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Properties

Label 44.44.116...625.1
Degree $44$
Signature $[44, 0]$
Discriminant $1.166\times 10^{83}$
Root discriminant $77.25$
Ramified primes $3, 5, 23$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2\times C_{22}$ (as 44T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^44 - x^43 - 67*x^42 + 66*x^41 + 2006*x^40 - 1940*x^39 - 35522*x^38 + 33582*x^37 + 415391*x^36 - 381809*x^35 - 3396482*x^34 + 3014673*x^33 + 20088042*x^32 - 17073369*x^31 - 87848856*x^30 + 70775487*x^29 + 288431096*x^28 - 217655609*x^27 - 718686410*x^26 + 501030801*x^25 + 1368730205*x^24 - 867699404*x^23 - 1999332041*x^22 + 1131696716*x^21 + 2238799988*x^20 - 1107666220*x^19 - 1911947202*x^18 + 806431252*x^17 + 1231950206*x^16 - 430200535*x^15 - 588270804*x^14 + 164470157*x^13 + 202516659*x^12 - 43750410*x^11 - 48240456*x^10 + 7822102*x^9 + 7480922*x^8 - 908640*x^7 - 688078*x^6 + 65581*x^5 + 31906*x^4 - 2504*x^3 - 504*x^2 + 24*x + 1)
 
gp: K = bnfinit(x^44 - x^43 - 67*x^42 + 66*x^41 + 2006*x^40 - 1940*x^39 - 35522*x^38 + 33582*x^37 + 415391*x^36 - 381809*x^35 - 3396482*x^34 + 3014673*x^33 + 20088042*x^32 - 17073369*x^31 - 87848856*x^30 + 70775487*x^29 + 288431096*x^28 - 217655609*x^27 - 718686410*x^26 + 501030801*x^25 + 1368730205*x^24 - 867699404*x^23 - 1999332041*x^22 + 1131696716*x^21 + 2238799988*x^20 - 1107666220*x^19 - 1911947202*x^18 + 806431252*x^17 + 1231950206*x^16 - 430200535*x^15 - 588270804*x^14 + 164470157*x^13 + 202516659*x^12 - 43750410*x^11 - 48240456*x^10 + 7822102*x^9 + 7480922*x^8 - 908640*x^7 - 688078*x^6 + 65581*x^5 + 31906*x^4 - 2504*x^3 - 504*x^2 + 24*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 24, -504, -2504, 31906, 65581, -688078, -908640, 7480922, 7822102, -48240456, -43750410, 202516659, 164470157, -588270804, -430200535, 1231950206, 806431252, -1911947202, -1107666220, 2238799988, 1131696716, -1999332041, -867699404, 1368730205, 501030801, -718686410, -217655609, 288431096, 70775487, -87848856, -17073369, 20088042, 3014673, -3396482, -381809, 415391, 33582, -35522, -1940, 2006, 66, -67, -1, 1]);
 

\( x^{44} - x^{43} - 67 x^{42} + 66 x^{41} + 2006 x^{40} - 1940 x^{39} - 35522 x^{38} + 33582 x^{37} + 415391 x^{36} - 381809 x^{35} - 3396482 x^{34} + 3014673 x^{33} + 20088042 x^{32} - 17073369 x^{31} - 87848856 x^{30} + 70775487 x^{29} + 288431096 x^{28} - 217655609 x^{27} - 718686410 x^{26} + 501030801 x^{25} + 1368730205 x^{24} - 867699404 x^{23} - 1999332041 x^{22} + 1131696716 x^{21} + 2238799988 x^{20} - 1107666220 x^{19} - 1911947202 x^{18} + 806431252 x^{17} + 1231950206 x^{16} - 430200535 x^{15} - 588270804 x^{14} + 164470157 x^{13} + 202516659 x^{12} - 43750410 x^{11} - 48240456 x^{10} + 7822102 x^{9} + 7480922 x^{8} - 908640 x^{7} - 688078 x^{6} + 65581 x^{5} + 31906 x^{4} - 2504 x^{3} - 504 x^{2} + 24 x + 1 \)

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

Invariants

Degree:  $44$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[44, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(116\!\cdots\!625\)\(\medspace = 3^{22}\cdot 5^{22}\cdot 23^{42}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $77.25$
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:  $3, 5, 23$
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:  \(345=3\cdot 5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{345}(256,·)$, $\chi_{345}(1,·)$, $\chi_{345}(259,·)$, $\chi_{345}(4,·)$, $\chi_{345}(134,·)$, $\chi_{345}(11,·)$, $\chi_{345}(14,·)$, $\chi_{345}(271,·)$, $\chi_{345}(16,·)$, $\chi_{345}(149,·)$, $\chi_{345}(151,·)$, $\chi_{345}(281,·)$, $\chi_{345}(154,·)$, $\chi_{345}(31,·)$, $\chi_{345}(289,·)$, $\chi_{345}(296,·)$, $\chi_{345}(169,·)$, $\chi_{345}(44,·)$, $\chi_{345}(301,·)$, $\chi_{345}(176,·)$, $\chi_{345}(49,·)$, $\chi_{345}(56,·)$, $\chi_{345}(314,·)$, $\chi_{345}(191,·)$, $\chi_{345}(64,·)$, $\chi_{345}(194,·)$, $\chi_{345}(139,·)$, $\chi_{345}(196,·)$, $\chi_{345}(329,·)$, $\chi_{345}(74,·)$, $\chi_{345}(331,·)$, $\chi_{345}(334,·)$, $\chi_{345}(211,·)$, $\chi_{345}(341,·)$, $\chi_{345}(86,·)$, $\chi_{345}(344,·)$, $\chi_{345}(89,·)$, $\chi_{345}(221,·)$, $\chi_{345}(94,·)$, $\chi_{345}(224,·)$, $\chi_{345}(206,·)$, $\chi_{345}(121,·)$, $\chi_{345}(251,·)$, $\chi_{345}(124,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $\frac{1}{139} a^{41} + \frac{25}{139} a^{40} - \frac{43}{139} a^{39} + \frac{65}{139} a^{38} - \frac{15}{139} a^{37} - \frac{57}{139} a^{36} + \frac{3}{139} a^{35} + \frac{66}{139} a^{34} - \frac{58}{139} a^{33} - \frac{5}{139} a^{32} + \frac{64}{139} a^{31} + \frac{67}{139} a^{30} + \frac{64}{139} a^{29} + \frac{41}{139} a^{28} + \frac{30}{139} a^{27} + \frac{26}{139} a^{26} + \frac{38}{139} a^{25} + \frac{14}{139} a^{24} - \frac{10}{139} a^{23} - \frac{43}{139} a^{21} - \frac{8}{139} a^{20} + \frac{57}{139} a^{19} - \frac{32}{139} a^{18} + \frac{5}{139} a^{17} - \frac{54}{139} a^{15} - \frac{40}{139} a^{14} - \frac{23}{139} a^{13} + \frac{23}{139} a^{12} - \frac{29}{139} a^{11} + \frac{19}{139} a^{10} - \frac{9}{139} a^{9} + \frac{62}{139} a^{8} + \frac{53}{139} a^{7} + \frac{24}{139} a^{6} + \frac{26}{139} a^{5} - \frac{4}{139} a^{4} - \frac{3}{139} a^{3} - \frac{65}{139} a^{2} + \frac{54}{139} a - \frac{8}{139}$, $\frac{1}{139} a^{42} + \frac{27}{139} a^{40} + \frac{28}{139} a^{39} + \frac{28}{139} a^{38} + \frac{40}{139} a^{37} + \frac{38}{139} a^{36} - \frac{9}{139} a^{35} - \frac{40}{139} a^{34} + \frac{55}{139} a^{33} + \frac{50}{139} a^{32} - \frac{4}{139} a^{31} + \frac{57}{139} a^{30} - \frac{30}{139} a^{29} - \frac{22}{139} a^{28} - \frac{29}{139} a^{27} - \frac{56}{139} a^{26} + \frac{37}{139} a^{25} + \frac{57}{139} a^{24} - \frac{28}{139} a^{23} - \frac{43}{139} a^{22} - \frac{45}{139} a^{21} - \frac{21}{139} a^{20} - \frac{67}{139} a^{19} - \frac{29}{139} a^{18} + \frac{14}{139} a^{17} - \frac{54}{139} a^{16} + \frac{59}{139} a^{15} + \frac{4}{139} a^{14} + \frac{42}{139} a^{13} - \frac{48}{139} a^{12} + \frac{49}{139} a^{11} - \frac{67}{139} a^{10} + \frac{9}{139} a^{9} + \frac{32}{139} a^{8} - \frac{50}{139} a^{7} - \frac{18}{139} a^{6} + \frac{41}{139} a^{5} - \frac{42}{139} a^{4} + \frac{10}{139} a^{3} + \frac{11}{139} a^{2} + \frac{32}{139} a + \frac{61}{139}$, $\frac{1}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{43} - \frac{845827948188354429440730587949801639534152578419209914002933360978556699203005894946723541}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{42} - \frac{9837007070721806637489122836262505883374751465201304662258389125564311271537105438516496}{5049283569677163579117885657063562738361422804138338180594722842839070505838506596941378943} a^{41} - \frac{97372738700385508966368382437706721183683781674058628615098136035821004293600328555107353128}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{40} + \frac{35408466319123262735033400491321832326435781414047721930341132085288111631901176232308694868}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{39} + \frac{1910256784497440869658569843674733266704948082645190957953041091647158817440220603809625193}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{38} - \frac{14384343485715623871627977176319455494678218793206776194652802064913424899165099517994742269}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{37} - \frac{52288139675485276513316042672850560893046184561942330471484189222305596683877460524990768988}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{36} + \frac{41054836813678109292079229903236639973524201963761212714864308299382665625366116003740368570}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{35} - \frac{20464510864988849961508848326578007487195021773236542025860595805709373552835799035970217048}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{34} + \frac{89672443547974098437144000461019277614436749949070535197856444326045124916704549914177481571}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{33} + \frac{44398303465289914699450544794312667670797869457677270438437822158651397978025851818774023579}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{32} - \frac{43336763778072443493520647460228400676349168214509237328941009768801855640543334027825305616}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{31} + \frac{35308379321048609226526723469017255891227098625062163433299429224319852244996454464508506849}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{30} - \frac{55041949850802857844063872706744015816968232103712784819051233868374735388513227843104976453}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{29} - \frac{105957764701329297337445354248934003795679054778329380567464067253080850642883796900011284192}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{28} - \frac{92682967350907597603670781998487125175834273804716744233551553956065294518192984821632578298}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{27} + \frac{116782079919726531092796449324638931422584951144127112789592261087526788215273207103236664513}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{26} - \frac{31447051823789612194875124512017942659052563888715361301066317360825588178917890938500346599}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{25} - \frac{56519990457449894547740200201099208399442690864677520644607297620990386787254665216237072689}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{24} - \frac{75784842220017330080104144860169798307948246986573711350986544725136668628481016922316961147}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{23} - \frac{93990038637050443269129848864377805459112478540726365079534647548475575030792838316518734512}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{22} - \frac{325379102149899341620917883211317569919629568654371277869904338678381173734718947848139061}{1707311710610263944018277884043075170525085408593538809265841536787311609887840360116869139} a^{21} + \frac{76913292708462631017892738853137220024457778994111968297959609231741597149978709593990718274}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{20} - \frac{84693598760290046732351046360268238404623171246520799431914684222737930140033528350720381080}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{19} + \frac{84332615223176174542685159998226314266127275697021844809899338182044629536862369423519225451}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{18} - \frac{16637717870547379628052599745304074599565528050027811592209357619593853700416414123665643221}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{17} + \frac{31133242292554042847522804043420934228636553287267837910810986556503603300838431106221402465}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{16} - \frac{114865368603423657457749344803648208347526115666808646909260051259513525466365579488549165420}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{15} - \frac{42813467385375360504060673472897634954225610986072548667970337718919356891654546297971677412}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{14} + \frac{51755651468793801095249443622352176510259847176707008509998970997005847404760107479711518003}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{13} - \frac{92813747800752121093639689460597014420202452734410527047414526732386339253027731067564038038}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{12} - \frac{93341117393346141842584641098335719340702471896189055226711721283776661299379955821000106270}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{11} - \frac{51934078459879938202882061573448119559497736167014083804931358893751841126991893972063710845}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{10} + \frac{106002624091796226019497100269734611066368918308755674948788145003742264489688849338482385929}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{9} + \frac{110661451192137779903298165095497738771255277586396769658158603951637438974684588831854846652}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{8} + \frac{15402936844927520467833811235541908675332814600276543393646057561054257879899278504093553712}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{7} + \frac{75552735514444660130679688927268945340349849882340677942076552856490688767224106051971988729}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{6} - \frac{93878136417475168210173439941329536326023307652216870661499308801274923693800688500442774043}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{5} + \frac{75807752374755017780184601550655334230832355251373470322573573668560104168734617911681064946}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{4} + \frac{110243893854618596832145728159972322424331471044619624074158129875104545707135235408539935282}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{3} + \frac{86116774695881419118798587509600929723039833119729729538092462593836692721215288192379857169}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a^{2} + \frac{88832265348859679266819550702558048892885036519341821807191388500260673329307052848259419501}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321} a + \frac{5964664815647874412497316026856807073800947997829182892868259580311998664217033782926205684}{237316327774826688218540625881987448702986871794501894487951973613436313774409810056244810321}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $43$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
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:  \( 4751041378621857000000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{44}\cdot(2\pi)^{0}\cdot 4751041378621857000000000000 \cdot 1}{2\sqrt{116567320065927752512435466812933331534234135648947894549180382317450046539306640625}}\approx 0.122402480927051$ (assuming GRH)

Galois group

$C_2\times C_{22}$ (as 44T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
An abelian group of order 44
The 44 conjugacy class representatives for $C_2\times C_{22}$
Character table for $C_2\times C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{345}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{69}) \), \(\Q(\sqrt{5}, \sqrt{69})\), \(\Q(\zeta_{23})^+\), 22.22.341419566026798986253349758444608447265625.1, 22.22.83796671451884098775580820361328125.1, \(\Q(\zeta_{69})^+\)

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 $22^{2}$ R R $22^{2}$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ $22^{2}$ R $22^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{22}$ $22^{2}$ $22^{2}$

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
3Data not computed
5Data not computed
23Data not computed