Properties

Label 32.0.845...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $8.452\times 10^{47}$
Root discriminant $31.46$
Ramified primes $2, 3, 5, 17$
Class number $16$ (GRH)
Class group $[4, 4]$ (GRH)
Galois group $C_2^3\times D_4$ (as 32T273)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 6*x^30 - 64*x^29 + 18*x^28 + 120*x^27 + 884*x^26 + 1360*x^25 + 2453*x^24 - 592*x^23 + 11300*x^22 + 12152*x^21 - 15370*x^20 + 39248*x^19 + 6590*x^18 - 25344*x^17 + 75876*x^16 - 25344*x^15 + 6590*x^14 + 39248*x^13 - 15370*x^12 + 12152*x^11 + 11300*x^10 - 592*x^9 + 2453*x^8 + 1360*x^7 + 884*x^6 + 120*x^5 + 18*x^4 - 64*x^3 - 6*x^2 + 1)
 
gp: K = bnfinit(x^32 - 6*x^30 - 64*x^29 + 18*x^28 + 120*x^27 + 884*x^26 + 1360*x^25 + 2453*x^24 - 592*x^23 + 11300*x^22 + 12152*x^21 - 15370*x^20 + 39248*x^19 + 6590*x^18 - 25344*x^17 + 75876*x^16 - 25344*x^15 + 6590*x^14 + 39248*x^13 - 15370*x^12 + 12152*x^11 + 11300*x^10 - 592*x^9 + 2453*x^8 + 1360*x^7 + 884*x^6 + 120*x^5 + 18*x^4 - 64*x^3 - 6*x^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -6, -64, 18, 120, 884, 1360, 2453, -592, 11300, 12152, -15370, 39248, 6590, -25344, 75876, -25344, 6590, 39248, -15370, 12152, 11300, -592, 2453, 1360, 884, 120, 18, -64, -6, 0, 1]);
 

\(x^{32} - 6 x^{30} - 64 x^{29} + 18 x^{28} + 120 x^{27} + 884 x^{26} + 1360 x^{25} + 2453 x^{24} - 592 x^{23} + 11300 x^{22} + 12152 x^{21} - 15370 x^{20} + 39248 x^{19} + 6590 x^{18} - 25344 x^{17} + 75876 x^{16} - 25344 x^{15} + 6590 x^{14} + 39248 x^{13} - 15370 x^{12} + 12152 x^{11} + 11300 x^{10} - 592 x^{9} + 2453 x^{8} + 1360 x^{7} + 884 x^{6} + 120 x^{5} + 18 x^{4} - 64 x^{3} - 6 x^{2} + 1\)  Toggle raw display

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:  \(845222867573683465013147373404160000000000000000\)\(\medspace = 2^{64}\cdot 3^{16}\cdot 5^{16}\cdot 17^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $31.46$
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:  $2, 3, 5, 17$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $16$
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{5}$, $\frac{1}{56} a^{18} - \frac{3}{14} a^{17} - \frac{1}{7} a^{16} + \frac{1}{28} a^{15} + \frac{3}{14} a^{13} - \frac{5}{56} a^{12} - \frac{1}{7} a^{10} - \frac{3}{7} a^{9} - \frac{1}{7} a^{8} + \frac{9}{56} a^{6} + \frac{3}{14} a^{5} + \frac{1}{28} a^{3} - \frac{1}{7} a^{2} - \frac{3}{14} a - \frac{13}{56}$, $\frac{1}{56} a^{19} - \frac{3}{14} a^{17} - \frac{5}{28} a^{16} - \frac{1}{14} a^{15} + \frac{3}{14} a^{14} - \frac{1}{56} a^{13} - \frac{1}{14} a^{12} - \frac{1}{7} a^{11} - \frac{1}{7} a^{10} - \frac{2}{7} a^{9} + \frac{2}{7} a^{8} + \frac{9}{56} a^{7} + \frac{1}{7} a^{6} + \frac{1}{14} a^{5} - \frac{13}{28} a^{4} - \frac{3}{14} a^{3} + \frac{1}{14} a^{2} - \frac{17}{56} a + \frac{3}{14}$, $\frac{1}{112} a^{20} - \frac{1}{112} a^{18} + \frac{13}{56} a^{17} - \frac{1}{14} a^{16} - \frac{11}{56} a^{15} - \frac{1}{112} a^{14} - \frac{3}{28} a^{13} - \frac{1}{16} a^{12} - \frac{1}{14} a^{11} - \frac{3}{7} a^{10} - \frac{3}{14} a^{9} + \frac{33}{112} a^{8} - \frac{3}{7} a^{7} + \frac{47}{112} a^{6} - \frac{3}{56} a^{5} - \frac{5}{14} a^{4} + \frac{13}{56} a^{3} - \frac{7}{16} a^{2} + \frac{5}{28} a + \frac{25}{112}$, $\frac{1}{112} a^{21} - \frac{1}{112} a^{19} + \frac{3}{14} a^{17} + \frac{9}{56} a^{16} + \frac{3}{112} a^{15} - \frac{3}{28} a^{14} + \frac{17}{112} a^{13} + \frac{5}{56} a^{12} - \frac{3}{7} a^{11} - \frac{5}{14} a^{10} - \frac{15}{112} a^{9} + \frac{3}{7} a^{8} + \frac{47}{112} a^{7} - \frac{1}{7} a^{6} + \frac{5}{14} a^{5} - \frac{15}{56} a^{4} - \frac{45}{112} a^{3} + \frac{1}{28} a^{2} + \frac{1}{112} a + \frac{1}{56}$, $\frac{1}{112} a^{22} - \frac{1}{112} a^{18} - \frac{1}{28} a^{17} + \frac{19}{112} a^{16} - \frac{13}{56} a^{15} + \frac{1}{7} a^{14} - \frac{5}{56} a^{13} + \frac{9}{112} a^{12} - \frac{3}{7} a^{11} + \frac{17}{112} a^{10} + \frac{5}{14} a^{9} + \frac{3}{7} a^{8} + \frac{3}{7} a^{7} - \frac{17}{112} a^{6} + \frac{3}{28} a^{5} - \frac{29}{112} a^{4} + \frac{19}{56} a^{3} + \frac{2}{7} a^{2} + \frac{15}{56} a - \frac{55}{112}$, $\frac{1}{112} a^{23} - \frac{1}{112} a^{19} + \frac{27}{112} a^{17} - \frac{1}{56} a^{16} + \frac{3}{14} a^{15} - \frac{5}{56} a^{14} + \frac{1}{112} a^{13} - \frac{3}{28} a^{12} + \frac{17}{112} a^{11} + \frac{1}{14} a^{10} - \frac{3}{7} a^{9} + \frac{1}{7} a^{8} - \frac{17}{112} a^{7} + \frac{3}{7} a^{6} - \frac{37}{112} a^{5} - \frac{9}{56} a^{4} + \frac{5}{14} a^{3} - \frac{1}{56} a^{2} - \frac{47}{112} a + \frac{1}{28}$, $\frac{1}{112} a^{24} - \frac{1}{4} a^{15} - \frac{1}{4} a^{12} - \frac{1}{2} a^{9} + \frac{1}{4} a^{3} - \frac{29}{112}$, $\frac{1}{112} a^{25} - \frac{1}{4} a^{16} - \frac{1}{4} a^{13} - \frac{1}{2} a^{10} + \frac{1}{4} a^{4} - \frac{29}{112} a$, $\frac{1}{224} a^{26} - \frac{1}{224} a^{24} - \frac{1}{8} a^{17} - \frac{1}{8} a^{15} + \frac{1}{8} a^{14} - \frac{1}{4} a^{13} - \frac{1}{8} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{3} + \frac{27}{224} a^{2} + \frac{1}{4} a - \frac{27}{224}$, $\frac{1}{5152} a^{27} - \frac{3}{2576} a^{26} + \frac{19}{5152} a^{25} - \frac{1}{644} a^{24} - \frac{1}{368} a^{23} - \frac{5}{1288} a^{22} - \frac{3}{1288} a^{21} + \frac{9}{2576} a^{20} + \frac{3}{2576} a^{19} + \frac{19}{2576} a^{18} + \frac{321}{2576} a^{17} + \frac{297}{1288} a^{16} - \frac{171}{1288} a^{15} - \frac{37}{368} a^{14} + \frac{431}{2576} a^{13} + \frac{3}{2576} a^{12} - \frac{855}{2576} a^{11} - \frac{311}{1288} a^{10} + \frac{3}{184} a^{9} - \frac{1055}{2576} a^{8} - \frac{773}{2576} a^{7} + \frac{799}{2576} a^{6} + \frac{15}{368} a^{5} + \frac{31}{1288} a^{4} - \frac{813}{5152} a^{3} - \frac{7}{23} a^{2} - \frac{2421}{5152} a + \frac{5}{368}$, $\frac{1}{81091593856} a^{28} - \frac{1111753}{20272898464} a^{27} - \frac{338535}{20272898464} a^{26} - \frac{3747431}{881430368} a^{25} - \frac{80899163}{81091593856} a^{24} + \frac{4339765}{1448064176} a^{23} - \frac{178418691}{40545796928} a^{22} + \frac{29115297}{10136449232} a^{21} - \frac{10236391}{2534112308} a^{20} + \frac{1454717}{194931716} a^{19} + \frac{202137853}{40545796928} a^{18} - \frac{2297265381}{10136449232} a^{17} + \frac{4877298729}{40545796928} a^{16} - \frac{284774923}{1267056154} a^{15} + \frac{28668265}{266748664} a^{14} + \frac{61062193}{5068224616} a^{13} + \frac{10039962313}{40545796928} a^{12} - \frac{4996645883}{10136449232} a^{11} + \frac{1778246291}{5792256704} a^{10} + \frac{5721727}{97465858} a^{9} + \frac{540657589}{2534112308} a^{8} + \frac{2484528465}{10136449232} a^{7} + \frac{4323172117}{40545796928} a^{6} + \frac{704762111}{1448064176} a^{5} + \frac{29368319025}{81091593856} a^{4} - \frac{8270901473}{20272898464} a^{3} + \frac{3186976635}{20272898464} a^{2} - \frac{8894114573}{20272898464} a - \frac{37791327027}{81091593856}$, $\frac{1}{81091593856} a^{29} - \frac{460923}{10136449232} a^{27} - \frac{1321595}{1066994656} a^{26} + \frac{287359505}{81091593856} a^{25} + \frac{5382685}{2896128352} a^{24} + \frac{1839051}{445558208} a^{23} - \frac{167829}{724032088} a^{22} - \frac{1709339}{5068224616} a^{21} + \frac{1793945}{5068224616} a^{20} + \frac{8418663}{1762860736} a^{19} - \frac{4700739}{633528077} a^{18} - \frac{6877769571}{40545796928} a^{17} + \frac{1166494529}{10136449232} a^{16} + \frac{251010219}{5068224616} a^{15} - \frac{426674951}{5068224616} a^{14} - \frac{700302207}{3118907456} a^{13} - \frac{15585905}{110178796} a^{12} + \frac{14764061969}{40545796928} a^{11} + \frac{4548494245}{10136449232} a^{10} + \frac{84710701}{266748664} a^{9} - \frac{4670451529}{10136449232} a^{8} + \frac{14845826313}{40545796928} a^{7} + \frac{1915780785}{5068224616} a^{6} - \frac{2616019239}{81091593856} a^{5} + \frac{8127778}{27544699} a^{4} + \frac{3079714913}{10136449232} a^{3} + \frac{626917365}{2896128352} a^{2} - \frac{16544892471}{81091593856} a + \frac{7335871601}{20272898464}$, $\frac{1}{2857622031826505216} a^{30} + \frac{7557813}{1428811015913252608} a^{29} - \frac{10877665}{2857622031826505216} a^{28} - \frac{30063986275}{408698803178848} a^{27} + \frac{147726305923767}{408231718832357888} a^{26} - \frac{3753261647483415}{1428811015913252608} a^{25} + \frac{10552949579894697}{2857622031826505216} a^{24} - \frac{1746753062829013}{714405507956626304} a^{23} + \frac{333003251230917}{204115859416178944} a^{22} + \frac{37445089707585}{51028964854044736} a^{21} + \frac{309169187703741}{1428811015913252608} a^{20} + \frac{95300854747025}{54954269842817408} a^{19} + \frac{2243703231320651}{357202753978313152} a^{18} - \frac{154562974572713771}{714405507956626304} a^{17} + \frac{5199026856205393}{46090677932685568} a^{16} + \frac{1808830912262347}{89300688494578288} a^{15} - \frac{3124694560078001}{46090677932685568} a^{14} + \frac{309905414443333}{8027028179287936} a^{13} - \frac{56506682964527247}{357202753978313152} a^{12} - \frac{247235924789838483}{714405507956626304} a^{11} - \frac{4171579863897565}{8874602583312128} a^{10} - \frac{138908262911127279}{357202753978313152} a^{9} + \frac{42970402637792811}{204115859416178944} a^{8} - \frac{186209450362627121}{714405507956626304} a^{7} - \frac{25106462550943229}{219817079371269632} a^{6} + \frac{360351479244494121}{1428811015913252608} a^{5} + \frac{1049766112180212751}{2857622031826505216} a^{4} - \frac{1663409196355825}{7765277260398112} a^{3} + \frac{294246258922823537}{2857622031826505216} a^{2} - \frac{143153873608074723}{1428811015913252608} a + \frac{837251644659234383}{2857622031826505216}$, $\frac{1}{134308235495845745152} a^{31} + \frac{1}{134308235495845745152} a^{30} - \frac{213940715}{134308235495845745152} a^{29} + \frac{220682697}{134308235495845745152} a^{28} + \frac{2576441744470865}{134308235495845745152} a^{27} - \frac{38584408073228855}{134308235495845745152} a^{26} - \frac{594076212624305}{618931960810349056} a^{25} - \frac{33901823380287689}{10331402730449672704} a^{24} + \frac{57676022324789437}{67154117747922872576} a^{23} + \frac{170706917901480641}{67154117747922872576} a^{22} + \frac{1760963526035657}{919919421204422912} a^{21} - \frac{5184893738845981}{2919744249909690112} a^{20} - \frac{13120292747606999}{1975121110233025664} a^{19} + \frac{9141844003432157}{1767213624945338752} a^{18} + \frac{54728212052014767}{737957337889262336} a^{17} + \frac{10147613037298776873}{67154117747922872576} a^{16} + \frac{652457099918357207}{2919744249909690112} a^{15} + \frac{1050533910678523079}{9593445392560410368} a^{14} - \frac{26886492109286451}{459959710602211456} a^{13} + \frac{2075000263966150659}{33577058873961436288} a^{12} + \frac{25560203727769855033}{67154117747922872576} a^{11} - \frac{1718300521073974547}{5165701365224836352} a^{10} - \frac{27720020322200391527}{67154117747922872576} a^{9} + \frac{7150490362531648537}{67154117747922872576} a^{8} + \frac{5332445055830891965}{19186890785120820736} a^{7} - \frac{55498703234344568141}{134308235495845745152} a^{6} - \frac{53125331466494798531}{134308235495845745152} a^{5} + \frac{62384158746191320889}{134308235495845745152} a^{4} - \frac{51947126542941615903}{134308235495845745152} a^{3} + \frac{222781091810137447}{1128640634418871808} a^{2} - \frac{28093785124115018507}{134308235495845745152} a - \frac{2868581203943125971}{10331402730449672704}$  Toggle raw display

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

Class group and class number

$C_{4}\times C_{4}$, which has order $16$ (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:  \( \frac{4689709082965751773}{16788529436980718144} a^{31} + \frac{137330444560895671}{1475914675778524672} a^{30} - \frac{112427100269645412029}{67154117747922872576} a^{29} - \frac{1431214514979260061}{77679719777817088} a^{28} - \frac{15763122802287323847}{16788529436980718144} a^{27} + \frac{4701631034306733264861}{134308235495845745152} a^{26} + \frac{17285813354663541881447}{67154117747922872576} a^{25} + \frac{62136172616210342802389}{134308235495845745152} a^{24} + \frac{27359139605703927634185}{33577058873961436288} a^{23} + \frac{5068993295620531954471}{67154117747922872576} a^{22} + \frac{325445315530141484321}{104276580353917504} a^{21} + \frac{300291824516369256227481}{67154117747922872576} a^{20} - \frac{105790582833901725307985}{33577058873961436288} a^{19} + \frac{162606792401010161338191}{16788529436980718144} a^{18} + \frac{26841638801142444672805}{4796722696280205184} a^{17} - \frac{63155204188786321786795}{9593445392560410368} a^{16} + \frac{23153940802969166122039}{1199180674070051296} a^{15} - \frac{786171987849048597059}{67154117747922872576} a^{14} - \frac{23144614610867763688777}{33577058873961436288} a^{13} + \frac{208568707183896889075829}{16788529436980718144} a^{12} - \frac{4614049092466058863499}{4796722696280205184} a^{11} + \frac{140781568582344069447511}{67154117747922872576} a^{10} + \frac{4670122682691082425955}{987560555116512832} a^{9} + \frac{47048325903156833396441}{67154117747922872576} a^{8} + \frac{3764659978284758030489}{4796722696280205184} a^{7} + \frac{4369682326649553862205}{5839488499819380224} a^{6} + \frac{3410347376610787077089}{9593445392560410368} a^{5} + \frac{2814498244283265128245}{19186890785120820736} a^{4} + \frac{596657558119808319585}{16788529436980718144} a^{3} - \frac{145571178965217497269}{19186890785120820736} a^{2} - \frac{369875046445990575197}{67154117747922872576} a - \frac{35660673875419454845}{134308235495845745152} \) (order $24$)  Toggle raw display
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:  \( 124577213214.25456 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{16}\cdot 124577213214.25456 \cdot 16}{24\sqrt{845222867573683465013147373404160000000000000000}}\approx 0.533015240485038$ (assuming GRH)

Galois group

$C_2^3\times D_4$ (as 32T273):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 64
The 40 conjugacy class representatives for $C_2^3\times D_4$
Character table for $C_2^3\times D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-1}) \), 4.0.979200.3, 4.0.39168.3, 4.4.9792.1, 4.4.244800.1, 4.0.1088.2, 4.0.27200.2, 4.4.108800.1, 4.4.4352.1, \(\Q(i, \sqrt{5})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{-5}, \sqrt{-6})\), \(\Q(\sqrt{-5}, \sqrt{6})\), \(\Q(i, \sqrt{30})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{15})\), \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(\sqrt{-3}, \sqrt{-10})\), \(\Q(\sqrt{-2}, \sqrt{-15})\), \(\Q(\sqrt{2}, \sqrt{-15})\), \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{3}, \sqrt{-10})\), \(\Q(\sqrt{-2}, \sqrt{15})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-6}, \sqrt{10})\), \(\Q(\sqrt{-6}, \sqrt{-10})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{6}, \sqrt{-10})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{10})\), \(\Q(i, \sqrt{15})\), \(\Q(\zeta_{12})\), 8.0.3317760000.9, 8.0.40960000.1, 8.0.12960000.1, 8.8.3317760000.1, 8.0.207360000.2, 8.0.207360000.1, 8.0.3317760000.7, 8.0.3317760000.3, 8.0.3317760000.1, 8.0.3317760000.6, 8.0.3317760000.8, 8.0.3317760000.4, 8.0.3317760000.2, \(\Q(\zeta_{24})\), 8.0.3317760000.5, 8.0.958832640000.35, 8.8.59927040000.2, 8.0.739840000.6, 8.8.11837440000.1, 8.0.958832640000.21, 8.0.958832640000.65, 8.0.11837440000.5, 8.0.11837440000.9, 8.0.958832640000.41, 8.0.958832640000.20, 8.8.958832640000.3, 8.8.958832640000.5, 8.0.59927040000.32, 8.0.59927040000.35, 8.0.958832640000.11, 8.0.958832640000.26, 8.0.95883264.1, 8.0.59927040000.12, 8.0.958832640000.42, 8.0.1534132224.10, 8.0.1534132224.4, 8.0.958832640000.29, 8.8.958832640000.4, 8.8.1534132224.1, 8.0.1534132224.8, 8.0.958832640000.82, 8.0.11837440000.20, 8.0.18939904.2, 16.0.11007531417600000000.1, Deg 16, Deg 16, Deg 16, Deg 16, 16.0.3591250123161600000000.1, Deg 16, Deg 16, Deg 16, 16.0.919360031529369600000000.1, Deg 16, Deg 16, Deg 16, 16.0.2353561680715186176.2, 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 R ${\href{/LocalNumberField/7.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{16}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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
$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}$
$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}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$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}$