Properties

Label 32.0.115...176.1
Degree $32$
Signature $[0, 16]$
Discriminant $1.157\times 10^{48}$
Root discriminant $31.77$
Ramified primes $2, 3, 13, 17$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group $C_2\times D_4^2$ (as 32T1016)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 10*x^30 - 28*x^29 + 56*x^28 - 208*x^27 + 604*x^26 - 876*x^25 + 2547*x^24 - 5760*x^23 + 7684*x^22 - 11384*x^21 + 37676*x^20 - 21776*x^19 + 12262*x^18 - 135456*x^17 + 96553*x^16 + 443172*x^15 + 1107228*x^14 + 758160*x^13 - 662400*x^12 - 2309040*x^11 - 1681344*x^10 + 1407456*x^9 + 4596912*x^8 + 3872448*x^7 - 62208*x^6 - 4385664*x^5 - 3545856*x^4 + 3919104*x^2 + 3359232*x + 1679616)
 
gp: K = bnfinit(x^32 + 10*x^30 - 28*x^29 + 56*x^28 - 208*x^27 + 604*x^26 - 876*x^25 + 2547*x^24 - 5760*x^23 + 7684*x^22 - 11384*x^21 + 37676*x^20 - 21776*x^19 + 12262*x^18 - 135456*x^17 + 96553*x^16 + 443172*x^15 + 1107228*x^14 + 758160*x^13 - 662400*x^12 - 2309040*x^11 - 1681344*x^10 + 1407456*x^9 + 4596912*x^8 + 3872448*x^7 - 62208*x^6 - 4385664*x^5 - 3545856*x^4 + 3919104*x^2 + 3359232*x + 1679616, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1679616, 3359232, 3919104, 0, -3545856, -4385664, -62208, 3872448, 4596912, 1407456, -1681344, -2309040, -662400, 758160, 1107228, 443172, 96553, -135456, 12262, -21776, 37676, -11384, 7684, -5760, 2547, -876, 604, -208, 56, -28, 10, 0, 1]);
 

\( x^{32} + 10 x^{30} - 28 x^{29} + 56 x^{28} - 208 x^{27} + 604 x^{26} - 876 x^{25} + 2547 x^{24} - 5760 x^{23} + 7684 x^{22} - 11384 x^{21} + 37676 x^{20} - 21776 x^{19} + 12262 x^{18} - 135456 x^{17} + 96553 x^{16} + 443172 x^{15} + 1107228 x^{14} + 758160 x^{13} - 662400 x^{12} - 2309040 x^{11} - 1681344 x^{10} + 1407456 x^{9} + 4596912 x^{8} + 3872448 x^{7} - 62208 x^{6} - 4385664 x^{5} - 3545856 x^{4} + 3919104 x^{2} + 3359232 x + 1679616 \)

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

Invariants

Degree:  $32$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 16]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(1156745857256138299057661301456078800598048178176\)\(\medspace = 2^{72}\cdot 3^{16}\cdot 13^{8}\cdot 17^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $31.77$
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:  $2, 3, 13, 17$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{6} a^{18} - \frac{1}{3} a^{16} + \frac{1}{3} a^{15} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{2} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{6} a^{2}$, $\frac{1}{6} a^{19} - \frac{1}{3} a^{17} + \frac{1}{3} a^{16} + \frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{2} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{3}$, $\frac{1}{36} a^{20} - \frac{1}{18} a^{18} + \frac{2}{9} a^{17} + \frac{2}{9} a^{16} - \frac{4}{9} a^{15} + \frac{1}{9} a^{14} + \frac{5}{12} a^{12} + \frac{4}{9} a^{10} - \frac{2}{9} a^{9} + \frac{2}{9} a^{8} - \frac{2}{9} a^{7} - \frac{1}{18} a^{6} - \frac{11}{36} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{36} a^{21} - \frac{1}{18} a^{19} + \frac{1}{18} a^{18} + \frac{2}{9} a^{17} - \frac{1}{9} a^{16} - \frac{2}{9} a^{15} - \frac{1}{3} a^{14} + \frac{1}{12} a^{13} + \frac{1}{3} a^{12} + \frac{4}{9} a^{11} + \frac{5}{18} a^{10} + \frac{2}{9} a^{9} + \frac{1}{9} a^{8} + \frac{5}{18} a^{7} - \frac{1}{3} a^{6} + \frac{1}{36} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a^{2}$, $\frac{1}{216} a^{22} - \frac{1}{108} a^{20} + \frac{1}{27} a^{19} + \frac{1}{27} a^{18} + \frac{7}{27} a^{17} + \frac{19}{54} a^{16} - \frac{1}{2} a^{15} + \frac{17}{72} a^{14} + \frac{1}{3} a^{13} + \frac{11}{27} a^{12} - \frac{11}{54} a^{11} - \frac{25}{54} a^{10} + \frac{8}{27} a^{9} - \frac{37}{108} a^{8} + \frac{1}{3} a^{7} + \frac{25}{216} a^{6} + \frac{1}{18} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{216} a^{23} - \frac{1}{108} a^{21} + \frac{1}{108} a^{20} + \frac{1}{27} a^{19} - \frac{1}{54} a^{18} + \frac{7}{54} a^{17} - \frac{1}{18} a^{16} + \frac{1}{72} a^{15} - \frac{4}{9} a^{14} - \frac{7}{27} a^{13} + \frac{5}{108} a^{12} - \frac{25}{54} a^{11} - \frac{4}{27} a^{10} - \frac{13}{108} a^{9} - \frac{2}{9} a^{8} + \frac{1}{216} a^{7} + \frac{4}{9} a^{6} + \frac{17}{36} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{1296} a^{24} - \frac{1}{648} a^{22} + \frac{1}{162} a^{21} + \frac{1}{162} a^{20} + \frac{7}{162} a^{19} + \frac{19}{324} a^{18} - \frac{1}{12} a^{17} + \frac{161}{432} a^{16} - \frac{1}{9} a^{15} - \frac{35}{81} a^{14} - \frac{11}{324} a^{13} - \frac{79}{324} a^{12} + \frac{4}{81} a^{11} + \frac{179}{648} a^{10} - \frac{1}{9} a^{9} + \frac{241}{1296} a^{8} - \frac{35}{108} a^{7} + \frac{2}{9} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4}$, $\frac{1}{2592} a^{25} - \frac{1}{1296} a^{23} - \frac{1}{648} a^{22} - \frac{7}{648} a^{21} + \frac{1}{324} a^{20} + \frac{13}{648} a^{19} + \frac{7}{216} a^{18} - \frac{13}{32} a^{17} - \frac{13}{54} a^{16} + \frac{1}{162} a^{15} + \frac{38}{81} a^{14} - \frac{107}{324} a^{13} + \frac{119}{324} a^{12} + \frac{155}{1296} a^{11} + \frac{2}{27} a^{10} - \frac{239}{2592} a^{9} - \frac{19}{72} a^{8} - \frac{11}{36} a^{7} + \frac{43}{108} a^{6} - \frac{35}{72} a^{5} - \frac{1}{36} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{7776} a^{26} - \frac{1}{3888} a^{24} + \frac{1}{972} a^{23} + \frac{1}{972} a^{22} + \frac{7}{972} a^{21} + \frac{19}{1944} a^{20} - \frac{1}{72} a^{19} + \frac{161}{2592} a^{18} + \frac{4}{27} a^{17} - \frac{35}{486} a^{16} - \frac{659}{1944} a^{15} - \frac{403}{1944} a^{14} + \frac{85}{486} a^{13} + \frac{1475}{3888} a^{12} + \frac{17}{54} a^{11} - \frac{1055}{7776} a^{10} - \frac{35}{648} a^{9} + \frac{1}{27} a^{8} + \frac{1}{24} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{2}$, $\frac{1}{15552} a^{27} - \frac{1}{7776} a^{25} - \frac{1}{3888} a^{24} - \frac{7}{3888} a^{23} + \frac{1}{1944} a^{22} + \frac{13}{3888} a^{21} + \frac{7}{1296} a^{20} - \frac{13}{192} a^{19} - \frac{13}{324} a^{18} - \frac{161}{972} a^{17} - \frac{62}{243} a^{16} - \frac{431}{1944} a^{15} + \frac{119}{1944} a^{14} + \frac{155}{7776} a^{13} - \frac{26}{81} a^{12} - \frac{2831}{15552} a^{11} + \frac{197}{432} a^{10} - \frac{47}{216} a^{9} + \frac{259}{648} a^{8} - \frac{107}{432} a^{7} + \frac{35}{216} a^{6} + \frac{4}{9} a^{5} - \frac{2}{9} a^{4} + \frac{5}{12} a^{3} + \frac{1}{6} a^{2}$, $\frac{1}{51554880} a^{28} - \frac{101}{8592480} a^{27} - \frac{193}{25777440} a^{26} + \frac{229}{1982880} a^{25} - \frac{3547}{12888720} a^{24} - \frac{1837}{991440} a^{23} + \frac{1969}{991440} a^{22} + \frac{5351}{859248} a^{21} + \frac{15331}{5728320} a^{20} + \frac{7561}{190944} a^{19} + \frac{500699}{6444360} a^{18} + \frac{1021219}{5155488} a^{17} - \frac{17369}{644436} a^{16} - \frac{224971}{6444360} a^{15} + \frac{550619}{1982880} a^{14} + \frac{683311}{1432080} a^{13} - \frac{2472623}{51554880} a^{12} - \frac{146213}{954720} a^{11} + \frac{240473}{537030} a^{10} + \frac{1069177}{2864160} a^{9} + \frac{13993}{110160} a^{8} + \frac{37}{39780} a^{7} + \frac{67471}{238680} a^{6} - \frac{2503}{8840} a^{5} - \frac{313}{39780} a^{4} + \frac{1711}{6630} a^{3} + \frac{233}{663} a^{2} + \frac{29}{442} a - \frac{213}{1105}$, $\frac{1}{10620305280} a^{29} - \frac{31}{5310152640} a^{28} - \frac{13115}{1062030528} a^{27} - \frac{281}{9833616} a^{26} + \frac{89831}{1327538160} a^{25} - \frac{41909}{165942270} a^{24} + \frac{100537}{68078880} a^{23} - \frac{3080477}{2655076320} a^{22} - \frac{10846247}{3540101760} a^{21} - \frac{2059253}{590016960} a^{20} + \frac{208001983}{2655076320} a^{19} - \frac{13387781}{331884540} a^{18} - \frac{42335}{6555744} a^{17} - \frac{199972777}{442512720} a^{16} - \frac{1157883029}{5310152640} a^{15} + \frac{1322824483}{2655076320} a^{14} + \frac{3995785801}{10620305280} a^{13} - \frac{556027477}{5310152640} a^{12} - \frac{34752703}{177005088} a^{11} + \frac{174579797}{442512720} a^{10} + \frac{69499889}{147504240} a^{9} + \frac{31051717}{73752120} a^{8} + \frac{4142287}{24584040} a^{7} - \frac{567947}{3073005} a^{6} + \frac{608783}{1365780} a^{5} + \frac{86779}{1365780} a^{4} - \frac{26161}{52530} a^{3} - \frac{5147}{10506} a^{2} - \frac{62433}{227630} a - \frac{1913}{6695}$, $\frac{1}{5691232462535268480} a^{30} + \frac{10556179}{1897077487511756160} a^{29} + \frac{11515517803}{1422808115633817120} a^{28} + \frac{628349476111}{33477838014913344} a^{27} - \frac{55141026504313}{1422808115633817120} a^{26} - \frac{893889347059}{355702028908454280} a^{25} - \frac{9029550811823}{1422808115633817120} a^{24} - \frac{1301059037125}{2964183574237119} a^{23} + \frac{1873212109485829}{1897077487511756160} a^{22} + \frac{668770898016763}{632359162503918720} a^{21} - \frac{16139437209096787}{2845616231267634240} a^{20} + \frac{114125039789061889}{1422808115633817120} a^{19} + \frac{6312778282101481}{177851014454227140} a^{18} - \frac{19887438619059121}{83694595037283360} a^{17} + \frac{173496966638486251}{569123246253526848} a^{16} + \frac{20025544263409991}{63235916250391872} a^{15} + \frac{1950710959160900149}{5691232462535268480} a^{14} + \frac{153867258191753419}{379415497502351232} a^{13} - \frac{87768109241455601}{948538743755878080} a^{12} + \frac{25899800460481243}{158089790625979680} a^{11} - \frac{7855235470685423}{31617958125195936} a^{10} - \frac{216197970145843}{2634829843766328} a^{9} + \frac{245588390197741}{1013396093756280} a^{8} + \frac{522636676664147}{2195691536471940} a^{7} - \frac{6517614342479}{109784576823597} a^{6} + \frac{363758997961751}{731897178823980} a^{5} - \frac{91316574805}{827002461948} a^{4} - \frac{8374457789739}{20330477189555} a^{3} - \frac{8693239495658}{20330477189555} a^{2} + \frac{8383053890937}{40660954379110} a + \frac{1897953289512}{20330477189555}$, $\frac{1}{9278588148848302755311118537028683298013758072170240} a^{31} + \frac{80773350484898781029731078931}{289955879651509461103472454282146353062929939755320} a^{30} + \frac{6930530148625419969232410323741142739109}{1159823518606037844413889817128585412251719759021280} a^{29} - \frac{216360118744201270832489028190546228597181}{77321567907069189627592654475239027483447983934752} a^{28} + \frac{21978538306572577066295434641686578233490773367}{2319647037212075688827779634257170824503439518042560} a^{27} + \frac{25793122272042812021461102574963288773967430151}{579911759303018922206944908564292706125859879510640} a^{26} + \frac{87321629877848613682337914237245234791057900059}{773215679070691896275926544752390274834479839347520} a^{25} - \frac{37592347764082578656458258605071264388188843347}{463929407442415137765555926851434164900687903608512} a^{24} + \frac{1985405476278162968451242874101235703269960049609}{3092862716282767585103706179009561099337919357390080} a^{23} + \frac{712461575018233747838466987692876841716563217683}{386607839535345948137963272376195137417239919673760} a^{22} - \frac{45845488026028397320381944281638088473402686971737}{4639294074424151377655559268514341649006879036085120} a^{21} - \frac{43273750178690160562404488730324120433695683849}{39316051478170774386911519224697810584804059627840} a^{20} - \frac{4598693338750491093162025484019857420941725572279}{773215679070691896275926544752390274834479839347520} a^{19} + \frac{2572573330147282849087017815042215692726514226603}{64434639922557658022993878729365856236206653278960} a^{18} - \frac{8297823966169132302340913970566786451843731851669}{71373754991140790425470142592528333061644292862848} a^{17} + \frac{1650698283285155356070205061967538079621888615183}{57991175930301892220694490856429270612585987951064} a^{16} + \frac{1460456244149680454386530746256949399261880420002669}{9278588148848302755311118537028683298013758072170240} a^{15} - \frac{89680195898803326686119308664598170874928013276429}{463929407442415137765555926851434164900687903608512} a^{14} - \frac{3892353588544989069134768552238235693965807692827}{515477119380461264183951029834926849889653226231680} a^{13} - \frac{102551927440435222473374401437753693008289119443357}{257738559690230632091975514917463424944826613115840} a^{12} + \frac{77399189663945793840181748692072377826983757541}{495651076327366600176875990225891201816974255992} a^{11} - \frac{2290355944846786308572633801946267110201528593}{21057071870116881706860744682799299423596945516} a^{10} - \frac{6385526078416277141820366646390058255745081589399}{42956426615038438681995919152910570824137768852640} a^{9} + \frac{10420452735907507899223798438258342978880556412097}{21478213307519219340997959576455285412068884426320} a^{8} + \frac{15926725982723115497405282562252959933949153725}{59661703631997831502772109934598015033524678962} a^{7} - \frac{557277330180377570870947325888092697993498049679}{1193234072639956630055442198691960300670493579240} a^{6} + \frac{66408209763027289362992320140167636868583895021}{238646814527991326011088439738392060134098715848} a^{5} - \frac{208638423653071133608102138585332823249162468}{2924593315294011348175103428166569364388464655} a^{4} - \frac{69619932506781625166307355503558728106286265273}{198872345439992771675907033115326716778415596540} a^{3} - \frac{7014201647037662982354518507566594023252402982}{16572695453332730972992252759610559731534633045} a^{2} + \frac{2121136116642975340463333073797681201359339161}{16572695453332730972992252759610559731534633045} a - \frac{224152501093487781634039115523303792603964722}{3314539090666546194598450551922111946306926609}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{2382040416463893287579988002935126354438348091}{515477119380461264183951029834926849889653226231680} a^{31} + \frac{3682789795177403138638270834516944070227363203}{773215679070691896275926544752390274834479839347520} a^{30} - \frac{20967387111652544489826270846085005876635986313}{515477119380461264183951029834926849889653226231680} a^{29} + \frac{31243637747961692447976143633798921308980754199}{193303919767672974068981636188097568708619959836880} a^{28} - \frac{63129155110638127433087581505365914669819533501}{193303919767672974068981636188097568708619959836880} a^{27} + \frac{360364649717040292041352515912294909958117011551}{386607839535345948137963272376195137417239919673760} a^{26} - \frac{574748503058125071600352040260274794414533356249}{193303919767672974068981636188097568708619959836880} a^{25} + \frac{1849545817408832893584982932525108163474351617499}{386607839535345948137963272376195137417239919673760} a^{24} - \frac{545420148945689144129036656432652336071582454333}{57275235486717918242661225537214094432183691803520} a^{23} + \frac{6286740477284044237740641871516511148186478106421}{257738559690230632091975514917463424944826613115840} a^{22} - \frac{17737972569223884991294154622025789970222656832869}{515477119380461264183951029834926849889653226231680} a^{21} + \frac{4791806337799625457048373758245893324477026896171}{193303919767672974068981636188097568708619959836880} a^{20} - \frac{82060165737750402102115092497127928595837112474109}{773215679070691896275926544752390274834479839347520} a^{19} + \frac{2168959825369738001735847882358002015822799622527}{19330391976767297406898163618809756870861995983688} a^{18} + \frac{106766355635967619589371429383246572278464215296367}{773215679070691896275926544752390274834479839347520} a^{17} + \frac{69272108479477756457391396122135452470665863441249}{386607839535345948137963272376195137417239919673760} a^{16} - \frac{391365008615154854387254896451022970580844462880821}{515477119380461264183951029834926849889653226231680} a^{15} - \frac{5016225456734192320077182764748934845515474208641}{2621070098544718292460767948313187372320270641856} a^{14} - \frac{55874833171130120593100750802215200986053984886629}{34365141292030750945596735322328456659310215082112} a^{13} + \frac{21273441970050799169440881522061949394143181274337}{8591285323007687736399183830582114164827553770528} a^{12} + \frac{475317329405255026660982790262069946590404977681371}{85912853230076877363991838305821141648275537705280} a^{11} + \frac{11162586685496737716708344994002636716670682759527}{3790272936621038707234934042903873896247450192880} a^{10} - \frac{1066467453304509977653620069842689783439158389185}{178985110895993494508316329803794045100574036886} a^{9} - \frac{7019431277068432227719095506554039001473885868985}{715940443583973978033265319215176180402296147544} a^{8} - \frac{11238824635581134298565346081203266552079755987179}{2386468145279913260110884397383920601340987158480} a^{7} + \frac{87157418284967221709688369452342959796978943055}{13258156362666184778393802207688447785227706436} a^{6} + \frac{2263822084368440320266593843967294364176777920747}{198872345439992771675907033115326716778415596540} a^{5} + \frac{47799669900353767156719485395406138420344795301}{11698373261176045392700413712666277457553858620} a^{4} - \frac{494815471693783692344320667845565975084605181541}{66290781813330923891969011038442238926138532180} a^{3} - \frac{143399542719934548484754326616288733192503896177}{16572695453332730972992252759610559731534633045} a^{2} - \frac{67967154683506895197862508698305376868873332343}{33145390906665461945984505519221119463069266090} a + \frac{57098645926766317052443680808417814698553965638}{16572695453332730972992252759610559731534633045} \) (order $24$)
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:  \( 97772762708.42017 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{16}\cdot 97772762708.42017 \cdot 12}{24\sqrt{1156745857256138299057661301456078800598048178176}}\approx 0.268192611147582$ (assuming GRH)

Galois group

$C_2\times D_4^2$ (as 32T1016):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 128
The 50 conjugacy class representatives for $C_2\times D_4^2$ are not computed
Character table for $C_2\times D_4^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{6}) \), 4.0.1088.2, 4.4.4352.1, 4.0.39168.3, 4.4.9792.1, \(\Q(i, \sqrt{6})\), 4.0.29952.1, \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), 4.0.7488.1, \(\Q(\zeta_{8})\), \(\Q(\zeta_{12})\), 4.4.29952.1, \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{3})\), 4.4.7488.1, 8.0.897122304.10, 8.0.3588489216.11, 8.0.3588489216.5, 8.0.3588489216.16, \(\Q(\zeta_{24})\), 8.0.56070144.2, 8.8.3588489216.1, 8.0.1534132224.10, 8.0.95883264.1, 8.0.18939904.2, 8.0.1534132224.8, 8.8.1037073383424.1, 8.0.1534132224.4, Deg 8, 8.8.1534132224.1, 16.0.12877254853348294656.1, 16.0.1075521202606502917963776.1, Deg 16, Deg 16, 16.0.2353561680715186176.2, 16.16.1075521202606502917963776.1, Deg 16

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 32 siblings: data not computed

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{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}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$