Properties

Label 32.0.184...576.1
Degree $32$
Signature $[0, 16]$
Discriminant $1.841\times 10^{50}$
Root discriminant $37.22$
Ramified primes $2, 3, 7, 17$
Class number not computed
Class group not computed
Galois group $C_2^3\times D_4$ (as 32T273)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 28*x^30 + 502*x^28 - 5304*x^26 + 40319*x^24 - 205348*x^22 + 769870*x^20 - 2005384*x^18 + 3886261*x^16 - 5310988*x^14 + 5324752*x^12 - 3494768*x^10 + 1644300*x^8 - 452512*x^6 + 81152*x^4 - 1184*x^2 + 16)
 
gp: K = bnfinit(x^32 - 28*x^30 + 502*x^28 - 5304*x^26 + 40319*x^24 - 205348*x^22 + 769870*x^20 - 2005384*x^18 + 3886261*x^16 - 5310988*x^14 + 5324752*x^12 - 3494768*x^10 + 1644300*x^8 - 452512*x^6 + 81152*x^4 - 1184*x^2 + 16, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 0, -1184, 0, 81152, 0, -452512, 0, 1644300, 0, -3494768, 0, 5324752, 0, -5310988, 0, 3886261, 0, -2005384, 0, 769870, 0, -205348, 0, 40319, 0, -5304, 0, 502, 0, -28, 0, 1]);
 

\( x^{32} - 28 x^{30} + 502 x^{28} - 5304 x^{26} + 40319 x^{24} - 205348 x^{22} + 769870 x^{20} - 2005384 x^{18} + 3886261 x^{16} - 5310988 x^{14} + 5324752 x^{12} - 3494768 x^{10} + 1644300 x^{8} - 452512 x^{6} + 81152 x^{4} - 1184 x^{2} + 16 \)

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:  \(184085596562491198850839682056048266622946116632576\)\(\medspace = 2^{64}\cdot 3^{16}\cdot 7^{16}\cdot 17^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $37.22$
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:  $2, 3, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{16} a^{16} - \frac{1}{4} a^{15} + \frac{1}{8} a^{14} + \frac{1}{16} a^{12} - \frac{1}{4} a^{11} + \frac{1}{8} a^{10} - \frac{1}{2} a^{9} - \frac{5}{16} a^{8} + \frac{1}{4} a^{7} + \frac{3}{8} a^{6} - \frac{1}{2} a^{5} + \frac{3}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{16} a^{17} + \frac{1}{8} a^{15} + \frac{1}{16} a^{13} + \frac{1}{8} a^{11} - \frac{1}{2} a^{10} - \frac{5}{16} a^{9} + \frac{3}{8} a^{7} - \frac{1}{2} a^{6} + \frac{3}{8} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{18} - \frac{3}{16} a^{14} - \frac{1}{2} a^{11} + \frac{7}{16} a^{10} - \frac{1}{2} a^{7} - \frac{3}{8} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{32} a^{19} + \frac{5}{32} a^{15} - \frac{1}{4} a^{12} + \frac{15}{32} a^{11} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{7}{16} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a$, $\frac{1}{32} a^{20} - \frac{1}{32} a^{16} - \frac{1}{4} a^{15} + \frac{1}{8} a^{14} - \frac{1}{4} a^{13} - \frac{7}{32} a^{12} - \frac{1}{4} a^{11} - \frac{3}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{3}{8} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{32} a^{21} - \frac{1}{32} a^{17} + \frac{1}{8} a^{15} - \frac{1}{4} a^{14} - \frac{7}{32} a^{13} - \frac{3}{8} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{3}{8} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{22} - \frac{1}{32} a^{18} - \frac{1}{4} a^{15} + \frac{1}{32} a^{14} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{32} a^{23} + \frac{3}{16} a^{15} - \frac{1}{4} a^{12} - \frac{9}{32} a^{11} - \frac{1}{2} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{7}{16} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} - \frac{3}{8} a^{3} + \frac{1}{4} a$, $\frac{1}{160} a^{24} + \frac{1}{80} a^{22} - \frac{1}{160} a^{20} + \frac{3}{160} a^{16} - \frac{1}{4} a^{15} - \frac{1}{10} a^{14} + \frac{1}{16} a^{12} - \frac{1}{4} a^{11} + \frac{37}{80} a^{10} - \frac{1}{2} a^{9} - \frac{13}{80} a^{8} - \frac{1}{4} a^{7} - \frac{7}{20} a^{6} - \frac{1}{2} a^{5} - \frac{3}{8} a^{4} + \frac{2}{5} a^{2} - \frac{1}{2} a + \frac{3}{20}$, $\frac{1}{160} a^{25} + \frac{1}{80} a^{23} - \frac{1}{160} a^{21} + \frac{3}{160} a^{17} - \frac{1}{10} a^{15} + \frac{1}{16} a^{13} + \frac{37}{80} a^{11} - \frac{1}{2} a^{10} - \frac{13}{80} a^{9} - \frac{1}{2} a^{8} - \frac{7}{20} a^{7} - \frac{1}{2} a^{6} - \frac{3}{8} a^{5} - \frac{1}{2} a^{4} + \frac{2}{5} a^{3} - \frac{1}{2} a^{2} + \frac{3}{20} a$, $\frac{1}{160} a^{26} + \frac{1}{80} a^{20} - \frac{1}{80} a^{18} - \frac{1}{80} a^{16} - \frac{1}{4} a^{15} + \frac{7}{160} a^{14} - \frac{3}{80} a^{12} + \frac{1}{4} a^{11} - \frac{7}{80} a^{10} - \frac{2}{5} a^{8} - \frac{1}{4} a^{7} - \frac{17}{40} a^{6} - \frac{1}{2} a^{5} + \frac{3}{20} a^{4} - \frac{2}{5} a^{2} - \frac{3}{10}$, $\frac{1}{160} a^{27} + \frac{1}{80} a^{21} - \frac{1}{80} a^{19} - \frac{1}{80} a^{17} + \frac{7}{160} a^{15} - \frac{3}{80} a^{13} - \frac{7}{80} a^{11} - \frac{2}{5} a^{9} - \frac{17}{40} a^{7} - \frac{1}{2} a^{6} + \frac{3}{20} a^{5} - \frac{2}{5} a^{3} - \frac{3}{10} a$, $\frac{1}{175840} a^{28} + \frac{1}{1099} a^{26} - \frac{13}{43960} a^{24} - \frac{8}{785} a^{22} + \frac{67}{17584} a^{20} + \frac{409}{87920} a^{18} - \frac{2659}{175840} a^{16} - \frac{1}{4} a^{15} - \frac{8177}{87920} a^{14} + \frac{9809}{43960} a^{12} + \frac{1}{4} a^{11} - \frac{35291}{87920} a^{10} - \frac{1}{2} a^{9} + \frac{15287}{87920} a^{8} + \frac{1}{4} a^{7} + \frac{389}{785} a^{6} - \frac{1}{2} a^{5} + \frac{1429}{43960} a^{4} - \frac{851}{10990} a^{2} - \frac{1}{2} a + \frac{8569}{21980}$, $\frac{1}{175840} a^{29} + \frac{1}{1099} a^{27} - \frac{13}{43960} a^{25} - \frac{8}{785} a^{23} + \frac{67}{17584} a^{21} + \frac{409}{87920} a^{19} - \frac{2659}{175840} a^{17} - \frac{8177}{87920} a^{15} + \frac{9809}{43960} a^{13} - \frac{35291}{87920} a^{11} - \frac{1}{2} a^{10} + \frac{15287}{87920} a^{9} - \frac{1}{2} a^{8} + \frac{389}{785} a^{7} - \frac{1}{2} a^{6} + \frac{1429}{43960} a^{5} - \frac{851}{10990} a^{3} - \frac{1}{2} a^{2} + \frac{8569}{21980} a$, $\frac{1}{190788337649711815645073658881787777760} a^{30} - \frac{345279915742872498713546968513941}{190788337649711815645073658881787777760} a^{28} + \frac{51340448332752303941002362577605425}{19078833764971181564507365888178777776} a^{26} - \frac{4703809618338991096136690375909967}{190788337649711815645073658881787777760} a^{24} - \frac{32609254834868263382482321597309411}{95394168824855907822536829440893888880} a^{22} - \frac{2881941860999584209684645275361626191}{190788337649711815645073658881787777760} a^{20} - \frac{56765329744331961628800578533135857}{2298654670478455610181610347973346720} a^{18} + \frac{175985088746743384284376084913685}{40081583539855423454847407328106676} a^{16} - \frac{1}{4} a^{15} - \frac{16939187225412775217071406152000730217}{95394168824855907822536829440893888880} a^{14} + \frac{4479584252661388560116053010516988}{70142771194746991045982962824186683} a^{12} - \frac{1}{4} a^{11} - \frac{3863925225707118773652159289935706337}{19078833764971181564507365888178777776} a^{10} + \frac{23039887897604817841482787391718497213}{95394168824855907822536829440893888880} a^{8} + \frac{1}{4} a^{7} + \frac{4857301798447588673638910232821392001}{11924271103106988477817103680111736110} a^{6} - \frac{1}{2} a^{5} + \frac{3190302612004962558075331382732057047}{6813869201775421987324059245778134920} a^{4} + \frac{496676804428185645325100279857400866}{5962135551553494238908551840055868055} a^{2} - \frac{1}{2} a + \frac{8926784699240658874266066678781473123}{23848542206213976955634207360223472220}$, $\frac{1}{381576675299423631290147317763575555520} a^{31} + \frac{184932775912029565899800329757587}{95394168824855907822536829440893888880} a^{29} + \frac{9814374949001165162572333596902007}{5451095361420337589859247396622507936} a^{27} + \frac{141412840960503542130663845835370077}{47697084412427953911268414720446944440} a^{25} - \frac{5586839587350488516174540678507505543}{381576675299423631290147317763575555520} a^{23} - \frac{836852897079579811679203572679531543}{95394168824855907822536829440893888880} a^{21} + \frac{12880434919492601610180312777209761}{2298654670478455610181610347973346720} a^{19} - \frac{76429007722801748987264265503451023}{2805710847789879641839318512967467320} a^{17} - \frac{731141051404079756788399063723570273}{54510953614203375898592473966225079360} a^{15} - \frac{1}{4} a^{14} + \frac{805234728295057212259097882333734273}{5611421695579759283678637025934934640} a^{13} - \frac{1}{4} a^{12} - \frac{1566675829260203015299608968878875067}{190788337649711815645073658881787777760} a^{11} + \frac{1}{4} a^{10} - \frac{4402512485969662767141081800282072543}{47697084412427953911268414720446944440} a^{9} - \frac{1}{4} a^{8} - \frac{15363841165020546080527258405397147359}{95394168824855907822536829440893888880} a^{7} - \frac{1}{4} a^{6} + \frac{3405682453048675023802962180171609}{9539416882485590782253682944089388888} a^{5} + \frac{1}{4} a^{4} + \frac{4839717672597433543048893239059306237}{23848542206213976955634207360223472220} a^{3} - \frac{1}{2} a^{2} - \frac{4983355851385076103122681255402267497}{11924271103106988477817103680111736110} a - \frac{1}{2}$

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

Class group and class number

not computed

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{514996232529175576668093054984868617}{95394168824855907822536829440893888880} a^{31} + \frac{988106520635483486035703712791161}{1362773840355084397464811849155626984} a^{30} - \frac{3604382449022270539473796127595209947}{23848542206213976955634207360223472220} a^{29} - \frac{96823285094289232263717895106937049}{4769708441242795391126841472044694444} a^{28} + \frac{258462223445647671992516308044925698831}{95394168824855907822536829440893888880} a^{27} + \frac{13886130096013375056443394420900028831}{38157667529942363129014731776357555552} a^{26} - \frac{2730362074569507354458069031754085364851}{95394168824855907822536829440893888880} a^{25} - \frac{146695531565362803444888021738262891657}{38157667529942363129014731776357555552} a^{24} + \frac{4150351026456753812894507152519997081587}{19078833764971181564507365888178777776} a^{23} + \frac{39819189815535103881481029638740300655}{1362773840355084397464811849155626984} a^{22} - \frac{52829924586599207128709148866690083750073}{47697084412427953911268414720446944440} a^{21} - \frac{5676793647105181999500713932237637857833}{38157667529942363129014731776357555552} a^{20} + \frac{4771180308057980188798357914271404306931}{1149327335239227805090805173986673360} a^{19} + \frac{8009974033143960501828165962871708212}{14366591690490347563635064674833417} a^{18} - \frac{12129576781381995809678117950782616810659}{1122284339115951856735727405186986928} a^{17} - \frac{3257615084062207937618450952890564871835}{2244568678231903713471454810373973856} a^{16} + \frac{1996934928789693615039128479885794027651887}{95394168824855907822536829440893888880} a^{15} + \frac{107217631326153839452799035893432493557651}{38157667529942363129014731776357555552} a^{14} - \frac{80196186032143145663190025351933459682693}{2805710847789879641839318512967467320} a^{13} - \frac{4302705717676639163954075810640278224847}{1122284339115951856735727405186986928} a^{12} + \frac{390162646975655668015937163945657317473881}{13627738403550843974648118491556269840} a^{11} + \frac{18287419839542968524085243536523061079151}{4769708441242795391126841472044694444} a^{10} - \frac{1789268649131026700171573847451747328813087}{95394168824855907822536829440893888880} a^{9} - \frac{23885063909469021611976909738013364013113}{9539416882485590782253682944089388888} a^{8} + \frac{105086668555474653712982687262953750190429}{11924271103106988477817103680111736110} a^{7} + \frac{397654790784460289549565150933093256049}{340693460088771099366202962288906746} a^{6} - \frac{23052120519913087963677428519103333252099}{9539416882485590782253682944089388888} a^{5} - \frac{1500435983134946677055644497324633547337}{4769708441242795391126841472044694444} a^{4} + \frac{1035161254463375616980583511991251656371}{2384854220621397695563420736022347222} a^{3} + \frac{252139166040173027919746581252542158743}{4769708441242795391126841472044694444} a^{2} - \frac{151027767921222955513495610139025474777}{23848542206213976955634207360223472220} a - \frac{152300882045682185285006785323835843}{2384854220621397695563420736022347222} \) (order $24$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{14}) \), 4.4.9792.1, 4.4.53312.1, 4.4.4352.1, 4.4.1919232.1, 4.0.1088.2, 4.0.479808.2, 4.0.39168.3, 4.0.213248.3, \(\Q(\sqrt{6}, \sqrt{14})\), \(\Q(\sqrt{-6}, \sqrt{-14})\), \(\Q(i, \sqrt{21})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{-14})\), \(\Q(\sqrt{-6}, \sqrt{14})\), \(\Q(i, \sqrt{14})\), \(\Q(\sqrt{2}, \sqrt{21})\), \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{-2}, \sqrt{21})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{6}, \sqrt{7})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{6}, \sqrt{-7})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-6}, \sqrt{-7})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{-6}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{-21})\), \(\Q(\sqrt{-2}, \sqrt{-21})\), \(\Q(\sqrt{3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{7})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{42})\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{-14})\), \(\Q(\sqrt{3}, \sqrt{-14})\), \(\Q(\sqrt{-2}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{7})\), \(\Q(\sqrt{3}, \sqrt{14})\), \(\Q(\sqrt{-3}, \sqrt{14})\), \(\Q(\sqrt{-2}, \sqrt{-7})\), 8.0.12745506816.1, 8.8.12745506816.1, 8.0.796594176.1, 8.0.796594176.2, 8.0.12745506816.3, 8.0.12745506816.8, 8.0.49787136.1, \(\Q(\zeta_{24})\), 8.0.12745506816.7, 8.0.12745506816.6, 8.0.12745506816.5, 8.0.12745506816.4, 8.0.12745506816.2, 8.0.157351936.1, 8.0.12745506816.9, 8.8.230215716864.1, 8.8.3683451469824.3, 8.0.230215716864.18, 8.0.3683451469824.25, 8.8.1534132224.1, 8.8.3683451469824.4, 8.0.1534132224.4, 8.0.3683451469824.32, 8.0.95883264.1, 8.0.230215716864.17, 8.0.1534132224.10, 8.0.3683451469824.82, 8.0.3683451469824.39, 8.0.3683451469824.20, 8.0.3683451469824.74, 8.0.3683451469824.21, 8.0.18939904.2, 8.0.3683451469824.60, 8.0.1534132224.8, 8.0.45474709504.3, 8.0.2842169344.2, 8.0.230215716864.9, 8.0.3683451469824.5, 8.0.45474709504.20, 8.8.3683451469824.5, 8.8.45474709504.1, 8.0.45474709504.2, 8.0.3683451469824.8, 16.0.162447943996702457856.1, Deg 16, Deg 16, 16.0.52999276291205413994496.1, Deg 16, Deg 16, Deg 16, 16.0.2353561680715186176.2, Deg 16, Deg 16, Deg 16, Deg 16, Deg 16, Deg 16, 16.0.2067949204473187926016.1

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{16}$ 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}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$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}$