Properties

Label 45.45.295...689.1
Degree $45$
Signature $[45, 0]$
Discriminant $2.954\times 10^{91}$
Root discriminant $107.81$
Ramified primes $3, 7, 11$
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 - 3*x^44 - 108*x^43 + 296*x^42 + 5217*x^41 - 13005*x^40 - 149828*x^39 + 337863*x^38 + 2868057*x^37 - 5809689*x^36 - 38881407*x^35 + 70098207*x^34 + 387120467*x^33 - 613625013*x^32 - 2896408122*x^31 + 3972622452*x^30 + 16519787829*x^29 - 19205853294*x^28 - 72406643558*x^27 + 69489283683*x^26 + 244599321027*x^25 - 187254237300*x^24 - 635787323022*x^23 + 371102281773*x^22 + 1263897395760*x^21 - 528600088722*x^20 - 1901669780781*x^19 + 520023114157*x^18 + 2134290033054*x^17 - 327330093672*x^16 - 1754567507744*x^15 + 107410900356*x^14 + 1034842778925*x^13 + 1036802542*x^12 - 428135520330*x^11 - 14111627571*x^10 + 121049080943*x^9 + 4346148948*x^8 - 22551409551*x^7 - 290753084*x^6 + 2597605155*x^5 - 77625633*x^4 - 161647682*x^3 + 13050165*x^2 + 3741168*x - 437977)
 
gp: K = bnfinit(x^45 - 3*x^44 - 108*x^43 + 296*x^42 + 5217*x^41 - 13005*x^40 - 149828*x^39 + 337863*x^38 + 2868057*x^37 - 5809689*x^36 - 38881407*x^35 + 70098207*x^34 + 387120467*x^33 - 613625013*x^32 - 2896408122*x^31 + 3972622452*x^30 + 16519787829*x^29 - 19205853294*x^28 - 72406643558*x^27 + 69489283683*x^26 + 244599321027*x^25 - 187254237300*x^24 - 635787323022*x^23 + 371102281773*x^22 + 1263897395760*x^21 - 528600088722*x^20 - 1901669780781*x^19 + 520023114157*x^18 + 2134290033054*x^17 - 327330093672*x^16 - 1754567507744*x^15 + 107410900356*x^14 + 1034842778925*x^13 + 1036802542*x^12 - 428135520330*x^11 - 14111627571*x^10 + 121049080943*x^9 + 4346148948*x^8 - 22551409551*x^7 - 290753084*x^6 + 2597605155*x^5 - 77625633*x^4 - 161647682*x^3 + 13050165*x^2 + 3741168*x - 437977, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-437977, 3741168, 13050165, -161647682, -77625633, 2597605155, -290753084, -22551409551, 4346148948, 121049080943, -14111627571, -428135520330, 1036802542, 1034842778925, 107410900356, -1754567507744, -327330093672, 2134290033054, 520023114157, -1901669780781, -528600088722, 1263897395760, 371102281773, -635787323022, -187254237300, 244599321027, 69489283683, -72406643558, -19205853294, 16519787829, 3972622452, -2896408122, -613625013, 387120467, 70098207, -38881407, -5809689, 2868057, 337863, -149828, -13005, 5217, 296, -108, -3, 1]);
 

\( x^{45} - 3 x^{44} - 108 x^{43} + 296 x^{42} + 5217 x^{41} - 13005 x^{40} - 149828 x^{39} + 337863 x^{38} + 2868057 x^{37} - 5809689 x^{36} - 38881407 x^{35} + 70098207 x^{34} + 387120467 x^{33} - 613625013 x^{32} - 2896408122 x^{31} + 3972622452 x^{30} + 16519787829 x^{29} - 19205853294 x^{28} - 72406643558 x^{27} + 69489283683 x^{26} + 244599321027 x^{25} - 187254237300 x^{24} - 635787323022 x^{23} + 371102281773 x^{22} + 1263897395760 x^{21} - 528600088722 x^{20} - 1901669780781 x^{19} + 520023114157 x^{18} + 2134290033054 x^{17} - 327330093672 x^{16} - 1754567507744 x^{15} + 107410900356 x^{14} + 1034842778925 x^{13} + 1036802542 x^{12} - 428135520330 x^{11} - 14111627571 x^{10} + 121049080943 x^{9} + 4346148948 x^{8} - 22551409551 x^{7} - 290753084 x^{6} + 2597605155 x^{5} - 77625633 x^{4} - 161647682 x^{3} + 13050165 x^{2} + 3741168 x - 437977 \)

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:  \(295\!\cdots\!689\)\(\medspace = 3^{60}\cdot 7^{30}\cdot 11^{36}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $107.81$
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:  $3, 7, 11$
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:  \(693=3^{2}\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{693}(256,·)$, $\chi_{693}(1,·)$, $\chi_{693}(130,·)$, $\chi_{693}(4,·)$, $\chi_{693}(520,·)$, $\chi_{693}(394,·)$, $\chi_{693}(268,·)$, $\chi_{693}(526,·)$, $\chi_{693}(16,·)$, $\chi_{693}(529,·)$, $\chi_{693}(148,·)$, $\chi_{693}(25,·)$, $\chi_{693}(289,·)$, $\chi_{693}(163,·)$, $\chi_{693}(676,·)$, $\chi_{693}(37,·)$, $\chi_{693}(295,·)$, $\chi_{693}(169,·)$, $\chi_{693}(298,·)$, $\chi_{693}(562,·)$, $\chi_{693}(58,·)$, $\chi_{693}(445,·)$, $\chi_{693}(190,·)$, $\chi_{693}(64,·)$, $\chi_{693}(67,·)$, $\chi_{693}(652,·)$, $\chi_{693}(331,·)$, $\chi_{693}(463,·)$, $\chi_{693}(592,·)$, $\chi_{693}(466,·)$, $\chi_{693}(214,·)$, $\chi_{693}(478,·)$, $\chi_{693}(421,·)$, $\chi_{693}(400,·)$, $\chi_{693}(610,·)$, $\chi_{693}(100,·)$, $\chi_{693}(487,·)$, $\chi_{693}(232,·)$, $\chi_{693}(361,·)$, $\chi_{693}(235,·)$, $\chi_{693}(625,·)$, $\chi_{693}(499,·)$, $\chi_{693}(631,·)$, $\chi_{693}(379,·)$, $\chi_{693}(247,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $\frac{1}{918611} a^{41} + \frac{48432}{918611} a^{40} - \frac{119150}{918611} a^{39} - \frac{189450}{918611} a^{38} - \frac{80821}{918611} a^{37} + \frac{400851}{918611} a^{36} + \frac{443389}{918611} a^{35} + \frac{296379}{918611} a^{34} - \frac{393315}{918611} a^{33} + \frac{56893}{918611} a^{32} + \frac{105872}{918611} a^{31} + \frac{309100}{918611} a^{30} + \frac{378563}{918611} a^{29} - \frac{296702}{918611} a^{28} - \frac{250752}{918611} a^{27} + \frac{315796}{918611} a^{26} + \frac{311471}{918611} a^{25} + \frac{131068}{918611} a^{24} - \frac{240853}{918611} a^{23} + \frac{427054}{918611} a^{22} - \frac{324831}{918611} a^{21} - \frac{10968}{918611} a^{20} + \frac{251358}{918611} a^{19} - \frac{345328}{918611} a^{18} - \frac{430339}{918611} a^{17} - \frac{349810}{918611} a^{16} + \frac{453931}{918611} a^{15} - \frac{6865}{918611} a^{14} - \frac{143700}{918611} a^{13} - \frac{218516}{918611} a^{12} + \frac{336716}{918611} a^{11} - \frac{193344}{918611} a^{10} + \frac{177268}{918611} a^{9} + \frac{234258}{918611} a^{8} - \frac{172995}{918611} a^{7} + \frac{1378}{918611} a^{6} + \frac{438791}{918611} a^{5} - \frac{345426}{918611} a^{4} - \frac{247340}{918611} a^{3} - \frac{223479}{918611} a^{2} + \frac{204043}{918611} a + \frac{255631}{918611}$, $\frac{1}{115617299071} a^{42} + \frac{19520}{115617299071} a^{41} - \frac{38827353107}{115617299071} a^{40} - \frac{21913581305}{115617299071} a^{39} + \frac{55698679560}{115617299071} a^{38} - \frac{46297843181}{115617299071} a^{37} + \frac{39992884149}{115617299071} a^{36} - \frac{57583799030}{115617299071} a^{35} - \frac{57793073398}{115617299071} a^{34} + \frac{13279535220}{115617299071} a^{33} - \frac{14648641860}{115617299071} a^{32} + \frac{26997208367}{115617299071} a^{31} + \frac{5588756495}{115617299071} a^{30} - \frac{9691406143}{115617299071} a^{29} + \frac{9253176557}{115617299071} a^{28} - \frac{40650912975}{115617299071} a^{27} + \frac{10101138804}{115617299071} a^{26} - \frac{29455236566}{115617299071} a^{25} + \frac{24645924636}{115617299071} a^{24} - \frac{1125576683}{2688774397} a^{23} - \frac{4812862657}{115617299071} a^{22} - \frac{11229476824}{115617299071} a^{21} - \frac{40027118335}{115617299071} a^{20} - \frac{12444899420}{115617299071} a^{19} - \frac{36998666798}{115617299071} a^{18} + \frac{9649952529}{115617299071} a^{17} + \frac{5234499019}{115617299071} a^{16} - \frac{29818896251}{115617299071} a^{15} - \frac{39946800742}{115617299071} a^{14} - \frac{1179335891}{2688774397} a^{13} + \frac{42352425005}{115617299071} a^{12} - \frac{8788238395}{115617299071} a^{11} + \frac{27867374357}{115617299071} a^{10} - \frac{32155985444}{115617299071} a^{9} - \frac{23449504385}{115617299071} a^{8} - \frac{20479717711}{115617299071} a^{7} + \frac{48298827486}{115617299071} a^{6} - \frac{51177390329}{115617299071} a^{5} - \frac{55181392390}{115617299071} a^{4} - \frac{10299063955}{115617299071} a^{3} + \frac{5906587847}{115617299071} a^{2} + \frac{20032434334}{115617299071} a - \frac{56246210896}{115617299071}$, $\frac{1}{115617299071} a^{43} - \frac{38926}{115617299071} a^{41} + \frac{56648461818}{115617299071} a^{40} - \frac{16871372864}{115617299071} a^{39} + \frac{24290153390}{115617299071} a^{38} - \frac{25219503771}{115617299071} a^{37} - \frac{35329236214}{115617299071} a^{36} + \frac{46954935247}{115617299071} a^{35} + \frac{6931473493}{115617299071} a^{34} + \frac{19344193029}{115617299071} a^{33} + \frac{7085683943}{115617299071} a^{32} - \frac{53917767279}{115617299071} a^{31} - \frac{12318397852}{115617299071} a^{30} + \frac{30911988955}{115617299071} a^{29} + \frac{34325196374}{115617299071} a^{28} - \frac{19394985896}{115617299071} a^{27} - \frac{41213883718}{115617299071} a^{26} - \frac{20118570993}{115617299071} a^{25} - \frac{51668822958}{115617299071} a^{24} - \frac{7382775373}{115617299071} a^{23} - \frac{24731221966}{115617299071} a^{22} + \frac{33613919936}{115617299071} a^{21} + \frac{34427057674}{115617299071} a^{20} - \frac{27620256682}{115617299071} a^{19} + \frac{49417991123}{115617299071} a^{18} + \frac{56334500237}{115617299071} a^{17} - \frac{23611053389}{115617299071} a^{16} - \frac{5295898323}{115617299071} a^{15} - \frac{20439205574}{115617299071} a^{14} + \frac{37513287135}{115617299071} a^{13} - \frac{10730082757}{115617299071} a^{12} - \frac{32983359959}{115617299071} a^{11} - \frac{42637785636}{115617299071} a^{10} + \frac{25963237579}{115617299071} a^{9} - \frac{16129087084}{115617299071} a^{8} - \frac{35335132121}{115617299071} a^{7} + \frac{50604173417}{115617299071} a^{6} - \frac{53830063263}{115617299071} a^{5} - \frac{15882517086}{115617299071} a^{4} + \frac{56632016847}{115617299071} a^{3} + \frac{40490911259}{115617299071} a^{2} - \frac{22414829316}{115617299071} a + \frac{28841832325}{115617299071}$, $\frac{1}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{44} - \frac{131498866797068121759194746224735234588610365273730645989439055975187065010214674853037479458921382013272935482340206023035750510861250515519785722731456107106977485107}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{43} - \frac{122832257165527957674243199493460271224068202343840757183379327661623668881555452145209698237615514123692528026041599489019061775772981315816961907682938433188353623388}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{42} + \frac{14039124544586721320944697151210892667462714226489268449123454191633591077546286096967793543345649573021691591179519285098551781541944061166053500559471885319652031617156932}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{41} + \frac{7235611791103217803913433455169582874012550384875167734289522525831358905286490842230417365515765471907874665097924769218758541474004946437947522231291593854991938584155862385839}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{40} - \frac{1890298526089202056388876888394801832178955101869225450602442266599134882936261210445557075048662175506159649163878513275433683707744301419322062926778299915430184125584846058192}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{39} - \frac{5605261773569153693517050015875698086248916184284273068922573381366499965424760864548205567597066519046221893081817483398748091404014908521623317134592737239696129455074443435745}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{38} + \frac{7760715376668123612914160040101544037533668151071047563631908239454685155355135835829063300027155127534969570832121042593039950445042550460800130303521726361081188861073249236804}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{37} + \frac{13258857899937063642614228823998614544212918383422834128582475599262373703158677540429084867022592612170869112446450522158075247791068468921076839601443086244364024191762021702324}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{36} + \frac{5305374596194679361484881252201962317469044673630753404031774064699379041674627992850465970131052470214897590031798248348184464737941926586541241327620989365205065633204763560007}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{35} + \frac{23204764494282278604516317872660693604241118591219360280011507924064065943382544130720218054384524849252394752446167873606293939967733445258471922454290367122367246622210693687}{170342767073896232003467933367654868013754140405630904079917620026436576277330055913788097521424343742754742350092827149588680021712064182894155864408908504369283718552198163137} a^{34} + \frac{48637989012576656352384228461576145811320811970912528139177168257708526546883801715878624931930915428902878456833506237691627315847960183163743196810617513944280937156350179149}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{33} + \frac{1137534147317495853442893258170220921271949271073200905885018823561211695325872795726259483056055897690805350929091627499575053078007138112104467028987550898553932205974562514642}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{32} + \frac{3344915254710845482136481491952360393028612575968657965809738220753484910771540968001172775081637099611327457697873439553722279299828758936476410878303513047770158642105344903643}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{31} - \frac{12660449333307055251255140640163035446722784809079952964179733342547807312847666623139936365981986254339770157332797141339652020533572280746843432263271109260354466009488310859082}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{30} - \frac{11368372531716344561157983582747497904242636498832466497341897070080306056669940252654084736845963994263738872709953274253633141047619961247018259741841837946362684222166496963033}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{29} + \frac{3020666560295776729733805383541154147955290363071507758632689456261725365226776301508743874207738263611603637517207265860488008631401059433684945561415112868089018124664172760090}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{28} + \frac{15599935694502395623942851215067117443249957483353409065301046594050045437133912444120401516824312417459172086381699256296042488264955865167899024430118853357192608810808148991313}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{27} - \frac{15769688994823929269476435753239559915509568635599045747868133097667648311669970956418196375318296286165765805280736093147385269737945222596295282404822274560672647916802461523233}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{26} - \frac{11293900313218994996535442477162318891354347960379395475124970015551145488170298810526932804468867901154757050516408703887577889042486747424058138014645551433565720154264668345855}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{25} - \frac{13257801825071630711602327412941714628661886129401281978305421962155129101772666595487187084747081269526865801576432871879291721595278028741370354498961273042956334780010602274597}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{24} - \frac{449923420023690721170277253293227786548364591384292850357871415992293088096064349682612841936418409514838575366882477008373834543148039798659004545095757566836938651007190096653}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{23} - \frac{1916228047241884211019538094359131421777466909321848542442929730104776765117746389832880237575651149661100977886505034670071213477339120334301461686568971229778438603841569963378}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{22} + \frac{12666892713634896833322270425097830211287816405844701799097004094422454235726000770320239215673236165141872436738088253651166544938036684530529010310881818268222349882653325045819}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{21} - \frac{3043675595894086155529240938704490243507067960013077363922413256950951383064964500880320503119658008517904925124300167651952522347674591015167377649636996960892104515198068927166}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{20} + \frac{11050578985440559514938573591431042371012886955383641993784024400388359512077043916567409530063051754227489686015393205960957366195893003205880994202156588298677791274006933064383}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{19} - \frac{3569783002876215972467086287975333256659175753393230185735925707743718179688024870522029011336504220225397994405169091169596509895061432536747759727901619216115983909358069071836}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{18} - \frac{2784166061451493599313131975905575562494170861356629648613046546704668403610998939858257367674688843234610950916322832531009638519898506804734760608254668208690076919583969570032}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{17} + \frac{11201189396062710963624198106508254027424638319179440953013260329787980862705666522198105681393925112246193221024554919762391683008335032963658619785595791758294815858816593787811}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{16} - \frac{8529792353690466808563579022388812463156013301193391570854056741114283403642766875407751234000622020334647249393934514525033400539571114149752307446965208254424785985601134682591}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{15} - \frac{11659767414490023182636431766038850608002045440629500408083521194607320063259054575765145374215553774295224313165049517472549603324579424975660798431407382350673085147164867358244}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{14} + \frac{6423998104485382906349810589893487908379734153547427669100256431867009225727295461286837183337298040396412694161651827181613402425304010496009674589506720720059613632968035496265}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{13} + \frac{15145889826281914297420075739952060517619965269605567925804199789204938833198251386503144649194583113347678397011658167459385738822281647698831824738744094150123014579420018311581}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{12} + \frac{15821070563379774509583685018226975083936481315515240680234155762269395204551560487784788200996033956031907573267665661969492516569070090026137753657282696064758227325128596648651}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{11} + \frac{5848729567685401769127790225077693930965399441434421466795941692149740078938801743658719218494990415393039522787479201796785113854286595654614344722784338303132153624260940136374}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{10} + \frac{7813326698233931644178838875198934853139328449412302758317091790317452245857323851175972814177550765257990665909100264817592740124734552841136509686643350226363668540190826021857}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{9} + \frac{10651987512765403124865016141456363274860644934376721063997786146566434123460169726809760809689537699206828059726816456381327355617403853040053571239792503753192095368483822061981}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{8} - \frac{12578046376604184144538409972680107837307873472808742492711103439389863334077754443107290478217937338569324012426114015842622025642332653090134505451497710406770388055275958383513}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{7} - \frac{10095527478036402261315643360219428861258324779951621647959566206102700793546918731400312668587970522208822178639918443281139892445294248816831097690558411580111572984270839281068}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{6} + \frac{7983655971655260742487074954839657360392258961286276545058579814356010422734082487069458262704641011572292080800209676387450304522986032509451213296042188281633668103565583649037}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{5} + \frac{9725085981705848198286356112458433381772577210924642191671531223378696941225322168095309249045699134721168133401636641769748508308941967865452724834368944081945034560028636051699}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{4} - \frac{8494119794628595645740387232179759131962264667107928068932902389168084561394810616040926972664978785461365937045866112517203755441460269367891626121012199197572527205915939631967}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{3} + \frac{13684396338204256114326228989793200708054056495701503673015254123466911227614348996387280932487242791433921791229376375939449188096391682387854543318825062362042309926200858915777}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a^{2} - \frac{15503858971654911168526323386700802590833257068106609166572078253117503442104636949373755933632166742016881874598587840422426805273596985962863223288257582192532609374088186508862}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989} a + \frac{85293757512097163437292418800191121233357093332147653304768243628902918516423202454009614210571395790887306809139478652519986057893957394210267012006043795469024291123023749732}{33557525113557557704683182873428008998709565659909288103743771145208005526634021015016255211720595717322684242968286948468969964277276644030148705288554975360748892554783038137989}$

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:  \( 36716059087827736000000000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{45}\cdot(2\pi)^{0}\cdot 36716059087827736000000000000000 \cdot 1}{2\sqrt{29536099970750111921739666309496858571020526814137892348544513245275107329742003872827344689}}\approx 0.118850029720553$ (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

3.3.3969.2, 3.3.3969.1, \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{11})^+\), 9.9.62523502209.1, 15.15.3091133177133909578645502426129.1, 15.15.3091133177133909578645502426129.2, 15.15.10943023107606534329121.1, 15.15.886528337182930278529.1

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 $15^{3}$ R $15^{3}$ R R $15^{3}$ $15^{3}$ $15^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{15}$ $15^{3}$ $15^{3}$ $15^{3}$ $15^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{15}$ $15^{3}$ $15^{3}$ $15^{3}$

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
3Data not computed
7Data not computed
$11$11.15.12.1$x^{15} + 165 x^{10} + 5324 x^{5} + 323433$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$
11.15.12.1$x^{15} + 165 x^{10} + 5324 x^{5} + 323433$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$
11.15.12.1$x^{15} + 165 x^{10} + 5324 x^{5} + 323433$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$