Properties

Label 44.0.342...125.1
Degree $44$
Signature $[0, 22]$
Discriminant $3.429\times 10^{77}$
Root discriminant $57.83$
Ramified primes $5, 23$
Class number $45013$ (GRH)
Class group $[45013]$ (GRH)
Galois group $C_{44}$ (as 44T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - x^43 + 11*x^42 - 12*x^41 + 77*x^40 - 74*x^39 + 424*x^38 - 368*x^37 + 2052*x^36 - 1675*x^35 + 8154*x^34 - 6259*x^33 + 27860*x^32 - 18614*x^31 + 82991*x^30 - 50150*x^29 + 218995*x^28 - 122605*x^27 + 487495*x^26 - 239465*x^25 + 942321*x^24 - 367121*x^23 + 1562162*x^22 - 556097*x^21 + 2230394*x^20 - 742219*x^19 + 2512113*x^18 - 607127*x^17 + 2369760*x^16 - 263798*x^15 + 1750478*x^14 - 273124*x^13 + 1076097*x^12 - 237409*x^11 + 402111*x^10 - 41911*x^9 + 123922*x^8 + 19642*x^7 + 17600*x^6 + 1439*x^5 + 2686*x^4 - 361*x^3 + 51*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^44 - x^43 + 11*x^42 - 12*x^41 + 77*x^40 - 74*x^39 + 424*x^38 - 368*x^37 + 2052*x^36 - 1675*x^35 + 8154*x^34 - 6259*x^33 + 27860*x^32 - 18614*x^31 + 82991*x^30 - 50150*x^29 + 218995*x^28 - 122605*x^27 + 487495*x^26 - 239465*x^25 + 942321*x^24 - 367121*x^23 + 1562162*x^22 - 556097*x^21 + 2230394*x^20 - 742219*x^19 + 2512113*x^18 - 607127*x^17 + 2369760*x^16 - 263798*x^15 + 1750478*x^14 - 273124*x^13 + 1076097*x^12 - 237409*x^11 + 402111*x^10 - 41911*x^9 + 123922*x^8 + 19642*x^7 + 17600*x^6 + 1439*x^5 + 2686*x^4 - 361*x^3 + 51*x^2 - 6*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 51, -361, 2686, 1439, 17600, 19642, 123922, -41911, 402111, -237409, 1076097, -273124, 1750478, -263798, 2369760, -607127, 2512113, -742219, 2230394, -556097, 1562162, -367121, 942321, -239465, 487495, -122605, 218995, -50150, 82991, -18614, 27860, -6259, 8154, -1675, 2052, -368, 424, -74, 77, -12, 11, -1, 1]);
 

\( x^{44} - x^{43} + 11 x^{42} - 12 x^{41} + 77 x^{40} - 74 x^{39} + 424 x^{38} - 368 x^{37} + 2052 x^{36} - 1675 x^{35} + 8154 x^{34} - 6259 x^{33} + 27860 x^{32} - 18614 x^{31} + 82991 x^{30} - 50150 x^{29} + 218995 x^{28} - 122605 x^{27} + 487495 x^{26} - 239465 x^{25} + 942321 x^{24} - 367121 x^{23} + 1562162 x^{22} - 556097 x^{21} + 2230394 x^{20} - 742219 x^{19} + 2512113 x^{18} - 607127 x^{17} + 2369760 x^{16} - 263798 x^{15} + 1750478 x^{14} - 273124 x^{13} + 1076097 x^{12} - 237409 x^{11} + 402111 x^{10} - 41911 x^{9} + 123922 x^{8} + 19642 x^{7} + 17600 x^{6} + 1439 x^{5} + 2686 x^{4} - 361 x^{3} + 51 x^{2} - 6 x + 1 \)

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

Invariants

Degree:  $44$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 22]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(342\!\cdots\!125\)\(\medspace = 5^{33}\cdot 23^{40}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $57.83$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $5, 23$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $44$
This field is Galois and abelian over $\Q$.
Conductor:  \(115=5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{115}(1,·)$, $\chi_{115}(2,·)$, $\chi_{115}(3,·)$, $\chi_{115}(4,·)$, $\chi_{115}(6,·)$, $\chi_{115}(8,·)$, $\chi_{115}(9,·)$, $\chi_{115}(12,·)$, $\chi_{115}(13,·)$, $\chi_{115}(16,·)$, $\chi_{115}(18,·)$, $\chi_{115}(24,·)$, $\chi_{115}(26,·)$, $\chi_{115}(27,·)$, $\chi_{115}(29,·)$, $\chi_{115}(31,·)$, $\chi_{115}(32,·)$, $\chi_{115}(36,·)$, $\chi_{115}(39,·)$, $\chi_{115}(41,·)$, $\chi_{115}(47,·)$, $\chi_{115}(48,·)$, $\chi_{115}(49,·)$, $\chi_{115}(52,·)$, $\chi_{115}(54,·)$, $\chi_{115}(58,·)$, $\chi_{115}(59,·)$, $\chi_{115}(62,·)$, $\chi_{115}(64,·)$, $\chi_{115}(71,·)$, $\chi_{115}(72,·)$, $\chi_{115}(73,·)$, $\chi_{115}(77,·)$, $\chi_{115}(78,·)$, $\chi_{115}(81,·)$, $\chi_{115}(82,·)$, $\chi_{115}(87,·)$, $\chi_{115}(93,·)$, $\chi_{115}(94,·)$, $\chi_{115}(96,·)$, $\chi_{115}(98,·)$, $\chi_{115}(101,·)$, $\chi_{115}(104,·)$, $\chi_{115}(108,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $\frac{1}{9136101057353736188751549898001468069294153521} a^{41} - \frac{91469180207132320080395725856741575384897350}{9136101057353736188751549898001468069294153521} a^{40} + \frac{2289129945262956177859247996641404220316711707}{9136101057353736188751549898001468069294153521} a^{39} - \frac{3203821747334279378663205255208819974165685198}{9136101057353736188751549898001468069294153521} a^{38} - \frac{2136404593961558289722526005092216209013734481}{9136101057353736188751549898001468069294153521} a^{37} - \frac{1406136227782890777704883543524925529089672813}{9136101057353736188751549898001468069294153521} a^{36} - \frac{570145232256253541880002479533867780584301332}{9136101057353736188751549898001468069294153521} a^{35} + \frac{4187413449841243524958293866906604900531362708}{9136101057353736188751549898001468069294153521} a^{34} + \frac{4116407449743690372987503140087237522686191713}{9136101057353736188751549898001468069294153521} a^{33} + \frac{1293099043434934388272235428387218306788688402}{9136101057353736188751549898001468069294153521} a^{32} - \frac{2552182248598676316811903091390125553373255984}{9136101057353736188751549898001468069294153521} a^{31} - \frac{2395501667648985635880662163375027555470418658}{9136101057353736188751549898001468069294153521} a^{30} + \frac{4079640432159846622150927392123979849709099559}{9136101057353736188751549898001468069294153521} a^{29} + \frac{2099127501227683328277110849450451588535116139}{9136101057353736188751549898001468069294153521} a^{28} - \frac{164540272791394335953031962046222545047953956}{9136101057353736188751549898001468069294153521} a^{27} + \frac{1366742080384721839785200078061376639502294412}{9136101057353736188751549898001468069294153521} a^{26} - \frac{4189230470914512193878956854475010161345069029}{9136101057353736188751549898001468069294153521} a^{25} + \frac{1759128141727612470883254643664977417433230765}{9136101057353736188751549898001468069294153521} a^{24} + \frac{3916464714661264454012823818256777634506277990}{9136101057353736188751549898001468069294153521} a^{23} - \frac{3184633854648360241305689179897294364342203157}{9136101057353736188751549898001468069294153521} a^{22} - \frac{1015505512066771080024287419983807202011282961}{9136101057353736188751549898001468069294153521} a^{21} + \frac{4287719644557700820914104231650678108439489985}{9136101057353736188751549898001468069294153521} a^{20} - \frac{306147141228741607699174042993665651864463143}{9136101057353736188751549898001468069294153521} a^{19} - \frac{1231348862884712351400587937249897551279731502}{9136101057353736188751549898001468069294153521} a^{18} + \frac{4174365182456468004649790236151204701645903080}{9136101057353736188751549898001468069294153521} a^{17} + \frac{2212130220418054149121137730326153445597322283}{9136101057353736188751549898001468069294153521} a^{16} - \frac{1114407700791493483946544220664915776534496991}{9136101057353736188751549898001468069294153521} a^{15} - \frac{4530347128877176843085036811289415434060293182}{9136101057353736188751549898001468069294153521} a^{14} + \frac{1950612659225373502496096100334147867229955342}{9136101057353736188751549898001468069294153521} a^{13} + \frac{711060184499673743057033430973832784676867975}{9136101057353736188751549898001468069294153521} a^{12} + \frac{4485722226768852360812919543410547561797574082}{9136101057353736188751549898001468069294153521} a^{11} - \frac{383854775203526757684319264923548072389845732}{9136101057353736188751549898001468069294153521} a^{10} + \frac{242381488506597388885175169034623315420007871}{9136101057353736188751549898001468069294153521} a^{9} - \frac{149378775012994690504326583567297407577649653}{9136101057353736188751549898001468069294153521} a^{8} - \frac{973643525434843872515508512793313234813990815}{9136101057353736188751549898001468069294153521} a^{7} + \frac{3306324454897637996608334785017184483729684173}{9136101057353736188751549898001468069294153521} a^{6} + \frac{3115477542344041122376504207097545491186697236}{9136101057353736188751549898001468069294153521} a^{5} - \frac{909606681250045857498807044816441060199696530}{9136101057353736188751549898001468069294153521} a^{4} - \frac{4482724494774010415869898130247492926996226272}{9136101057353736188751549898001468069294153521} a^{3} + \frac{2913805838646425958909220361360480704350691960}{9136101057353736188751549898001468069294153521} a^{2} - \frac{2266590740732167390625607531936040526081888584}{9136101057353736188751549898001468069294153521} a - \frac{2050581202786043264480017475916088163156792615}{9136101057353736188751549898001468069294153521}$, $\frac{1}{9136101057353736188751549898001468069294153521} a^{42} + \frac{2565125985016985637884896413421912466822318889}{9136101057353736188751549898001468069294153521} a^{40} - \frac{2123712611314610591495483268313115622027543315}{9136101057353736188751549898001468069294153521} a^{39} + \frac{366669289423258404089797708527836082368271606}{9136101057353736188751549898001468069294153521} a^{38} - \frac{3053145937995254171254947666007524863690253553}{9136101057353736188751549898001468069294153521} a^{37} + \frac{359618157450933596681518234150868352604023312}{9136101057353736188751549898001468069294153521} a^{36} - \frac{1130578256422052890277692741698837950499190131}{9136101057353736188751549898001468069294153521} a^{35} + \frac{352549567703600624591958531885904105856767397}{9136101057353736188751549898001468069294153521} a^{34} - \frac{1876459210387661425607979545802798630695275146}{9136101057353736188751549898001468069294153521} a^{33} + \frac{3344709247143556921917304109088839821394122336}{9136101057353736188751549898001468069294153521} a^{32} - \frac{4034242288434141612996363365417933643667183402}{9136101057353736188751549898001468069294153521} a^{31} + \frac{3229091622988478326698886761060617256226908363}{9136101057353736188751549898001468069294153521} a^{30} + \frac{300925773876833097975908151791703481339854981}{9136101057353736188751549898001468069294153521} a^{29} - \frac{3910865301089576151999720482820314060128807508}{9136101057353736188751549898001468069294153521} a^{28} + \frac{3014829150976953864257994275811650485603550678}{9136101057353736188751549898001468069294153521} a^{27} + \frac{1529913130882469948920943280111145085858710303}{9136101057353736188751549898001468069294153521} a^{26} + \frac{2690564548575430733773007017108624416938332092}{9136101057353736188751549898001468069294153521} a^{25} + \frac{2802914258878304333182499315152059783785003913}{9136101057353736188751549898001468069294153521} a^{24} + \frac{2159285686723778209417723268465419941361722675}{9136101057353736188751549898001468069294153521} a^{23} + \frac{352188530849166906643445846701550595446783459}{9136101057353736188751549898001468069294153521} a^{22} - \frac{420320337290451013360218519281877468473194911}{9136101057353736188751549898001468069294153521} a^{21} - \frac{1008197737873853180007625288484189476354344118}{9136101057353736188751549898001468069294153521} a^{20} + \frac{1276836662488979405693962811689912820306663384}{9136101057353736188751549898001468069294153521} a^{19} + \frac{4374407174031424766271179600760029444762819235}{9136101057353736188751549898001468069294153521} a^{18} - \frac{1481843141949093014852883345911291633774141968}{9136101057353736188751549898001468069294153521} a^{17} - \frac{2660491640480176168479839133007898735390985380}{9136101057353736188751549898001468069294153521} a^{16} + \frac{2158952680562424755864762257646363025165181665}{9136101057353736188751549898001468069294153521} a^{15} + \frac{4223526245230505936865822546093969927931459033}{9136101057353736188751549898001468069294153521} a^{14} + \frac{3959674643514324438316951018802700367886483844}{9136101057353736188751549898001468069294153521} a^{13} + \frac{3322565402947303450191247546144897285812884856}{9136101057353736188751549898001468069294153521} a^{12} + \frac{2529888588389964844172502643274337008656685664}{9136101057353736188751549898001468069294153521} a^{11} - \frac{2054191400133822478222784018297859011565014526}{9136101057353736188751549898001468069294153521} a^{10} + \frac{1736999454604164268737855520349007319424320534}{9136101057353736188751549898001468069294153521} a^{9} - \frac{3446739313466830609300057692506237100480274554}{9136101057353736188751549898001468069294153521} a^{8} - \frac{3231566454293615672559971162191958875241794947}{9136101057353736188751549898001468069294153521} a^{7} + \frac{210020904333470975386116166981193553813278233}{9136101057353736188751549898001468069294153521} a^{6} - \frac{2398893836333408437352597183093915393453613209}{9136101057353736188751549898001468069294153521} a^{5} - \frac{2605714005855302571679974586489262503205099696}{9136101057353736188751549898001468069294153521} a^{4} - \frac{1308892853473412436445281837297345928329532434}{9136101057353736188751549898001468069294153521} a^{3} - \frac{357368663937514162735094207139100674531087032}{9136101057353736188751549898001468069294153521} a^{2} - \frac{2947878783727484063966141711316468456535345778}{9136101057353736188751549898001468069294153521} a + \frac{1113265702785061866908101173632355800454033983}{9136101057353736188751549898001468069294153521}$, $\frac{1}{9136101057353736188751549898001468069294153521} a^{43} + \frac{778853916321883783209172491300052506120230468}{9136101057353736188751549898001468069294153521} a^{40} - \frac{1725569335705265494688516174695913975001225181}{9136101057353736188751549898001468069294153521} a^{39} - \frac{4157476959008393455145693805450773795538042458}{9136101057353736188751549898001468069294153521} a^{38} - \frac{1487914494372331847279750616952170092545400333}{9136101057353736188751549898001468069294153521} a^{37} - \frac{1756471869444257665153993153157363956523545905}{9136101057353736188751549898001468069294153521} a^{36} + \frac{3453406756880450445397364653795973842616417108}{9136101057353736188751549898001468069294153521} a^{35} + \frac{2806316228094019324414071710700756179568491687}{9136101057353736188751549898001468069294153521} a^{34} + \frac{3815659968481989482875381650346590597709190714}{9136101057353736188751549898001468069294153521} a^{33} + \frac{1126138993352448270974093680165922736071989704}{9136101057353736188751549898001468069294153521} a^{32} + \frac{2202035432089255165178470119231379354602798526}{9136101057353736188751549898001468069294153521} a^{31} - \frac{1032570273217799595949224386405665777431163558}{9136101057353736188751549898001468069294153521} a^{30} + \frac{3752095954298914965047972582995897056360668880}{9136101057353736188751549898001468069294153521} a^{29} + \frac{2757501875763710956872444374055646274467942628}{9136101057353736188751549898001468069294153521} a^{28} + \frac{1293587673063147701734302180774082215453342735}{9136101057353736188751549898001468069294153521} a^{27} + \frac{4243796187650733310139254502489828686818782305}{9136101057353736188751549898001468069294153521} a^{26} - \frac{3981550706792135928236605520344106934775223312}{9136101057353736188751549898001468069294153521} a^{25} + \frac{148128844098214611517952622208117191578182173}{9136101057353736188751549898001468069294153521} a^{24} + \frac{2086420580517019742329591932795930274276595431}{9136101057353736188751549898001468069294153521} a^{23} + \frac{4548903439360345507984550150604390839343122539}{9136101057353736188751549898001468069294153521} a^{22} + \frac{40882851737439233355760651031243965355184122}{9136101057353736188751549898001468069294153521} a^{21} - \frac{397923581245366218035225959915825537554965695}{9136101057353736188751549898001468069294153521} a^{20} - \frac{1077837391644957254531394716696733651043916157}{9136101057353736188751549898001468069294153521} a^{19} - \frac{4311442038922722267930570027102282195363304028}{9136101057353736188751549898001468069294153521} a^{18} + \frac{2013777970957155958627307413943222636149348414}{9136101057353736188751549898001468069294153521} a^{17} - \frac{732089115771088075888142872472955739515374648}{9136101057353736188751549898001468069294153521} a^{16} + \frac{2892817444729106636778410824517533477293210060}{9136101057353736188751549898001468069294153521} a^{15} - \frac{1575391186447599602191442457217192763106136732}{9136101057353736188751549898001468069294153521} a^{14} - \frac{1607548204304721272672178354276915832048763212}{9136101057353736188751549898001468069294153521} a^{13} + \frac{3918838900329499088716786340231241280997705848}{9136101057353736188751549898001468069294153521} a^{12} + \frac{2376898716427734958975018901662507199689190688}{9136101057353736188751549898001468069294153521} a^{11} + \frac{2691803664756483106814725858077108141972035527}{9136101057353736188751549898001468069294153521} a^{10} + \frac{2031438715918745993390007792958641692918509917}{9136101057353736188751549898001468069294153521} a^{9} + \frac{129590073799228900864144715682252234480811204}{9136101057353736188751549898001468069294153521} a^{8} + \frac{506274407467295168549891765184451017398176493}{9136101057353736188751549898001468069294153521} a^{7} + \frac{1761315323592523374710500119919852811749020410}{9136101057353736188751549898001468069294153521} a^{6} - \frac{2766403353705403321502951639840373407154534175}{9136101057353736188751549898001468069294153521} a^{5} - \frac{350743520587285782696802180384510603058724174}{9136101057353736188751549898001468069294153521} a^{4} + \frac{4083316489170854587131888300756468532800192672}{9136101057353736188751549898001468069294153521} a^{3} - \frac{1741231624333285250486487546793929412868793039}{9136101057353736188751549898001468069294153521} a^{2} + \frac{2085094185628224409863203562142023252283499025}{9136101057353736188751549898001468069294153521} a + \frac{4354767395481132534148420569526021396282549439}{9136101057353736188751549898001468069294153521}$

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

Class group and class number

$C_{45013}$, which has order $45013$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $21$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{143504380309978050139881567973335320096946001}{9136101057353736188751549898001468069294153521} a^{43} + \frac{81368225088904635639643242222958289840485360}{9136101057353736188751549898001468069294153521} a^{42} - \frac{1492231294800999393499889646283360262044835668}{9136101057353736188751549898001468069294153521} a^{41} + \frac{1019370527770824485026681601720312810602393430}{9136101057353736188751549898001468069294153521} a^{40} - \frac{10042664933624921435671222612358500651410850334}{9136101057353736188751549898001468069294153521} a^{39} + \frac{5598518894712457990681568250129869301219118290}{9136101057353736188751549898001468069294153521} a^{38} - \frac{54439443092304226137092628289942470954432716948}{9136101057353736188751549898001468069294153521} a^{37} + \frac{25046768213369137622452464058667318173761005651}{9136101057353736188751549898001468069294153521} a^{36} - \frac{261675475689135149812732129394105464547887305550}{9136101057353736188751549898001468069294153521} a^{35} + \frac{106006149934327707187186860862887429199777655855}{9136101057353736188751549898001468069294153521} a^{34} - \frac{1018020153523507803054954131696212731384989055983}{9136101057353736188751549898001468069294153521} a^{33} + \frac{360868913551724277449830863493937698706480336055}{9136101057353736188751549898001468069294153521} a^{32} - \frac{3419087929686705779392212790150592656405732168743}{9136101057353736188751549898001468069294153521} a^{31} + \frac{827520078509455974126373061859624784677633384470}{9136101057353736188751549898001468069294153521} a^{30} - \frac{10105632475745477178767443068810705542601985118826}{9136101057353736188751549898001468069294153521} a^{29} + \frac{1721655411985498108932461991190925395560931023439}{9136101057353736188751549898001468069294153521} a^{28} - \frac{26379968816467594647512817238674814140808086761957}{9136101057353736188751549898001468069294153521} a^{27} + \frac{3165153959303854936220694197994417993417747525060}{9136101057353736188751549898001468069294153521} a^{26} - \frac{57245272100169277205124608505170969150184789408770}{9136101057353736188751549898001468069294153521} a^{25} + \frac{2135469037678029195969847258311039512705177671261}{9136101057353736188751549898001468069294153521} a^{24} - \frac{109043960405876300104464918727943361473069774995405}{9136101057353736188751549898001468069294153521} a^{23} - \frac{9394367245362583061729845311873119449368514197033}{9136101057353736188751549898001468069294153521} a^{22} - \frac{179492447323669580307644301966947668061487024905260}{9136101057353736188751549898001468069294153521} a^{21} - \frac{21789532311278591125661166913986888157234519547176}{9136101057353736188751549898001468069294153521} a^{20} - \frac{249045211393859148621624433309241448155897060631260}{9136101057353736188751549898001468069294153521} a^{19} - \frac{38346669529210729755020489975763052316588856466547}{9136101057353736188751549898001468069294153521} a^{18} - \frac{262393676597008755852667438862819053492774408920205}{9136101057353736188751549898001468069294153521} a^{17} - \frac{76776809216200678040852496849808737961296716745800}{9136101057353736188751549898001468069294153521} a^{16} - \frac{244152822345396638525731568775136429978233150069260}{9136101057353736188751549898001468069294153521} a^{15} - \frac{112902416032775381176555799479769868186552088794493}{9136101057353736188751549898001468069294153521} a^{14} - \frac{179290617172262217286376867432743846823222431058505}{9136101057353736188751549898001468069294153521} a^{13} - \frac{65470244859845637792486912705750698987242722132550}{9136101057353736188751549898001468069294153521} a^{12} - \frac{95430799894192649077541424315906600459814682525175}{9136101057353736188751549898001468069294153521} a^{11} - \frac{31748909586203858691990761692951136137711540127758}{9136101057353736188751549898001468069294153521} a^{10} - \frac{17759400710147720196586546868252742926126088090524}{9136101057353736188751549898001468069294153521} a^{9} - \frac{20183709730699723087162238404965531540012392192071}{9136101057353736188751549898001468069294153521} a^{8} - \frac{6296016141516024306135728642564513083607941961208}{9136101057353736188751549898001468069294153521} a^{7} - \frac{10049103659438267859668632986449870027282711753670}{9136101057353736188751549898001468069294153521} a^{6} - \frac{858010346659704501968645248274200275556980986171}{9136101057353736188751549898001468069294153521} a^{5} - \frac{298361613164131089770410931908806186529504065289}{9136101057353736188751549898001468069294153521} a^{4} + \frac{39651548858446898930005679224874141617425029253}{9136101057353736188751549898001468069294153521} a^{3} - \frac{5840288335645027161315799389478167811794293879}{9136101057353736188751549898001468069294153521} a^{2} + \frac{67219294366979430088793002262755859468098366401}{9136101057353736188751549898001468069294153521} a - \frac{143806742197913588744033711164446545109789525}{9136101057353736188751549898001468069294153521} \) (order $10$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 36326958556899850 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{22}\cdot 36326958556899850 \cdot 45013}{10\sqrt{342865339180420288801608222738062084913425127327306009945459663867950439453125}}\approx 0.101382339536813$ (assuming GRH)

Galois group

$C_{44}$ (as 44T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 44
The 44 conjugacy class representatives for $C_{44}$
Character table for $C_{44}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), \(\Q(\zeta_{23})^+\), 22.22.83796671451884098775580820361328125.1

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 $44$ $44$ R $44$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{4}$ $44$ $44$ $22^{2}$ R $22^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{4}$ $44$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{4}$ $44$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{11}$ $44$ $22^{2}$

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

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
5Data not computed
23Data not computed