↑

Properties

Label 32.0.30424831247...0336.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{124}\cdot 3^{16}\cdot 7^{16}$
Root discriminant $67.24$
Ramified primes $2, 3, 7$
Class number $584$ (GRH)
Class group $[584]$ (GRH)
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![24964, -230048, 1107472, -3743168, 9918048, -21489824, 39138560, -61558848, 84840210, -102977936, 111890280, -109734560, 94104568, -70800080, 51652776, -34704128, 16043315, -5531224, 4810220, -3819200, 890618, 179256, 119568, -162808, 22459, 10616, 1136, -2264, 234, 80, 4, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 8*x^31 + 4*x^30 + 80*x^29 + 234*x^28 - 2264*x^27 + 1136*x^26 + 10616*x^25 + 22459*x^24 - 162808*x^23 + 119568*x^22 + 179256*x^21 + 890618*x^20 - 3819200*x^19 + 4810220*x^18 - 5531224*x^17 + 16043315*x^16 - 34704128*x^15 + 51652776*x^14 - 70800080*x^13 + 94104568*x^12 - 109734560*x^11 + 111890280*x^10 - 102977936*x^9 + 84840210*x^8 - 61558848*x^7 + 39138560*x^6 - 21489824*x^5 + 9918048*x^4 - 3743168*x^3 + 1107472*x^2 - 230048*x + 24964)
 
gp: K = bnfinit(x^32 - 8*x^31 + 4*x^30 + 80*x^29 + 234*x^28 - 2264*x^27 + 1136*x^26 + 10616*x^25 + 22459*x^24 - 162808*x^23 + 119568*x^22 + 179256*x^21 + 890618*x^20 - 3819200*x^19 + 4810220*x^18 - 5531224*x^17 + 16043315*x^16 - 34704128*x^15 + 51652776*x^14 - 70800080*x^13 + 94104568*x^12 - 109734560*x^11 + 111890280*x^10 - 102977936*x^9 + 84840210*x^8 - 61558848*x^7 + 39138560*x^6 - 21489824*x^5 + 9918048*x^4 - 3743168*x^3 + 1107472*x^2 - 230048*x + 24964, 1)
 

Normalized defining polynomial

\( x^{32} - 8 x^{31} + 4 x^{30} + 80 x^{29} + 234 x^{28} - 2264 x^{27} + 1136 x^{26} + 10616 x^{25} + 22459 x^{24} - 162808 x^{23} + 119568 x^{22} + 179256 x^{21} + 890618 x^{20} - 3819200 x^{19} + 4810220 x^{18} - 5531224 x^{17} + 16043315 x^{16} - 34704128 x^{15} + 51652776 x^{14} - 70800080 x^{13} + 94104568 x^{12} - 109734560 x^{11} + 111890280 x^{10} - 102977936 x^{9} + 84840210 x^{8} - 61558848 x^{7} + 39138560 x^{6} - 21489824 x^{5} + 9918048 x^{4} - 3743168 x^{3} + 1107472 x^{2} - 230048 x + 24964 \)

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:  \(30424831247408100720538934941471930111515239368421944590336=2^{124}\cdot 3^{16}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.24$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(672=2^{5}\cdot 3\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{672}(1,·)$, $\chi_{672}(643,·)$, $\chi_{672}(265,·)$, $\chi_{672}(139,·)$, $\chi_{672}(659,·)$, $\chi_{672}(281,·)$, $\chi_{672}(155,·)$, $\chi_{672}(545,·)$, $\chi_{672}(419,·)$, $\chi_{672}(41,·)$, $\chi_{672}(43,·)$, $\chi_{672}(433,·)$, $\chi_{672}(307,·)$, $\chi_{672}(449,·)$, $\chi_{672}(323,·)$, $\chi_{672}(587,·)$, $\chi_{672}(337,·)$, $\chi_{672}(211,·)$, $\chi_{672}(377,·)$, $\chi_{672}(601,·)$, $\chi_{672}(475,·)$, $\chi_{672}(97,·)$, $\chi_{672}(379,·)$, $\chi_{672}(209,·)$, $\chi_{672}(617,·)$, $\chi_{672}(491,·)$, $\chi_{672}(113,·)$, $\chi_{672}(83,·)$, $\chi_{672}(547,·)$, $\chi_{672}(169,·)$, $\chi_{672}(505,·)$, $\chi_{672}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{16} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{17} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{26} - \frac{1}{2} a^{18} - \frac{1}{2} a^{10}$, $\frac{1}{2686} a^{27} - \frac{122}{1343} a^{26} - \frac{419}{2686} a^{25} - \frac{106}{1343} a^{24} + \frac{607}{1343} a^{23} + \frac{497}{1343} a^{22} + \frac{57}{1343} a^{21} + \frac{461}{1343} a^{20} + \frac{93}{2686} a^{19} + \frac{97}{1343} a^{18} - \frac{1215}{2686} a^{17} - \frac{268}{1343} a^{16} - \frac{125}{1343} a^{15} - \frac{303}{1343} a^{14} + \frac{602}{1343} a^{13} + \frac{555}{1343} a^{12} + \frac{567}{2686} a^{11} - \frac{241}{1343} a^{10} - \frac{587}{2686} a^{9} - \frac{539}{1343} a^{8} - \frac{174}{1343} a^{7} + \frac{33}{79} a^{6} - \frac{299}{1343} a^{5} + \frac{24}{79} a^{4} - \frac{317}{1343} a^{3} + \frac{60}{1343} a^{2} - \frac{627}{1343} a - \frac{5}{17}$, $\frac{1}{18568318} a^{28} - \frac{7}{18568318} a^{27} + \frac{151261}{18568318} a^{26} - \frac{378121}{9284159} a^{25} - \frac{1285592}{9284159} a^{24} + \frac{102609}{299489} a^{23} - \frac{2099447}{9284159} a^{22} - \frac{4533428}{9284159} a^{21} + \frac{4064959}{18568318} a^{20} - \frac{8940947}{18568318} a^{19} + \frac{4124797}{18568318} a^{18} - \frac{4290758}{9284159} a^{17} - \frac{994340}{9284159} a^{16} + \frac{2838720}{9284159} a^{15} - \frac{2939857}{9284159} a^{14} + \frac{872478}{9284159} a^{13} + \frac{4606899}{18568318} a^{12} - \frac{6656311}{18568318} a^{11} - \frac{3252069}{18568318} a^{10} - \frac{3833856}{9284159} a^{9} + \frac{1843607}{9284159} a^{8} + \frac{2243766}{9284159} a^{7} - \frac{3637143}{9284159} a^{6} - \frac{4560104}{9284159} a^{5} - \frac{3110705}{9284159} a^{4} + \frac{1024848}{9284159} a^{3} - \frac{3633995}{9284159} a^{2} - \frac{26043}{117521} a - \frac{39265}{117521}$, $\frac{1}{18568318} a^{29} - \frac{437}{9284159} a^{27} + \frac{274933}{18568318} a^{26} + \frac{77101}{9284159} a^{25} + \frac{1018692}{9284159} a^{24} + \frac{124928}{546127} a^{23} - \frac{1974709}{9284159} a^{22} - \frac{31235}{235042} a^{21} + \frac{4634350}{9284159} a^{20} + \frac{833721}{9284159} a^{19} - \frac{9212621}{18568318} a^{18} - \frac{3626514}{9284159} a^{17} - \frac{499248}{9284159} a^{16} - \frac{1194703}{9284159} a^{15} - \frac{1476940}{9284159} a^{14} + \frac{824909}{18568318} a^{13} + \frac{2661533}{9284159} a^{12} - \frac{3050391}{9284159} a^{11} + \frac{5736621}{18568318} a^{10} + \frac{1075538}{9284159} a^{9} + \frac{4281779}{9284159} a^{8} + \frac{82091}{546127} a^{7} - \frac{3930443}{9284159} a^{6} + \frac{1158122}{9284159} a^{5} + \frac{756256}{9284159} a^{4} - \frac{3953751}{9284159} a^{3} + \frac{516114}{9284159} a^{2} + \frac{3580936}{9284159} a + \frac{15491}{117521}$, $\frac{1}{8337174782} a^{30} + \frac{7}{268941122} a^{29} + \frac{222773}{4168587391} a^{27} + \frac{18814374}{4168587391} a^{26} - \frac{811915875}{4168587391} a^{25} - \frac{1646425429}{8337174782} a^{24} - \frac{1997714440}{4168587391} a^{23} - \frac{3589516103}{8337174782} a^{22} + \frac{3697686633}{8337174782} a^{21} - \frac{1068260764}{4168587391} a^{20} + \frac{1963949}{9284159} a^{19} + \frac{858090527}{4168587391} a^{18} + \frac{120028816}{245211023} a^{17} + \frac{938071085}{8337174782} a^{16} - \frac{332207766}{4168587391} a^{15} - \frac{3067274577}{8337174782} a^{14} - \frac{2770052051}{8337174782} a^{13} - \frac{671928279}{4168587391} a^{12} - \frac{1690728793}{4168587391} a^{11} - \frac{732766362}{4168587391} a^{10} - \frac{1117789777}{4168587391} a^{9} + \frac{1225749567}{8337174782} a^{8} - \frac{663317918}{4168587391} a^{7} + \frac{1569029617}{4168587391} a^{6} + \frac{1031170998}{4168587391} a^{5} - \frac{505863062}{4168587391} a^{4} - \frac{80492595}{245211023} a^{3} + \frac{702965461}{4168587391} a^{2} - \frac{1741664407}{4168587391} a + \frac{6934090}{52766929}$, $\frac{1}{17384211314360424837269633850330604957316269675004164867709198391884861486600615752443742} a^{31} + \frac{564655088723008455942107517723054567975634185246354007038373356742472195707603}{17384211314360424837269633850330604957316269675004164867709198391884861486600615752443742} a^{30} + \frac{55818200787234062494612823324367495114430838592649827334087497638809493785050033}{8692105657180212418634816925165302478658134837502082433854599195942430743300307876221871} a^{29} + \frac{138051763272490812905298937827604930578349049787630657385085614608648528592924702}{8692105657180212418634816925165302478658134837502082433854599195942430743300307876221871} a^{28} + \frac{981134057146910618433558700217563770606169370380007577317961346446016455917707125}{369876836475753719941907103198523509730133397340514146121472306210316201842566292605186} a^{27} + \frac{2357073370009852141143286743870553097620224455914963494602932809153344902941742133026795}{17384211314360424837269633850330604957316269675004164867709198391884861486600615752443742} a^{26} + \frac{703322316739985907958101068002792188777842150802311829765167973408326662725157818361419}{17384211314360424837269633850330604957316269675004164867709198391884861486600615752443742} a^{25} + \frac{1291534627166746550245367666262317914100714675892151045106073319735885821450355057726627}{17384211314360424837269633850330604957316269675004164867709198391884861486600615752443742} a^{24} - \frac{1874751296580820972528166340377960331107264165196623948453166650691566542965617214866713}{17384211314360424837269633850330604957316269675004164867709198391884861486600615752443742} a^{23} - \frac{6300382062604997569965053895979620742111073069048818213379617746206914495264876832809559}{17384211314360424837269633850330604957316269675004164867709198391884861486600615752443742} a^{22} - \frac{605739159096302093634706370473945327673156627862426042182695466185337510237367733339285}{8692105657180212418634816925165302478658134837502082433854599195942430743300307876221871} a^{21} - \frac{613952143397106808663283959842198004892649050921415699827455774527608725177727427390913}{8692105657180212418634816925165302478658134837502082433854599195942430743300307876221871} a^{20} + \frac{7048321969553632256315857981277966541134895023505025429647009205325977379497103707699219}{17384211314360424837269633850330604957316269675004164867709198391884861486600615752443742} a^{19} + \frac{6346723976863438660201773423416338858617811701636192891417988411638216487920148075552267}{17384211314360424837269633850330604957316269675004164867709198391884861486600615752443742} a^{18} - \frac{7056686441597639644194067910650784520157172545930945232358570050324973362427013396104093}{17384211314360424837269633850330604957316269675004164867709198391884861486600615752443742} a^{17} - \frac{109114344453000144417972619767089347436534486398203065216475801829756464901864477200123}{560781010140658865718375285494535643784395795967876286055135431996285854406471475885282} a^{16} + \frac{6722427870259810148198046134248075743681367579973710486416217435328763159496975416183269}{17384211314360424837269633850330604957316269675004164867709198391884861486600615752443742} a^{15} + \frac{7711223107984073121807394971526949103283807101482896525708603733438369845746343184789345}{17384211314360424837269633850330604957316269675004164867709198391884861486600615752443742} a^{14} - \frac{39519450139873460848201000641814705672282309261539666801933142467540211152693381771898}{280390505070329432859187642747267821892197897983938143027567715998142927203235737942641} a^{13} + \frac{1586378889587862013862319840247442761062208967010390261563142646323639956342212880549506}{8692105657180212418634816925165302478658134837502082433854599195942430743300307876221871} a^{12} - \frac{6938999168978655096655612640574009818611147533230698387370657592847934719749502722050269}{17384211314360424837269633850330604957316269675004164867709198391884861486600615752443742} a^{11} + \frac{5581637205695353184605484002910114637892266905360373976440734053415137713065470808175919}{17384211314360424837269633850330604957316269675004164867709198391884861486600615752443742} a^{10} + \frac{2537363107994225922680568855152978532037596198646250839364628663857205647908392116996231}{17384211314360424837269633850330604957316269675004164867709198391884861486600615752443742} a^{9} - \frac{2567752757454329848037804371323558339803600347709892927854674948532646497937159601234699}{17384211314360424837269633850330604957316269675004164867709198391884861486600615752443742} a^{8} + \frac{1449029692251756632429575357245911078355054083271083845844143189637352533186291582434248}{8692105657180212418634816925165302478658134837502082433854599195942430743300307876221871} a^{7} + \frac{3900844415448458905795211596014394183132851680692523571441943829900391567528509483131972}{8692105657180212418634816925165302478658134837502082433854599195942430743300307876221871} a^{6} + \frac{2918313845777958248715256778191526066860432201617739113478977304965781525000404628018922}{8692105657180212418634816925165302478658134837502082433854599195942430743300307876221871} a^{5} - \frac{3717137252908963412666819959039532484236878086451058537708896611451488833794550446161737}{8692105657180212418634816925165302478658134837502082433854599195942430743300307876221871} a^{4} - \frac{304571991912004555828648000562379439981364351651751215065129481136541333708465347437990}{8692105657180212418634816925165302478658134837502082433854599195942430743300307876221871} a^{3} - \frac{4026783967704455833626154069132745429477127723074391507884846528926031188958572935792028}{8692105657180212418634816925165302478658134837502082433854599195942430743300307876221871} a^{2} + \frac{3142399582012115693062651759732251407181312511640331158998339283917167989491980897316743}{8692105657180212418634816925165302478658134837502082433854599195942430743300307876221871} a - \frac{1981399038196141205958583822237255114407696625028598090923018738524836484838920915340}{6472156111079830542542678276370292240251775753910709183808338939644401149143937361297}$

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

Class group and class number

$C_{584}$, which has order $584$ (assuming GRH)

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{1363503891982943661950762423072740726708283139943367320883785990680892398782544}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a^{31} - \frac{10125503926820395842107285731076455152580589050238249610497032557261436759590570}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a^{30} - \frac{388123584129043225420721959518586223266106466691919348603599205058017395253604}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a^{29} + \frac{109027492763268156510970989996241807380039638803282656575738818180877309652058330}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a^{28} + \frac{8127844857653351940779327744771200871004695595875413469527739875522022416649552}{59581885275982360457365452842393951573458707013143975879902250949409766023370689} a^{27} - \frac{2869698963011620631345459717222049123365816610386533936604081193741168958956868898}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a^{26} - \frac{110882085636143125774149579681472612103308314049616464703988452267313812904023988}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a^{25} + \frac{28908017893666901385304286952654282302038729071905910504052273242044118638132506189}{5600697215942341882992352567185031447905118459235533732710811589244518006196844766} a^{24} + \frac{39018239742844287165645919762122436924614386895829952628399752677422631182302592628}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a^{23} - \frac{199787217433066638480087761962754524155843440834523829864510220111844699379032182976}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a^{22} + \frac{2765707461059837341476930098732732900904300956969121321450547130707208910303233380}{164726388704186525970363310799559748467797601742221580373847399683662294299907199} a^{21} + \frac{273570882312808678560289967805168666017618011056021652495947106869556410803974471877}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a^{20} + \frac{1376029792704720459512945956221020576636914167251345773052707037757183769945851586308}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a^{19} - \frac{4418121394083305600119294406515650352746774788108376260068352037089536357394392201182}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a^{18} + \frac{3987001237686129770053768418660183526346783878816347285314924529599874285747863034912}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a^{17} - \frac{10443133048816137528777508496256582359354573902858653168593848865522684319701486250667}{5600697215942341882992352567185031447905118459235533732710811589244518006196844766} a^{16} + \frac{18892401288754413937901658666659807017913552376760577283381001715840863253453428866732}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a^{15} - \frac{36386417580554357217287417895346779471884352113056729069036900611494396464001601242178}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a^{14} + \frac{49300462946768972499502896339261893501347700316292436548252571936904609561224089209208}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a^{13} - \frac{68006419794992771567988708654377187313639092161967950545787904478712901874425896300651}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a^{12} + \frac{88960771644717317861640689542942416287376849116987081242729829543303759523653339400912}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a^{11} - \frac{98075199726837965437042916456083983870268951402727943087198368185543646775881396160988}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a^{10} + \frac{95787472575884298131482160574595036797768636879323976850513045938544931950484976312784}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a^{9} - \frac{170003425817568403113235513024684007643714220518812120337395117934358911666404891036521}{5600697215942341882992352567185031447905118459235533732710811589244518006196844766} a^{8} + \frac{66522356342902714646799462703861976651319191313314266897969442336733813815719973486408}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a^{7} - \frac{45502803799923385119211094486689393418649479334182707075104042801903916233501735491728}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a^{6} + \frac{27112065328241874996837693236642317887974817140304982194945330329979071797464601134976}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a^{5} - \frac{13682978954068269913495542561836003064893238777566740881528219980296878375037042886474}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a^{4} + \frac{5661756839578241123852171687117010284276749667856182185045417021583424269346284351952}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a^{3} - \frac{1864273349063501163131716855952581092678957095588576214890335001288565201827393970336}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a^{2} + \frac{451474081196540674381025802884882985966584975716363384067018696279033928470529911104}{2800348607971170941496176283592515723952559229617766866355405794622259003098422383} a - \frac{756977737424503626354895187803930936012995655963548623769140272025258801424199653}{35447450733812290398685775741677414227247585185035023624751972083826063330359777} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 94813681955904.33 \) (assuming GRH)
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

\(\Q(\sqrt{-14}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-6}, \sqrt{-14})\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{-14})\), \(\Q(\sqrt{2}, \sqrt{21})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-6}, \sqrt{-7})\), \(\Q(\zeta_{16})^+\), 4.0.100352.5, 4.4.903168.2, 4.0.18432.2, 8.0.796594176.2, 8.0.10070523904.2, 8.0.815712436224.2, 8.8.815712436224.1, 8.0.815712436224.1, 8.0.339738624.2, 8.0.815712436224.5, 8.0.417644767346688.52, 8.8.173946175488.1, 8.0.2147483648.1, 8.8.5156108238848.1, 16.0.665386778610493267378176.2, 16.0.174427151692069147083584569344.1, 16.0.26585452170716224216367104.2, 16.0.174427151692069147083584569344.2, 16.16.174427151692069147083584569344.2, 16.0.174427151692069147083584569344.4, 16.0.30257271966902092038144.2

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 R R ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
2Data not computed
3Data not computed
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$