↑

Properties

Label 32.0.81208878148...8801.1
Degree $32$
Signature $[0, 16]$
Discriminant $3^{16}\cdot 13^{16}\cdot 17^{28}$
Root discriminant $74.50$
Ramified primes $3, 13, 17$
Class number Not computed
Class group Not computed
Galois group $C_2^2\times C_8$ (as 32T37)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10201, -101101, 787679, -3082612, 11237683, -28239532, 79124517, -161825433, 331102182, -456787822, 629711197, -589256116, 676121303, -502558589, 483174616, -262803136, 215848229, -95340760, 69153169, -24484833, 15332596, -4605832, 2525043, -640430, 297875, -65115, 25964, -4738, 1535, -224, 60, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 6*x^31 + 60*x^30 - 224*x^29 + 1535*x^28 - 4738*x^27 + 25964*x^26 - 65115*x^25 + 297875*x^24 - 640430*x^23 + 2525043*x^22 - 4605832*x^21 + 15332596*x^20 - 24484833*x^19 + 69153169*x^18 - 95340760*x^17 + 215848229*x^16 - 262803136*x^15 + 483174616*x^14 - 502558589*x^13 + 676121303*x^12 - 589256116*x^11 + 629711197*x^10 - 456787822*x^9 + 331102182*x^8 - 161825433*x^7 + 79124517*x^6 - 28239532*x^5 + 11237683*x^4 - 3082612*x^3 + 787679*x^2 - 101101*x + 10201)
 
gp: K = bnfinit(x^32 - 6*x^31 + 60*x^30 - 224*x^29 + 1535*x^28 - 4738*x^27 + 25964*x^26 - 65115*x^25 + 297875*x^24 - 640430*x^23 + 2525043*x^22 - 4605832*x^21 + 15332596*x^20 - 24484833*x^19 + 69153169*x^18 - 95340760*x^17 + 215848229*x^16 - 262803136*x^15 + 483174616*x^14 - 502558589*x^13 + 676121303*x^12 - 589256116*x^11 + 629711197*x^10 - 456787822*x^9 + 331102182*x^8 - 161825433*x^7 + 79124517*x^6 - 28239532*x^5 + 11237683*x^4 - 3082612*x^3 + 787679*x^2 - 101101*x + 10201, 1)
 

Normalized defining polynomial

\( x^{32} - 6 x^{31} + 60 x^{30} - 224 x^{29} + 1535 x^{28} - 4738 x^{27} + 25964 x^{26} - 65115 x^{25} + 297875 x^{24} - 640430 x^{23} + 2525043 x^{22} - 4605832 x^{21} + 15332596 x^{20} - 24484833 x^{19} + 69153169 x^{18} - 95340760 x^{17} + 215848229 x^{16} - 262803136 x^{15} + 483174616 x^{14} - 502558589 x^{13} + 676121303 x^{12} - 589256116 x^{11} + 629711197 x^{10} - 456787822 x^{9} + 331102182 x^{8} - 161825433 x^{7} + 79124517 x^{6} - 28239532 x^{5} + 11237683 x^{4} - 3082612 x^{3} + 787679 x^{2} - 101101 x + 10201 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(812088781483990309128958174884497308062548741403852104148801=3^{16}\cdot 13^{16}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.50$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(663=3\cdot 13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{663}(1,·)$, $\chi_{663}(259,·)$, $\chi_{663}(389,·)$, $\chi_{663}(274,·)$, $\chi_{663}(404,·)$, $\chi_{663}(662,·)$, $\chi_{663}(25,·)$, $\chi_{663}(155,·)$, $\chi_{663}(157,·)$, $\chi_{663}(287,·)$, $\chi_{663}(545,·)$, $\chi_{663}(38,·)$, $\chi_{663}(298,·)$, $\chi_{663}(560,·)$, $\chi_{663}(53,·)$, $\chi_{663}(443,·)$, $\chi_{663}(64,·)$, $\chi_{663}(196,·)$, $\chi_{663}(586,·)$, $\chi_{663}(77,·)$, $\chi_{663}(467,·)$, $\chi_{663}(599,·)$, $\chi_{663}(220,·)$, $\chi_{663}(610,·)$, $\chi_{663}(103,·)$, $\chi_{663}(365,·)$, $\chi_{663}(625,·)$, $\chi_{663}(118,·)$, $\chi_{663}(376,·)$, $\chi_{663}(506,·)$, $\chi_{663}(508,·)$, $\chi_{663}(638,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{8}$, $\frac{1}{4} a^{24} - \frac{1}{4} a^{18} - \frac{1}{4} a^{12} - \frac{1}{2} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4}$, $\frac{1}{4} a^{25} - \frac{1}{4} a^{19} - \frac{1}{4} a^{13} - \frac{1}{2} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{4} a^{26} - \frac{1}{4} a^{20} - \frac{1}{4} a^{14} - \frac{1}{2} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{27} - \frac{1}{4} a^{21} - \frac{1}{4} a^{15} + \frac{1}{4} a^{9} - \frac{1}{2} a^{6} + \frac{1}{4} a^{3} - \frac{1}{2}$, $\frac{1}{412} a^{28} - \frac{6}{103} a^{27} + \frac{19}{412} a^{26} - \frac{5}{412} a^{25} + \frac{3}{412} a^{24} - \frac{12}{103} a^{23} - \frac{65}{412} a^{22} + \frac{21}{103} a^{21} - \frac{9}{412} a^{20} - \frac{77}{412} a^{19} + \frac{3}{412} a^{18} - \frac{35}{206} a^{17} - \frac{21}{412} a^{16} + \frac{27}{206} a^{15} - \frac{3}{412} a^{14} - \frac{61}{412} a^{13} - \frac{15}{412} a^{12} + \frac{79}{206} a^{11} - \frac{159}{412} a^{10} - \frac{93}{206} a^{9} + \frac{157}{412} a^{8} - \frac{119}{412} a^{7} + \frac{101}{412} a^{6} - \frac{45}{206} a^{5} + \frac{177}{412} a^{4} + \frac{33}{206} a^{3} - \frac{137}{412} a^{2} - \frac{191}{412} a - \frac{21}{412}$, $\frac{1}{41612} a^{29} + \frac{29}{41612} a^{28} - \frac{1665}{41612} a^{27} - \frac{3427}{41612} a^{26} - \frac{1367}{20806} a^{25} - \frac{410}{10403} a^{24} - \frac{5287}{41612} a^{23} + \frac{9823}{41612} a^{22} - \frac{7093}{41612} a^{21} - \frac{9515}{41612} a^{20} - \frac{1689}{10403} a^{19} + \frac{460}{10403} a^{18} + \frac{1625}{41612} a^{17} - \frac{2707}{41612} a^{16} + \frac{9657}{41612} a^{15} + \frac{5857}{41612} a^{14} - \frac{4199}{20806} a^{13} - \frac{2945}{20806} a^{12} + \frac{11305}{41612} a^{11} - \frac{6759}{41612} a^{10} + \frac{61}{412} a^{9} - \frac{187}{412} a^{8} - \frac{3148}{10403} a^{7} + \frac{1653}{20806} a^{6} - \frac{13451}{41612} a^{5} - \frac{18157}{41612} a^{4} - \frac{11883}{41612} a^{3} + \frac{11397}{41612} a^{2} + \frac{1623}{20806} a + \frac{49}{103}$, $\frac{1}{57195470783740875921368596} a^{30} - \frac{27705368035626342218}{14298867695935218980342149} a^{29} - \frac{48580425318833474833}{138823958212963291071283} a^{28} + \frac{2112962579320934974389663}{57195470783740875921368596} a^{27} - \frac{629839301188175490292040}{14298867695935218980342149} a^{26} - \frac{713845088400681724381673}{28597735391870437960684298} a^{25} + \frac{2171600926749033680128631}{57195470783740875921368596} a^{24} - \frac{2963531553426410716774846}{14298867695935218980342149} a^{23} + \frac{5144609306547962334384619}{28597735391870437960684298} a^{22} + \frac{3773091525776345973150819}{57195470783740875921368596} a^{21} + \frac{4657479431689182821604637}{28597735391870437960684298} a^{20} + \frac{117326989151959012603397}{14298867695935218980342149} a^{19} - \frac{8814940780085465723697563}{57195470783740875921368596} a^{18} - \frac{2676613642241667191775452}{14298867695935218980342149} a^{17} + \frac{3266653691552708714230832}{14298867695935218980342149} a^{16} - \frac{6324748959011337722853963}{57195470783740875921368596} a^{15} - \frac{5985293285418762401422239}{28597735391870437960684298} a^{14} - \frac{3289512480830608658550624}{14298867695935218980342149} a^{13} + \frac{4371471438672340434703273}{57195470783740875921368596} a^{12} + \frac{8953058179896614124284811}{28597735391870437960684298} a^{11} - \frac{12311932569565663795433659}{28597735391870437960684298} a^{10} + \frac{173786700637412682090789}{566291789938028474468996} a^{9} + \frac{1592191397289011854315195}{28597735391870437960684298} a^{8} - \frac{5673122286558957621145009}{28597735391870437960684298} a^{7} - \frac{1191777310318336162025059}{57195470783740875921368596} a^{6} - \frac{557371077626476904624400}{14298867695935218980342149} a^{5} - \frac{5701307458898393788394474}{14298867695935218980342149} a^{4} + \frac{6136780314634755778739759}{57195470783740875921368596} a^{3} + \frac{2488957278954652100221135}{28597735391870437960684298} a^{2} + \frac{3664585435830405245527397}{14298867695935218980342149} a + \frac{35529072886029767113489}{283145894969014237234498}$, $\frac{1}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{31} - \frac{35531195630243543392907580037885487738025111639296545383285698648210219652826217}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{30} + \frac{22697531729231727733154274271812414878568995979712977740022377263937886069363960004710196955675962769}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{29} - \frac{1064445199114071880639629167067178124898006992117176516968711041570791186658091704227852531241251650807}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{28} - \frac{141571267165642196809889685476791654574262412478088847723101452690548403261228923836425172697265384255455}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{27} + \frac{78795494082674368345791272471887244670425454434360752335661445429558526467599870755950131548544655757271}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{26} - \frac{780396229427830486479636994450696527077150404483289870395135903746922119625988043857925276035694153102255}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{25} - \frac{260711189352867708671234213664468332207197219661358443898917700593055826642882625453799164083964590797045}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{24} - \frac{434495308492831700431290494988647284023366436627974347847471368348921193709147855828423704895502155059677}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{23} - \frac{482281997829875869564779762003333516077763350144082670160904179856242111565365266944305219526436361239437}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{22} - \frac{1612829193110232253200136879788171768427981980688443255863277250863965450493393779236584817001722115517881}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{21} - \frac{1779353515430295405866596316687652022487426182637305506874855797300371190834639129513833730951937417578057}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{20} + \frac{1793036602804315680456121732581169025205561918404044984199208037326026640266506985203673971546063069323265}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{19} + \frac{1188012856048986865565549193974453531297998702320472331526091278769925496798719304717153827686823542256039}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{18} + \frac{181664483708977499499501831285120515873278778400716190056980646791472008800711491184828047645456325409773}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{17} + \frac{40223765902490591686925164496738171751337135910095845124682826276161525330010386958766299575478293759831}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{16} - \frac{1694580181935067623247708608854249140315249128202580038447603520593901808364388931619631053334489391060749}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{15} - \frac{940130272253101558044971612830517841098037618484445209816016933631928436077052280452775120276109209531319}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{14} - \frac{1128679009085131119575465417439295460267740141422571054117665718365258328271715828843600587888179384029235}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{13} - \frac{671305174724035340227991978150744671973413073060795879342761495388070380557063781082614438297532754882405}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{12} - \frac{2576200324726408299561320047349218592123688196357258396053462846179165310651840014708580177489566801927661}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{11} + \frac{177649278917604949677298715675034356892310138755181212351589431013510594955764599689278603437895767119733}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{10} + \frac{2110094100908070552302167685891354994830034173686115620242767023935870136408980711227142314478289099339469}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{9} + \frac{3131426581357030166188584796286536378232969212742359840372750866470304183682271223279475605422542228770251}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{8} + \frac{2803701024450011610503827545384454052602592347117179849190426786632321903245509704902083320697573536124935}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{7} + \frac{1489201855066490284372096207998033772989698085017131214264203639300079637353662218762842295216206299923433}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{6} - \frac{3147667836570508248169029902508179796846404236349492338775247585170496003191404321073998187661662780789981}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{5} + \frac{3588047843300780602540951776373777319504482446649883354712308723489845232921940423242522405908367387204017}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{4} - \frac{2019836869709212412316365420768087435728747712870999380412113424873372006277420149096214951747752935909463}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{3} + \frac{116084456569985256215738030583155090898670488845341349327034452874021871321880915685433679831074360934461}{7206571330537620430066436954536711547999670603607181030567078293517465225867198371068876005173971306821188} a^{2} - \frac{3319423785960848244021752289378552339037514159899006127938609997099001029514728833662258197014848066151}{9660283284902976447810237204472803683645671050411770818454528543589095477033777977304123331332401215578} a + \frac{10683190013282569782537839356204101723695999607333907740884374783549196224637954339701455191122747675985}{35676095695730794208249687893746096772275597047560302131520189571868639732015833520142950520663224291194}$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{25289314214689645909606386816494140363980941939354892334928909140865203209125938667}{1955665039595391102424895642049475450996891168666098673999582216420963838542192008761228} a^{31} + \frac{37666312859514383648178244808016666279050786294813267633187984573745761961064087173}{488916259898847775606223910512368862749222792166524668499895554105240959635548002190307} a^{30} - \frac{756121398875216999986062703429019407411821627227537415982842995982054030609949242639}{977832519797695551212447821024737725498445584333049336999791108210481919271096004380614} a^{29} + \frac{54449028708082330485972281693413882333428149440532602223064150388192511126926411797}{18987039219372729149756268369412383019387292899670860912617303072048192607205747657876} a^{28} - \frac{38656693773228599145896715801748111178871742162556905215132878246573337590786491712609}{1955665039595391102424895642049475450996891168666098673999582216420963838542192008761228} a^{27} + \frac{118459977930449093651269128638302570753360340159374537352615351091015185977045027840981}{1955665039595391102424895642049475450996891168666098673999582216420963838542192008761228} a^{26} - \frac{653505714804771978454866346068609488459387642215502582747854604723282481574168724239545}{1955665039595391102424895642049475450996891168666098673999582216420963838542192008761228} a^{25} + \frac{1624841871169948524254685568670376943670412439407717342155285870300634892697626577817445}{1955665039595391102424895642049475450996891168666098673999582216420963838542192008761228} a^{24} - \frac{3748205349762495395945361531924642431811099866859931086882506351172406723894614883863595}{977832519797695551212447821024737725498445584333049336999791108210481919271096004380614} a^{23} + \frac{15957618948020650824715161841418517671615519262491737748593460038873613461419176345489345}{1955665039595391102424895642049475450996891168666098673999582216420963838542192008761228} a^{22} - \frac{63547954596515776290695853006131263484106337989259978473236030084199483308097570874475993}{1955665039595391102424895642049475450996891168666098673999582216420963838542192008761228} a^{21} + \frac{114557883953719314650394438714544444431589269776957901836265349831775695999244404284137805}{1955665039595391102424895642049475450996891168666098673999582216420963838542192008761228} a^{20} - \frac{385994988454839258376521775720109183861836902014965766572812399531884462613709786603063507}{1955665039595391102424895642049475450996891168666098673999582216420963838542192008761228} a^{19} + \frac{608339662424585694685083607200557756494699127532385392449852609357066261557668767582669197}{1955665039595391102424895642049475450996891168666098673999582216420963838542192008761228} a^{18} - \frac{870864462192682981623783263327542948048851149475840842160909437510367352718161084207854897}{977832519797695551212447821024737725498445584333049336999791108210481919271096004380614} a^{17} + \frac{2366128360076392327685484056954810016384337991192747249525156375957363176632003722436948327}{1955665039595391102424895642049475450996891168666098673999582216420963838542192008761228} a^{16} - \frac{5441816327365276635087977911764618645364149620108696390897829026955229504597452395068024585}{1955665039595391102424895642049475450996891168666098673999582216420963838542192008761228} a^{15} + \frac{6525912301501682953659963534229480496995747729150674484052207119165021657000891213056894055}{1955665039595391102424895642049475450996891168666098673999582216420963838542192008761228} a^{14} - \frac{12198925738118940282446080359844791722428372143441901980106470237631133799461303676664158979}{1955665039595391102424895642049475450996891168666098673999582216420963838542192008761228} a^{13} + \frac{12490356556687193147264671010876270099337109866223098030777108619778842711946105848666775371}{1955665039595391102424895642049475450996891168666098673999582216420963838542192008761228} a^{12} - \frac{4282826354595057027084491991945829733727066671250616907079350048166421837177867049968473242}{488916259898847775606223910512368862749222792166524668499895554105240959635548002190307} a^{11} + \frac{142985296172780651248994237861315570228027347232706137895278433775197222271861594552392891}{18987039219372729149756268369412383019387292899670860912617303072048192607205747657876} a^{10} - \frac{16053496735075278970490390043994254376324473218585435230666634968796132741814566771223714195}{1955665039595391102424895642049475450996891168666098673999582216420963838542192008761228} a^{9} + \frac{11480278870913081546308823629121420679707605871901177358967604914439554978577524980732588679}{1955665039595391102424895642049475450996891168666098673999582216420963838542192008761228} a^{8} - \frac{8546829496896310564577081426917385034077959386335478995132498552199583276421249956700941809}{1955665039595391102424895642049475450996891168666098673999582216420963838542192008761228} a^{7} + \frac{4162958739602595846186787739862508800379149918799603942036985550629682587856779134519137519}{1955665039595391102424895642049475450996891168666098673999582216420963838542192008761228} a^{6} - \frac{531157012282595880799991515953156584073454091133037613288179211776723758243142305872945407}{488916259898847775606223910512368862749222792166524668499895554105240959635548002190307} a^{5} + \frac{732369927180712761010983441824207878530532117519671176875021079144618531733887137692258307}{1955665039595391102424895642049475450996891168666098673999582216420963838542192008761228} a^{4} - \frac{296301315463818147456051454318764388819638372829852810028173769291998069286452211455563163}{1955665039595391102424895642049475450996891168666098673999582216420963838542192008761228} a^{3} + \frac{69436872603561191861628070860532836554352596443509888934239165223467578738949802903966995}{1955665039595391102424895642049475450996891168666098673999582216420963838542192008761228} a^{2} - \frac{5299826892359598205429615627234690658635636448105597502603010809718183584062496631396610}{488916259898847775606223910512368862749222792166524668499895554105240959635548002190307} a + \frac{27047680455338304265230743347705139054053410174176657743322835673366197340070249362039}{19363020194013773291335600416331440108880110580852462118807744717039245926160316918428} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_8$ (as 32T37):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2^2\times C_8$
Character table for $C_2^2\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{-663}) \), \(\Q(\sqrt{221}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{-3}, \sqrt{221})\), \(\Q(\sqrt{17}, \sqrt{-39})\), \(\Q(\sqrt{13}, \sqrt{-51})\), \(\Q(\sqrt{13}, \sqrt{17})\), \(\Q(\sqrt{-39}, \sqrt{-51})\), \(\Q(\sqrt{-3}, \sqrt{17})\), \(\Q(\sqrt{-3}, \sqrt{13})\), 4.4.4913.1, 4.0.7472673.1, 4.4.830297.1, 4.0.44217.1, 8.0.193220905761.10, 8.0.55840841764929.8, 8.0.55840841764929.3, 8.8.689393108209.1, 8.0.55840841764929.5, 8.0.1955143089.1, 8.0.55840841764929.12, 8.0.33237432513.1, 8.8.11719682839553.1, 8.0.949294310003793.2, \(\Q(\zeta_{17})^+\), 16.0.3118199609015838915694375041.1, 16.0.901159687005577446635674386849.3, 16.0.901159687005577446635674386849.4, 16.0.901159687005577446635674386849.1, 16.16.137350965859713069141239809.1, 16.0.1104726920056229495169.1, 16.0.901159687005577446635674386849.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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

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

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17Data not computed