7} + \frac{3734491706700705469691521312726}{11285110078649419119243880198081} a^{6} - \frac{226329053623662461786204094514}{1025919098059038101749443654371} a^{5} + \frac{5872935521602027822134949902}{93265372550821645613585786761} a^{4} - \frac{2036608358491015507455460165}{8478670231892876873962344251} a^{3} + \frac{139738939892965074209865386}{770788202899352443087485841} a^{2} - \frac{24502282832757507622578070}{70071654809032040280680531} a + \frac{807806728804590724144908}{6370150437184730934607321}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{263784942664060407864840}{15020481514682376847713604543645811} a^{39} + \frac{586188761475689795255200}{15020481514682376847713604543645811} a^{38} - \frac{2960253245452233466038760}{15020481514682376847713604543645811} a^{37} + \frac{5861887614756897952552000}{15020481514682376847713604543645811} a^{36} - \frac{24326833601241126503090800}{15020481514682376847713604543645811} a^{35} + \frac{43993466548750519133902760}{15020481514682376847713604543645811} a^{34} - \frac{177615194727134007962325600}{15020481514682376847713604543645811} a^{33} + \frac{290134127492392664161561240}{15020481514682376847713604543645811} a^{32} - \frac{111376586708687315474438280}{1365498319516579713428509503967801} a^{31} + \frac{159443343121387624309414400}{1365498319516579713428509503967801} a^{30} - \frac{8215113088262980851114237640}{15020481514682376847713604543645811} a^{29} + \frac{2630258282179493895799845160}{1365498319516579713428509503967801} a^{28} - \frac{97013683144903259209430107560}{15020481514682376847713604543645811} a^{27} + \frac{24016417342601674972013408840}{1365498319516579713428509503967801} a^{26} - \frac{786541925166085072530577180400}{15020481514682376847713604543645811} a^{25} + \frac{16303990426742171904820267960}{124136210865143610311682682178891} a^{24} - \frac{5606884336619066779725388388440}{15020481514682376847713604543645811} a^{23} + \frac{1216096916944702150880531191240}{1365498319516579713428509503967801} a^{22} - \frac{37432018364414163675028717806760}{15020481514682376847713604543645811} a^{21} + \frac{7787599980716545537936199183239}{1365498319516579713428509503967801} a^{20} - \frac{240481885653766470350463633644040}{15020481514682376847713604543645811} a^{19} + \frac{47946670828463476374806745594760}{1365498319516579713428509503967801} a^{18} - \frac{97446441687572779006295621001600}{1365498319516579713428509503967801} a^{17} + \frac{2177743168177670814620139058556440}{15020481514682376847713604543645811} a^{16} - \frac{4432919741038060601282437753749160}{15020481514682376847713604543645811} a^{15} + \frac{8583863357323312169888806584632760}{15020481514682376847713604543645811} a^{14} - \frac{17089433068870513990087724700173640}{15020481514682376847713604543645811} a^{13} + \frac{30492750765483399146723514534005200}{15020481514682376847713604543645811} a^{12} - \frac{58280761017989548914406962282632360}{15020481514682376847713604543645811} a^{11} + \frac{84488123356949081455352198356156040}{15020481514682376847713604543645811} a^{10} - \frac{146582087528498194757414002254809640}{15020481514682376847713604543645811} a^{9} + \frac{388780615721051055604022489087160}{124136210865143610311682682178891} a^{8} + \frac{26349331375522625219170053800000}{11285110078649419119243880198081} a^{7} + \frac{1791235756481632529434762806400}{1025919098059038101749443654371} a^{6} + \frac{120759838159026567401341106440}{93265372550821645613585786761} a^{5} + \frac{8327397545523649231395371200}{8478670231892876873962344251} a^{4} + \frac{540495347518659775715057160}{770788202899352443087485841} a^{3} + \frac{41033213303298285667864000}{70071654809032040280680531} a^{2} + \frac{2051660665164914283393200}{6370150437184730934607321} a + \frac{2901634369304664486513240}{6370150437184730934607321} \) (order $22$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{20}$ (as 40T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2\times C_{20}$
Character table for $C_2\times C_{20}$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{5}, \sqrt{-11})\), 4.0.741125.2, 4.4.6125.1, \(\Q(\zeta_{11})^+\), 8.0.549266265625.1, \(\Q(\zeta_{11})\), 10.10.669871503125.1, 10.0.7368586534375.1, 20.0.54296067514572573056640625.1, 20.0.47929047471561558446078911407470703125.1, 20.20.396107830343483954099825714111328125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $20^{2}$ $20^{2}$ R R R $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ $20^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
5Data not computed
7Data not computed
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$