Properties

Label 32.0.779...216.1
Degree $32$
Signature $[0, 16]$
Discriminant $7.790\times 10^{48}$
Root discriminant $33.72$
Ramified primes $2, 3, 11, 17$
Class number $24$ (GRH)
Class group $[2, 12]$ (GRH)
Galois group $C_2\times D_4^2$ (as 32T1016)

Related objects

Downloads

Learn more about

Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 16*x^31 + 156*x^30 - 1096*x^29 + 6134*x^28 - 28580*x^27 + 114266*x^26 - 399684*x^25 + 1239490*x^24 - 3440780*x^23 + 8605010*x^22 - 19473044*x^21 + 39982438*x^20 - 74587072*x^19 + 126485556*x^18 - 194983784*x^17 + 273242634*x^16 - 348358772*x^15 + 405080242*x^14 - 431986068*x^13 + 426399702*x^12 - 394223580*x^11 + 344722538*x^10 - 285681476*x^9 + 222647745*x^8 - 160135088*x^7 + 103163456*x^6 - 57484000*x^5 + 26721528*x^4 - 9914240*x^3 + 2744800*x^2 - 506176*x + 48016)
 
gp: K = bnfinit(x^32 - 16*x^31 + 156*x^30 - 1096*x^29 + 6134*x^28 - 28580*x^27 + 114266*x^26 - 399684*x^25 + 1239490*x^24 - 3440780*x^23 + 8605010*x^22 - 19473044*x^21 + 39982438*x^20 - 74587072*x^19 + 126485556*x^18 - 194983784*x^17 + 273242634*x^16 - 348358772*x^15 + 405080242*x^14 - 431986068*x^13 + 426399702*x^12 - 394223580*x^11 + 344722538*x^10 - 285681476*x^9 + 222647745*x^8 - 160135088*x^7 + 103163456*x^6 - 57484000*x^5 + 26721528*x^4 - 9914240*x^3 + 2744800*x^2 - 506176*x + 48016, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![48016, -506176, 2744800, -9914240, 26721528, -57484000, 103163456, -160135088, 222647745, -285681476, 344722538, -394223580, 426399702, -431986068, 405080242, -348358772, 273242634, -194983784, 126485556, -74587072, 39982438, -19473044, 8605010, -3440780, 1239490, -399684, 114266, -28580, 6134, -1096, 156, -16, 1]);
 

\( x^{32} - 16 x^{31} + 156 x^{30} - 1096 x^{29} + 6134 x^{28} - 28580 x^{27} + 114266 x^{26} - 399684 x^{25} + 1239490 x^{24} - 3440780 x^{23} + 8605010 x^{22} - 19473044 x^{21} + 39982438 x^{20} - 74587072 x^{19} + 126485556 x^{18} - 194983784 x^{17} + 273242634 x^{16} - 348358772 x^{15} + 405080242 x^{14} - 431986068 x^{13} + 426399702 x^{12} - 394223580 x^{11} + 344722538 x^{10} - 285681476 x^{9} + 222647745 x^{8} - 160135088 x^{7} + 103163456 x^{6} - 57484000 x^{5} + 26721528 x^{4} - 9914240 x^{3} + 2744800 x^{2} - 506176 x + 48016 \)

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:  \(7790452574060543254749837160138952789467349385216\)\(\medspace = 2^{64}\cdot 3^{24}\cdot 11^{8}\cdot 17^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $33.72$
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:  $2, 3, 11, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{20} - \frac{1}{4} a^{18} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{21} - \frac{1}{4} a^{19} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{22} - \frac{1}{4} a^{18} - \frac{1}{4} a^{14} + \frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{104} a^{23} - \frac{7}{104} a^{22} - \frac{5}{104} a^{21} + \frac{1}{104} a^{20} + \frac{1}{26} a^{19} + \frac{7}{52} a^{18} + \frac{11}{52} a^{17} - \frac{5}{52} a^{16} - \frac{21}{104} a^{15} - \frac{21}{104} a^{14} - \frac{11}{104} a^{13} - \frac{5}{104} a^{12} + \frac{1}{52} a^{11} - \frac{6}{13} a^{10} - \frac{19}{52} a^{9} + \frac{7}{52} a^{8} - \frac{5}{104} a^{7} - \frac{25}{104} a^{6} - \frac{1}{104} a^{5} + \frac{41}{104} a^{4} - \frac{3}{26} a^{3} + \frac{21}{52} a^{2} - \frac{11}{26} a + \frac{5}{13}$, $\frac{1}{6448} a^{24} - \frac{3}{1612} a^{23} - \frac{89}{3224} a^{22} - \frac{7}{124} a^{21} - \frac{547}{6448} a^{20} + \frac{239}{1612} a^{19} + \frac{67}{3224} a^{18} - \frac{121}{1612} a^{17} + \frac{81}{6448} a^{16} - \frac{369}{1612} a^{15} + \frac{567}{3224} a^{14} + \frac{357}{1612} a^{13} + \frac{417}{6448} a^{12} + \frac{76}{403} a^{11} + \frac{373}{806} a^{10} + \frac{185}{403} a^{9} + \frac{1849}{6448} a^{8} - \frac{1}{31} a^{7} + \frac{291}{1612} a^{6} + \frac{24}{403} a^{5} - \frac{971}{6448} a^{4} - \frac{23}{806} a^{3} + \frac{43}{806} a^{2} - \frac{1}{31} a - \frac{297}{1612}$, $\frac{1}{6448} a^{25} - \frac{3}{1612} a^{23} + \frac{83}{3224} a^{22} - \frac{17}{6448} a^{21} - \frac{231}{3224} a^{20} - \frac{25}{3224} a^{19} - \frac{123}{806} a^{18} + \frac{1093}{6448} a^{17} - \frac{95}{1612} a^{16} - \frac{5}{62} a^{15} + \frac{233}{3224} a^{14} + \frac{1247}{6448} a^{13} - \frac{83}{3224} a^{12} - \frac{287}{1612} a^{11} - \frac{119}{403} a^{10} - \frac{203}{6448} a^{9} - \frac{337}{806} a^{8} - \frac{1441}{3224} a^{7} - \frac{729}{3224} a^{6} - \frac{3121}{6448} a^{5} - \frac{369}{3224} a^{4} - \frac{295}{806} a^{3} - \frac{601}{1612} a^{2} + \frac{505}{1612} a - \frac{116}{403}$, $\frac{1}{12896} a^{26} - \frac{1}{12896} a^{25} + \frac{17}{6448} a^{23} + \frac{905}{12896} a^{22} + \frac{23}{12896} a^{21} - \frac{329}{3224} a^{20} - \frac{373}{6448} a^{19} + \frac{2073}{12896} a^{18} + \frac{2391}{12896} a^{17} - \frac{23}{124} a^{16} - \frac{303}{6448} a^{15} + \frac{1493}{12896} a^{14} - \frac{2009}{12896} a^{13} - \frac{407}{6448} a^{12} + \frac{319}{1612} a^{11} + \frac{3657}{12896} a^{10} - \frac{5661}{12896} a^{9} + \frac{1329}{6448} a^{8} - \frac{67}{806} a^{7} + \frac{2633}{12896} a^{6} - \frac{4293}{12896} a^{5} - \frac{2607}{6448} a^{4} - \frac{483}{1612} a^{3} - \frac{683}{3224} a^{2} + \frac{825}{3224} a + \frac{15}{52}$, $\frac{1}{12896} a^{27} - \frac{1}{12896} a^{25} - \frac{17}{12896} a^{23} + \frac{51}{1612} a^{22} - \frac{1441}{12896} a^{21} - \frac{237}{3224} a^{20} + \frac{2055}{12896} a^{19} + \frac{3}{248} a^{18} - \frac{657}{12896} a^{17} + \frac{45}{403} a^{16} + \frac{2339}{12896} a^{15} - \frac{189}{3224} a^{14} + \frac{2317}{12896} a^{13} - \frac{599}{3224} a^{12} + \frac{825}{12896} a^{11} + \frac{175}{3224} a^{10} + \frac{6221}{12896} a^{9} + \frac{431}{1612} a^{8} + \frac{2557}{12896} a^{7} - \frac{489}{1612} a^{6} + \frac{4593}{12896} a^{5} + \frac{869}{3224} a^{4} + \frac{783}{3224} a^{3} - \frac{123}{403} a^{2} - \frac{817}{3224} a + \frac{153}{806}$, $\frac{1}{25792} a^{28} - \frac{1}{25792} a^{26} + \frac{1}{25792} a^{24} + \frac{17}{6448} a^{23} + \frac{2671}{25792} a^{22} - \frac{27}{1612} a^{21} + \frac{1757}{25792} a^{20} + \frac{993}{6448} a^{19} - \frac{3205}{25792} a^{18} - \frac{111}{806} a^{17} + \frac{5037}{25792} a^{16} - \frac{84}{403} a^{15} - \frac{459}{25792} a^{14} - \frac{35}{806} a^{13} + \frac{5727}{25792} a^{12} - \frac{863}{6448} a^{11} + \frac{4629}{25792} a^{10} + \frac{213}{1612} a^{9} + \frac{1863}{25792} a^{8} + \frac{1465}{6448} a^{7} + \frac{2853}{25792} a^{6} - \frac{138}{403} a^{5} - \frac{1655}{12896} a^{4} + \frac{43}{1612} a^{3} + \frac{1847}{6448} a^{2} + \frac{5}{31} a + \frac{737}{3224}$, $\frac{1}{25792} a^{29} - \frac{1}{25792} a^{27} + \frac{1}{25792} a^{25} + \frac{15}{25792} a^{23} - \frac{339}{3224} a^{22} - \frac{1267}{25792} a^{21} - \frac{1}{26} a^{20} + \frac{1723}{25792} a^{19} + \frac{401}{3224} a^{18} + \frac{253}{25792} a^{17} - \frac{489}{6448} a^{16} + \frac{5173}{25792} a^{15} - \frac{665}{3224} a^{14} + \frac{4847}{25792} a^{13} - \frac{97}{1612} a^{12} - \frac{1179}{25792} a^{11} - \frac{441}{1612} a^{10} - \frac{3001}{25792} a^{9} + \frac{29}{62} a^{8} + \frac{8565}{25792} a^{7} - \frac{37}{806} a^{6} + \frac{3145}{12896} a^{5} - \frac{2789}{6448} a^{4} + \frac{883}{6448} a^{3} + \frac{81}{806} a^{2} + \frac{645}{3224} a - \frac{407}{1612}$, $\frac{1}{27368334734137351552} a^{30} + \frac{413668346333483}{27368334734137351552} a^{29} + \frac{437636904689185}{27368334734137351552} a^{28} - \frac{652940747591199}{27368334734137351552} a^{27} - \frac{427007964032757}{27368334734137351552} a^{26} - \frac{178448333508561}{27368334734137351552} a^{25} - \frac{6354824762865}{463870080239616128} a^{24} + \frac{66334949590416409}{27368334734137351552} a^{23} + \frac{77286664506076361}{882849507552817792} a^{22} - \frac{2329692348359166669}{27368334734137351552} a^{21} - \frac{2112238067575675287}{27368334734137351552} a^{20} - \frac{637211869054983987}{27368334734137351552} a^{19} + \frac{3453584297464929519}{27368334734137351552} a^{18} + \frac{1264877328663770383}{27368334734137351552} a^{17} + \frac{2493603214122858215}{27368334734137351552} a^{16} - \frac{6024246604807807501}{27368334734137351552} a^{15} + \frac{1986469139228112917}{27368334734137351552} a^{14} + \frac{276788580965461169}{27368334734137351552} a^{13} - \frac{100486818219061571}{463870080239616128} a^{12} + \frac{6650418084857908891}{27368334734137351552} a^{11} + \frac{12060792850474394973}{27368334734137351552} a^{10} - \frac{1132937095453043095}{27368334734137351552} a^{9} - \frac{12714938182679727153}{27368334734137351552} a^{8} - \frac{688803063213325}{6032253633268096} a^{7} + \frac{307567849734731351}{3421041841767168944} a^{6} + \frac{3801122546396902455}{13684167367068675776} a^{5} + \frac{89987853368491931}{427630230220896118} a^{4} + \frac{576314855732254645}{6842083683534337888} a^{3} - \frac{10352213789871345}{131578532375660344} a^{2} - \frac{36333969628432777}{263157064751320688} a - \frac{100241368968762695}{1710520920883584472}$, $\frac{1}{46197930647331808257414636659621250388647592260666945152} a^{31} - \frac{662825536326714997404606818697469189}{46197930647331808257414636659621250388647592260666945152} a^{30} + \frac{506540085879702455742274663320807502859644341452151}{46197930647331808257414636659621250388647592260666945152} a^{29} - \frac{5166279010210470975763549750615281369082851952179}{783015773683589970464654858637648311671993089163846528} a^{28} - \frac{1023730788181643842557308676851959339406204430418087}{46197930647331808257414636659621250388647592260666945152} a^{27} + \frac{492919705958202698388206063721191937803389062573249}{46197930647331808257414636659621250388647592260666945152} a^{26} - \frac{785649289330252584941959698650865744120925139291785}{46197930647331808257414636659621250388647592260666945152} a^{25} + \frac{473892893060222571749475550938430698492128642547135}{46197930647331808257414636659621250388647592260666945152} a^{24} - \frac{208196391242743430505777102564824020981183876953804731}{46197930647331808257414636659621250388647592260666945152} a^{23} + \frac{1944779144688446355030553673081100210007024806958297269}{46197930647331808257414636659621250388647592260666945152} a^{22} + \frac{303086779220932827532236004715035677050377014971705807}{3553686972871677558262664358432403876049814789282072704} a^{21} - \frac{3751657116022982563745901054423517269681770516781643973}{46197930647331808257414636659621250388647592260666945152} a^{20} + \frac{9801054078693944971145087364876247317385123367435210325}{46197930647331808257414636659621250388647592260666945152} a^{19} - \frac{7237295792136850931314610426121523973536916811979412295}{46197930647331808257414636659621250388647592260666945152} a^{18} - \frac{6926798633447675395538087308410880070926155526923799087}{46197930647331808257414636659621250388647592260666945152} a^{17} + \frac{3285212368680920405014836670342704823882012549543443}{114635063641021856718150463175238834711284348041357184} a^{16} + \frac{2153653799165611340160481720954688746613834357315300095}{46197930647331808257414636659621250388647592260666945152} a^{15} + \frac{2329273965726962377144174462057135110453788084807044279}{46197930647331808257414636659621250388647592260666945152} a^{14} - \frac{277616916178513638965917039432037807266935036255041075}{46197930647331808257414636659621250388647592260666945152} a^{13} - \frac{4382304418810678282397562152751126744716435447598122139}{46197930647331808257414636659621250388647592260666945152} a^{12} + \frac{850387138287143329434083738141919544652995054482656611}{3553686972871677558262664358432403876049814789282072704} a^{11} + \frac{19517425838283054042755008063058449880826597461951502399}{46197930647331808257414636659621250388647592260666945152} a^{10} - \frac{14670381252815470921550088261674867055451190088954488051}{46197930647331808257414636659621250388647592260666945152} a^{9} + \frac{13540669488872148490989102043597442340446816756375110613}{46197930647331808257414636659621250388647592260666945152} a^{8} - \frac{9161623237324442511725283181841252342789638422480315647}{23098965323665904128707318329810625194323796130333472576} a^{7} + \frac{2306117175318097478071668009373623453588680763515910085}{11549482661832952064353659164905312597161898065166736288} a^{6} + \frac{823186889358009879028927803076224155570704969832376529}{2887370665458238016088414791226328149290474516291684072} a^{5} - \frac{244345057012205465792588870448433142152039430826517937}{1443685332729119008044207395613164074645237258145842036} a^{4} + \frac{534529777022162547525833217777688368292650418810374957}{5774741330916476032176829582452656298580949032583368144} a^{3} + \frac{727697388328691517758017305024692839319748645552865}{8542516761710763361208327784693278548196670166543444} a^{2} - \frac{40974929721170506165567806478230528691031230542956803}{93140989208330258583497251329881553202918532783602712} a + \frac{244809961134494103844933725271787314623679213563769219}{2887370665458238016088414791226328149290474516291684072}$

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

Class group and class number

$C_{2}\times C_{12}$, which has order $24$ (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{5218015018014496867534361723302710370923454205}{6350164678325258493164145180012037041076628238752} a^{31} - \frac{315745854338833719442733331483793503445726527117}{25400658713301033972656580720048148164306512955008} a^{30} + \frac{2980839981596353439395188998874682480050899123241}{25400658713301033972656580720048148164306512955008} a^{29} - \frac{20278164767916066284221199433129250014311497513351}{25400658713301033972656580720048148164306512955008} a^{28} + \frac{110361102192611475372100436350751609753232597646863}{25400658713301033972656580720048148164306512955008} a^{27} - \frac{500368542207551763651134315120025412623398282148721}{25400658713301033972656580720048148164306512955008} a^{26} + \frac{1948994937006334771796687763686134111611506802477181}{25400658713301033972656580720048148164306512955008} a^{25} - \frac{214318720213926845621605882174484320171865483474501}{819376087525839805569567120001553166590532675968} a^{24} + \frac{20080323774413916833015730228294615126964501823445183}{25400658713301033972656580720048148164306512955008} a^{23} - \frac{54311556605545172695716263244594753944889858521779085}{25400658713301033972656580720048148164306512955008} a^{22} + \frac{782520816517091853336753220880511614891382428012449}{150299755700006118181399885917444663694121378432} a^{21} - \frac{291087686222432488575742779217223434780578069343205079}{25400658713301033972656580720048148164306512955008} a^{20} + \frac{580500634204775427302519893957289010282073992750102427}{25400658713301033972656580720048148164306512955008} a^{19} - \frac{33869691622656727417571353741362378870185136633797443}{819376087525839805569567120001553166590532675968} a^{18} + \frac{1722751497339918996445720898167596194277132940001884357}{25400658713301033972656580720048148164306512955008} a^{17} - \frac{197198895622318792282727043162313477183518074465418529}{1953896824100079536358198516926780628023577919616} a^{16} + \frac{3459935954441455347295804134153297764889556404948663061}{25400658713301033972656580720048148164306512955008} a^{15} - \frac{4240280630114003287526895544424952122819982059318129807}{25400658713301033972656580720048148164306512955008} a^{14} + \frac{4736137601391150557004530954068128448094967987198888187}{25400658713301033972656580720048148164306512955008} a^{13} - \frac{4857187545656762532619572120523286879470223403206980001}{25400658713301033972656580720048148164306512955008} a^{12} + \frac{356010074349391177063678588201422947880280226305115561}{1953896824100079536358198516926780628023577919616} a^{11} - \frac{4152306556979381305194473466956950698482303193598419207}{25400658713301033972656580720048148164306512955008} a^{10} + \frac{3533697003054614558621769212210639356146577124951674387}{25400658713301033972656580720048148164306512955008} a^{9} - \frac{2843480075467151158019891882860740640110502474058195009}{25400658713301033972656580720048148164306512955008} a^{8} + \frac{2134103279288069416474805017491267617438077086412289985}{25400658713301033972656580720048148164306512955008} a^{7} - \frac{726631963711313610143845507384798643892265410387898275}{12700329356650516986328290360024074082153256477504} a^{6} + \frac{431815038806499008060550352675790649263410083637225961}{12700329356650516986328290360024074082153256477504} a^{5} - \frac{107564903011568162950075090237039710879819387958732127}{6350164678325258493164145180012037041076628238752} a^{4} + \frac{42979004032676267715954153894812868818605172638662831}{6350164678325258493164145180012037041076628238752} a^{3} - \frac{37960575528666866688082203616434908985126601087639}{18787469462500764772674985739680582961765172304} a^{2} + \frac{1266337995317896992353826983637844073606627511371821}{3175082339162629246582072590006018520538314119376} a - \frac{16120194331840303759660323104265522360739299966987}{396885292395328655822759073750752315067289264922} \) (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:  \( 233422630699.0076 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{16}\cdot 233422630699.0076 \cdot 24}{24\sqrt{7790452574060543254749837160138952789467349385216}}\approx 0.493445859313154$ (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), 4.4.76032.1, 4.0.4752.1, 4.4.4752.1, 4.0.76032.2, \(\Q(\zeta_{8})\), 4.0.39168.3, 4.4.9792.1, \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), 4.4.4352.1, 4.0.1088.2, \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{6})\), 8.0.1534132224.8, 8.0.18939904.2, \(\Q(\zeta_{24})\), 8.0.1534132224.10, 8.0.1534132224.4, 8.8.1534132224.1, 8.0.95883264.1, 8.0.5780865024.3, 8.0.5780865024.13, Deg 8, 8.8.1670669991936.3, 8.8.5780865024.1, 8.0.5780865024.4, 8.0.5780865024.5, 8.0.22581504.2, 16.0.2353561680715186176.2, Deg 16, 16.0.33418400425706520576.1, Deg 16, 16.0.2791138221955434305028096.1, 16.16.2791138221955434305028096.1, 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{16}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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
$2$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
3Data not computed
$11$11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$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}$