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Properties

Label 45.45.977...089.1
Degree $45$
Signature $[45, 0]$
Discriminant $9.775\times 10^{87}$
Root discriminant $90.23$
Ramified primes $7, 31$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_3\times C_{15}$ (as 45T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^45 - x^44 - 72*x^43 + 67*x^42 + 2321*x^41 - 2007*x^40 - 44393*x^39 + 35655*x^38 + 562984*x^37 - 420187*x^36 - 5012298*x^35 + 3480849*x^34 + 32367724*x^33 - 20955723*x^32 - 154542085*x^31 + 93497512*x^30 + 551503498*x^29 - 312562651*x^28 - 1478859633*x^27 + 786888918*x^26 + 2984367290*x^25 - 1493106677*x^24 - 4527360022*x^23 + 2130398676*x^22 + 5146818792*x^21 - 2274599139*x^20 - 4361796744*x^19 + 1804001386*x^18 + 2733500368*x^17 - 1051995625*x^16 - 1251105602*x^15 + 444673277*x^14 + 410259354*x^13 - 133494819*x^12 - 93584992*x^11 + 27616610*x^10 + 14189768*x^9 - 3755862*x^8 - 1331682*x^7 + 310282*x^6 + 69020*x^5 - 13468*x^4 - 1660*x^3 + 232*x^2 + 16*x - 1)
 
gp: K = bnfinit(x^45 - x^44 - 72*x^43 + 67*x^42 + 2321*x^41 - 2007*x^40 - 44393*x^39 + 35655*x^38 + 562984*x^37 - 420187*x^36 - 5012298*x^35 + 3480849*x^34 + 32367724*x^33 - 20955723*x^32 - 154542085*x^31 + 93497512*x^30 + 551503498*x^29 - 312562651*x^28 - 1478859633*x^27 + 786888918*x^26 + 2984367290*x^25 - 1493106677*x^24 - 4527360022*x^23 + 2130398676*x^22 + 5146818792*x^21 - 2274599139*x^20 - 4361796744*x^19 + 1804001386*x^18 + 2733500368*x^17 - 1051995625*x^16 - 1251105602*x^15 + 444673277*x^14 + 410259354*x^13 - 133494819*x^12 - 93584992*x^11 + 27616610*x^10 + 14189768*x^9 - 3755862*x^8 - 1331682*x^7 + 310282*x^6 + 69020*x^5 - 13468*x^4 - 1660*x^3 + 232*x^2 + 16*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 16, 232, -1660, -13468, 69020, 310282, -1331682, -3755862, 14189768, 27616610, -93584992, -133494819, 410259354, 444673277, -1251105602, -1051995625, 2733500368, 1804001386, -4361796744, -2274599139, 5146818792, 2130398676, -4527360022, -1493106677, 2984367290, 786888918, -1478859633, -312562651, 551503498, 93497512, -154542085, -20955723, 32367724, 3480849, -5012298, -420187, 562984, 35655, -44393, -2007, 2321, 67, -72, -1, 1]);
 

\( x^{45} - x^{44} - 72 x^{43} + 67 x^{42} + 2321 x^{41} - 2007 x^{40} - 44393 x^{39} + 35655 x^{38} + 562984 x^{37} - 420187 x^{36} - 5012298 x^{35} + 3480849 x^{34} + 32367724 x^{33} - 20955723 x^{32} - 154542085 x^{31} + 93497512 x^{30} + 551503498 x^{29} - 312562651 x^{28} - 1478859633 x^{27} + 786888918 x^{26} + 2984367290 x^{25} - 1493106677 x^{24} - 4527360022 x^{23} + 2130398676 x^{22} + 5146818792 x^{21} - 2274599139 x^{20} - 4361796744 x^{19} + 1804001386 x^{18} + 2733500368 x^{17} - 1051995625 x^{16} - 1251105602 x^{15} + 444673277 x^{14} + 410259354 x^{13} - 133494819 x^{12} - 93584992 x^{11} + 27616610 x^{10} + 14189768 x^{9} - 3755862 x^{8} - 1331682 x^{7} + 310282 x^{6} + 69020 x^{5} - 13468 x^{4} - 1660 x^{3} + 232 x^{2} + 16 x - 1 \)

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

Invariants

Degree:  $45$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[45, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(977\!\cdots\!089\)\(\medspace = 7^{30}\cdot 31^{42}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $90.23$
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:  $7, 31$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $45$
This field is Galois and abelian over $\Q$.
Conductor:  \(217=7\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{217}(128,·)$, $\chi_{217}(1,·)$, $\chi_{217}(2,·)$, $\chi_{217}(4,·)$, $\chi_{217}(134,·)$, $\chi_{217}(8,·)$, $\chi_{217}(9,·)$, $\chi_{217}(142,·)$, $\chi_{217}(16,·)$, $\chi_{217}(18,·)$, $\chi_{217}(149,·)$, $\chi_{217}(25,·)$, $\chi_{217}(156,·)$, $\chi_{217}(32,·)$, $\chi_{217}(162,·)$, $\chi_{217}(163,·)$, $\chi_{217}(36,·)$, $\chi_{217}(165,·)$, $\chi_{217}(39,·)$, $\chi_{217}(169,·)$, $\chi_{217}(72,·)$, $\chi_{217}(50,·)$, $\chi_{217}(51,·)$, $\chi_{217}(183,·)$, $\chi_{217}(190,·)$, $\chi_{217}(191,·)$, $\chi_{217}(64,·)$, $\chi_{217}(193,·)$, $\chi_{217}(67,·)$, $\chi_{217}(71,·)$, $\chi_{217}(200,·)$, $\chi_{217}(204,·)$, $\chi_{217}(205,·)$, $\chi_{217}(78,·)$, $\chi_{217}(81,·)$, $\chi_{217}(211,·)$, $\chi_{217}(214,·)$, $\chi_{217}(95,·)$, $\chi_{217}(144,·)$, $\chi_{217}(100,·)$, $\chi_{217}(102,·)$, $\chi_{217}(107,·)$, $\chi_{217}(109,·)$, $\chi_{217}(113,·)$, $\chi_{217}(121,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $\frac{1}{5} a^{42} + \frac{1}{5} a^{41} - \frac{1}{5} a^{40} - \frac{2}{5} a^{39} + \frac{2}{5} a^{38} + \frac{2}{5} a^{36} + \frac{2}{5} a^{35} + \frac{1}{5} a^{34} + \frac{1}{5} a^{33} + \frac{1}{5} a^{32} - \frac{1}{5} a^{31} + \frac{2}{5} a^{29} + \frac{1}{5} a^{26} + \frac{1}{5} a^{25} - \frac{2}{5} a^{24} + \frac{2}{5} a^{23} - \frac{2}{5} a^{21} + \frac{2}{5} a^{20} + \frac{2}{5} a^{19} + \frac{1}{5} a^{18} - \frac{1}{5} a^{17} + \frac{1}{5} a^{16} - \frac{2}{5} a^{15} - \frac{1}{5} a^{14} - \frac{1}{5} a^{13} - \frac{1}{5} a^{12} + \frac{2}{5} a^{11} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{43} - \frac{2}{5} a^{41} - \frac{1}{5} a^{40} - \frac{1}{5} a^{39} - \frac{2}{5} a^{38} + \frac{2}{5} a^{37} - \frac{1}{5} a^{35} - \frac{2}{5} a^{32} + \frac{1}{5} a^{31} + \frac{2}{5} a^{30} - \frac{2}{5} a^{29} + \frac{1}{5} a^{27} + \frac{2}{5} a^{25} - \frac{1}{5} a^{24} - \frac{2}{5} a^{23} - \frac{2}{5} a^{22} - \frac{1}{5} a^{21} - \frac{1}{5} a^{19} - \frac{2}{5} a^{18} + \frac{2}{5} a^{17} + \frac{2}{5} a^{16} + \frac{1}{5} a^{15} - \frac{2}{5} a^{12} - \frac{2}{5} a^{11} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{44} + \frac{6161984712379635277588776628491359396102554884424692528136339662960838597936756996605781346470187175751695460709}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{43} - \frac{38646634055857772756328082711082701227484273134154754730838407869379794590016646822168844550283043987478338326412}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{42} - \frac{234116753854692466187006473936495383293663056123912360582075384614556627069886451041145113760350069371551586417744}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{41} + \frac{14296940773648455652786461312797186618094760785848128444855704663628120457450221268200784306872106541727293393917}{116701280011358303311760005633138781180260792208881034484658294442940632039582612732477838087454729691514583098103} a^{40} + \frac{22079356215630954961447571160347658850109703707283172621176883775555983375630287874888546182107551822348767316274}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{39} + \frac{237918766565315667258687789123129734700466894445923603481981194735206977747630402374397870212123902123302317870044}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{38} + \frac{251354993462740771074243902374392189719184527438459284344950429208837789392998219652494239311663268003895520543488}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{37} + \frac{24140065167729439308269446835707657215431413259843662233726767603885563324228110816138526571010517573414542190119}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{36} - \frac{258361881947126603888674383789838295778684221280842865212638561635652580648658836799523222823511467627964289165524}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{35} + \frac{15724988983195923483337444108106453232512286808803503060948702953256475443350585970990191757638218663464605680755}{116701280011358303311760005633138781180260792208881034484658294442940632039582612732477838087454729691514583098103} a^{34} + \frac{157095045123407486167249374691635864222180471619209978714299163963625590684809535193505218770541466295859631341823}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{33} + \frac{251052334268090301369243605995138666455673225693839511769466988205766771743604179503053552368372930731186764753578}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{32} + \frac{240831983639203953944507086783586933447971581706494466376517479347085489565318458879751070408932604298161157011846}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{31} + \frac{174794421334687182293280751730088828870229633035465307427362373604448640904097450033182576507851922043700752734586}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{30} + \frac{222675702435008622737848632844488091819218457646794743792864448481630302913812638602052131769458919701440618851377}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{29} + \frac{111216653789480159294238271978229137209641278320236828547768782166354034995873521682636348676754177859097495824211}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{28} + \frac{187733214466258264712042445692477730745683107088549055367628900427898702479760074406807285357860000068811537431164}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{27} - \frac{196916445175810534771786388562908284594639119889459084190463213250875627003102876459612586314389259018645199055293}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{26} - \frac{86540791974778638791664283042682391444377028025592567921557373579514725015169565306682300966733467978656231212968}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{25} + \frac{260263768151550467689800514133010869326395748746761168157864893681435599894979570598943942926889215575880242786574}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{24} + \frac{34620590107297856961823150068302171205265821340690560773079292492920006375812936165326999932891126217416463585289}{116701280011358303311760005633138781180260792208881034484658294442940632039582612732477838087454729691514583098103} a^{23} + \frac{99955577550634176567844281621179395947750902073040761720686116704069233896632778994077230738025253218009387654136}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{22} + \frac{249718220326537112198079496012857081049256548556711447595925221891056878699034078436236883658940237403810098913486}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{21} + \frac{73027503247831738662499366248011809776152185321033692469428206143335315954007124693735496820327095226850629743904}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{20} - \frac{145878125436396286093555186687169287524682048152226099741950050503149564368519970025863705850277346049711159167471}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{19} - \frac{13736860407883233871355500552592919010337918280235378736137797665754273718439866084721121115125001399849982679791}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{18} - \frac{14791139755274028877562394338036671705065143094399894758577311695142498994187483271550490938865715186106225958918}{116701280011358303311760005633138781180260792208881034484658294442940632039582612732477838087454729691514583098103} a^{17} - \frac{210094208961061184984747733351987375143661127514512753727227660547086530357487410741509132310404335818585040750056}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{16} + \frac{173265403267907676464765241480683653030018174211518300539255952752852338344462292468422433458222586581338063302729}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{15} - \frac{4931360678491495964776513511870659799849286531409576624411438290593017018864321364321751953446448578675446540544}{116701280011358303311760005633138781180260792208881034484658294442940632039582612732477838087454729691514583098103} a^{14} - \frac{37466718255184887979178734913618508868300438912206757776586993438981834922177476072550105194609445848471556445822}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{13} - \frac{22601361208949479403138993336132038587427589999118098487907393096458520513570061458250940115117323226719003873562}{116701280011358303311760005633138781180260792208881034484658294442940632039582612732477838087454729691514583098103} a^{12} - \frac{194304777853594342827223463390339339688499186389038991831043864417261117470922729742821046722732818986115822664168}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{11} + \frac{239040467137975311273856848408640262507718152299992125911089981456224141237206067077232010585653589849606975570376}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{10} + \frac{55874869105185412557694497326477093855092343315672984696337976966109014748632451902182489590447347955819878395377}{116701280011358303311760005633138781180260792208881034484658294442940632039582612732477838087454729691514583098103} a^{9} + \frac{123777769000561706960905140877231332149751642864983593424169067734697521627454811958831680394280811182545290611923}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{8} + \frac{62106345745306662195184594049988209676627988579719151087452905500198195232679316027434364508075267089632824783962}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{7} + \frac{221712353016122368356396635438535194419336250364799469207187456752452571330256590481249128027435291401287509478481}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{6} + \frac{203377501051508793869611939667723982833253288431860777974954261593962499007204432827399666560796270549683392377186}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{5} - \frac{274333935817067999061401350330572304135629866724342680738118822568283423931903197259431612385254468811666609554732}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{4} - \frac{216461445667138224782915854613502684375245118698830243273424589214770912384004540833972596730737486999776165627602}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{3} - \frac{267869754335052650449652294954885575600932528325932493121203415800644368308186430365149391537653443343244836239197}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a^{2} - \frac{104071529494640672026103406554621413928981276749999811049220780982615780273660746429034660954053943799120693007941}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515} a + \frac{257146582418819093317604848179818871567109570874714868547492355497053407067299471663605474978862422963774404335839}{583506400056791516558800028165693905901303961044405172423291472214703160197913063662389190437273648457572915490515}$

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:  $44$
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:  \( 604682720155280400000000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{45}\cdot(2\pi)^{0}\cdot 604682720155280400000000000000 \cdot 1}{2\sqrt{9775350573779289822992954615907439666894942099243693331564576241100496506992103462547089}}\approx 0.107592301246632$ (assuming GRH)

Galois group

$C_3\times C_{15}$ (as 45T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
An abelian group of order 45
The 45 conjugacy class representatives for $C_3\times C_{15}$
Character table for $C_3\times C_{15}$ is not computed

Intermediate fields

3.3.961.1, 3.3.47089.1, 3.3.47089.2, \(\Q(\zeta_{7})^+\), 5.5.923521.1, 9.9.104413920565969.1, \(\Q(\zeta_{31})^+\), 15.15.213817926580534310560958234929.1, 15.15.213817926580534310560958234929.2, 15.15.222495240978703757087365489.1

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 $15^{3}$ $15^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{15}$ R $15^{3}$ $15^{3}$ $15^{3}$ $15^{3}$ $15^{3}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{9}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{15}$ $15^{3}$ $15^{3}$ $15^{3}$ $15^{3}$ $15^{3}$

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
7Data not computed
31Data not computed