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Properties

Label 20.4.164...000.1
Degree $20$
Signature $[4, 8]$
Discriminant $1.645\times 10^{23}$
Root discriminant \(14.48\)
Ramified primes $2,5,89$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^8.\POPlus(4,5)$ (as 20T1013)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 15*x^18 - 18*x^17 + x^16 + 46*x^15 - 138*x^14 + 300*x^13 - 521*x^12 + 716*x^11 - 763*x^10 + 598*x^9 - 273*x^8 - 58*x^7 + 250*x^6 - 272*x^5 + 199*x^4 - 110*x^3 + 43*x^2 - 10*x + 1)
 
Copy content gp:K = bnfinit(y^20 - 6*y^19 + 15*y^18 - 18*y^17 + y^16 + 46*y^15 - 138*y^14 + 300*y^13 - 521*y^12 + 716*y^11 - 763*y^10 + 598*y^9 - 273*y^8 - 58*y^7 + 250*y^6 - 272*y^5 + 199*y^4 - 110*y^3 + 43*y^2 - 10*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 6*x^19 + 15*x^18 - 18*x^17 + x^16 + 46*x^15 - 138*x^14 + 300*x^13 - 521*x^12 + 716*x^11 - 763*x^10 + 598*x^9 - 273*x^8 - 58*x^7 + 250*x^6 - 272*x^5 + 199*x^4 - 110*x^3 + 43*x^2 - 10*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 6*x^19 + 15*x^18 - 18*x^17 + x^16 + 46*x^15 - 138*x^14 + 300*x^13 - 521*x^12 + 716*x^11 - 763*x^10 + 598*x^9 - 273*x^8 - 58*x^7 + 250*x^6 - 272*x^5 + 199*x^4 - 110*x^3 + 43*x^2 - 10*x + 1)
 

\( x^{20} - 6 x^{19} + 15 x^{18} - 18 x^{17} + x^{16} + 46 x^{15} - 138 x^{14} + 300 x^{13} - 521 x^{12} + \cdots + 1 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $20$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[4, 8]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(164475020247040000000000\) \(\medspace = 2^{28}\cdot 5^{10}\cdot 89^{4}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(14.48\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{115/48}5^{1/2}89^{2/3}\approx 234.5827309266611$
Ramified primes:   \(2\), \(5\), \(89\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{2888902}a^{19}-\frac{419709}{2888902}a^{18}-\frac{280955}{1444451}a^{17}-\frac{101029}{1444451}a^{16}+\frac{630565}{2888902}a^{15}-\frac{598831}{2888902}a^{14}-\frac{418043}{2888902}a^{13}-\frac{672539}{2888902}a^{12}-\frac{756769}{2888902}a^{11}+\frac{778835}{2888902}a^{10}+\frac{1318983}{2888902}a^{9}-\frac{22415}{61466}a^{8}+\frac{601017}{1444451}a^{7}+\frac{173503}{1444451}a^{6}+\frac{202009}{2888902}a^{5}-\frac{287703}{2888902}a^{4}-\frac{256694}{1444451}a^{3}-\frac{580459}{1444451}a^{2}+\frac{264}{1444451}a+\frac{421130}{1444451}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $11$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{64117}{6862}a^{19}-\frac{339237}{6862}a^{18}+\frac{360542}{3431}a^{17}-\frac{641313}{6862}a^{16}-\frac{197459}{3431}a^{15}+\frac{2672619}{6862}a^{14}-\frac{6949175}{6862}a^{13}+\frac{14295893}{6862}a^{12}-\frac{11619157}{3431}a^{11}+\frac{29361211}{6862}a^{10}-\frac{13995232}{3431}a^{9}+\frac{195375}{73}a^{8}-\frac{4367441}{6862}a^{7}-\frac{3441999}{3431}a^{6}+\frac{5574049}{3431}a^{5}-\frac{9488551}{6862}a^{4}+\frac{2991307}{3431}a^{3}-\frac{1387098}{3431}a^{2}+\frac{768643}{6862}a-\frac{100259}{6862}$, $\frac{138652157}{2888902}a^{19}-\frac{755613271}{2888902}a^{18}+\frac{831473326}{1444451}a^{17}-\frac{787421373}{1444451}a^{16}-\frac{740750239}{2888902}a^{15}+\frac{2990899812}{1444451}a^{14}-\frac{15834903167}{2888902}a^{13}+\frac{16417446477}{1444451}a^{12}-\frac{54049126959}{2888902}a^{11}+\frac{69286251565}{2888902}a^{10}-\frac{67262298279}{2888902}a^{9}+\frac{482947178}{30733}a^{8}-\frac{6199783206}{1444451}a^{7}-\frac{7579684396}{1444451}a^{6}+\frac{26371030423}{2888902}a^{5}-\frac{11532238808}{1444451}a^{4}+\frac{7352772866}{1444451}a^{3}-\frac{7005085065}{2888902}a^{2}+\frac{1007119004}{1444451}a-\frac{118125338}{1444451}$, $a$, $\frac{35202359}{2888902}a^{19}-\frac{90642241}{1444451}a^{18}+\frac{368339985}{2888902}a^{17}-\frac{145653959}{1444451}a^{16}-\frac{136453274}{1444451}a^{15}+\frac{718195639}{1444451}a^{14}-\frac{1796773576}{1444451}a^{13}+\frac{3636358233}{1444451}a^{12}-\frac{11573832953}{2888902}a^{11}+\frac{7089842118}{1444451}a^{10}-\frac{12908592349}{2888902}a^{9}+\frac{82403262}{30733}a^{8}-\frac{447712594}{1444451}a^{7}-\frac{2044168537}{1444451}a^{6}+\frac{2740489632}{1444451}a^{5}-\frac{2121784163}{1444451}a^{4}+\frac{2497016539}{2888902}a^{3}-\frac{527368823}{1444451}a^{2}+\frac{209095479}{2888902}a-\frac{2391237}{1444451}$, $a^{19}-5a^{18}+10a^{17}-8a^{16}-7a^{15}+39a^{14}-99a^{13}+201a^{12}-320a^{11}+396a^{10}-367a^{9}+231a^{8}-42a^{7}-100a^{6}+150a^{5}-122a^{4}+77a^{3}-33a^{2}+10a-1$, $\frac{19288349}{2888902}a^{19}-\frac{91152157}{2888902}a^{18}+\frac{160892087}{2888902}a^{17}-\frac{40211767}{1444451}a^{16}-\frac{101761950}{1444451}a^{15}+\frac{712215411}{2888902}a^{14}-\frac{822846698}{1444451}a^{13}+\frac{3203495397}{2888902}a^{12}-\frac{4781248143}{2888902}a^{11}+\frac{5342250309}{2888902}a^{10}-\frac{4198830263}{2888902}a^{9}+\frac{37368931}{61466}a^{8}+\frac{428262652}{1444451}a^{7}-\frac{1097776720}{1444451}a^{6}+\frac{1027264952}{1444451}a^{5}-\frac{1225803779}{2888902}a^{4}+\frac{583027723}{2888902}a^{3}-\frac{67959680}{1444451}a^{2}-\frac{55464867}{2888902}a+\frac{9104654}{1444451}$, $\frac{89118477}{1444451}a^{19}-\frac{979442935}{2888902}a^{18}+\frac{2177615683}{2888902}a^{17}-\frac{2102012541}{2888902}a^{16}-\frac{447812404}{1444451}a^{15}+\frac{3877853100}{1444451}a^{14}-\frac{10335223964}{1444451}a^{13}+\frac{42981973249}{2888902}a^{12}-\frac{35515022227}{1444451}a^{11}+\frac{45748679585}{1444451}a^{10}-\frac{89396056881}{2888902}a^{9}+\frac{1297090467}{61466}a^{8}-\frac{8722275215}{1444451}a^{7}-\frac{19396661001}{2888902}a^{6}+\frac{17392572659}{1444451}a^{5}-\frac{15387116335}{1444451}a^{4}+\frac{9871796873}{1444451}a^{3}-\frac{4750004219}{1444451}a^{2}+\frac{2789475041}{2888902}a-\frac{170000120}{1444451}$, $\frac{79056402}{1444451}a^{19}-\frac{443618237}{1444451}a^{18}+\frac{1012210487}{1444451}a^{17}-\frac{2046743783}{2888902}a^{16}-\frac{661899789}{2888902}a^{15}+\frac{3516606156}{1444451}a^{14}-\frac{9532069279}{1444451}a^{13}+\frac{39926473943}{2888902}a^{12}-\frac{66619492737}{2888902}a^{11}+\frac{86834883303}{2888902}a^{10}-\frac{86138426951}{2888902}a^{9}+\frac{1280324057}{61466}a^{8}-\frac{18989405163}{2888902}a^{7}-\frac{16989752133}{2888902}a^{6}+\frac{32926725029}{2888902}a^{5}-\frac{14959644509}{1444451}a^{4}+\frac{9743578613}{1444451}a^{3}-\frac{9563957189}{2888902}a^{2}+\frac{2936639399}{2888902}a-\frac{193493636}{1444451}$, $\frac{71617093}{1444451}a^{19}-\frac{794472825}{2888902}a^{18}+\frac{894914009}{1444451}a^{17}-\frac{887239853}{1444451}a^{16}-\frac{645746191}{2888902}a^{15}+\frac{3142233986}{1444451}a^{14}-\frac{16929919623}{2888902}a^{13}+\frac{35355466169}{2888902}a^{12}-\frac{29373235460}{1444451}a^{11}+\frac{38113321598}{1444451}a^{10}-\frac{75210406449}{2888902}a^{9}+\frac{554663835}{30733}a^{8}-\frac{7994077277}{1444451}a^{7}-\frac{15246918987}{2888902}a^{6}+\frac{28866714957}{2888902}a^{5}-\frac{13020609937}{1444451}a^{4}+\frac{16918972011}{2888902}a^{3}-\frac{4129501444}{1444451}a^{2}+\frac{1259103647}{1444451}a-\frac{331465035}{2888902}$, $\frac{78430296}{1444451}a^{19}-\frac{448821062}{1444451}a^{18}+\frac{2098895069}{2888902}a^{17}-\frac{2216440571}{2888902}a^{16}-\frac{506486417}{2888902}a^{15}+\frac{3555475702}{1444451}a^{14}-\frac{19633944693}{2888902}a^{13}+\frac{20707976880}{1444451}a^{12}-\frac{34876125690}{1444451}a^{11}+\frac{45996899943}{1444451}a^{10}-\frac{46321594929}{1444451}a^{9}+\frac{1409838931}{61466}a^{8}-\frac{11397606299}{1444451}a^{7}-\frac{8187705053}{1444451}a^{6}+\frac{34735120697}{2888902}a^{5}-\frac{16235671081}{1444451}a^{4}+\frac{21496751861}{2888902}a^{3}-\frac{5379388132}{1444451}a^{2}+\frac{3454143479}{2888902}a-\frac{485724645}{2888902}$, $\frac{135951149}{2888902}a^{19}-\frac{369640180}{1444451}a^{18}+\frac{1625130329}{2888902}a^{17}-\frac{1541156141}{2888902}a^{16}-\frac{715018835}{2888902}a^{15}+\frac{2920223304}{1444451}a^{14}-\frac{7745547171}{1444451}a^{13}+\frac{32137531539}{2888902}a^{12}-\frac{26456471034}{1444451}a^{11}+\frac{67894893481}{2888902}a^{10}-\frac{33017157275}{1444451}a^{9}+\frac{951978617}{61466}a^{8}-\frac{12461059769}{2888902}a^{7}-\frac{14611205737}{2888902}a^{6}+\frac{25772023377}{2888902}a^{5}-\frac{11337504552}{1444451}a^{4}+\frac{14525791167}{2888902}a^{3}-\frac{6944510403}{2888902}a^{2}+\frac{1013389490}{1444451}a-\frac{126017604}{1444451}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 5058.99177989 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 5058.99177989 \cdot 1}{2\cdot\sqrt{164475020247040000000000}}\cr\approx \mathstrut & 0.242405758937 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 15*x^18 - 18*x^17 + x^16 + 46*x^15 - 138*x^14 + 300*x^13 - 521*x^12 + 716*x^11 - 763*x^10 + 598*x^9 - 273*x^8 - 58*x^7 + 250*x^6 - 272*x^5 + 199*x^4 - 110*x^3 + 43*x^2 - 10*x + 1) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^20 - 6*x^19 + 15*x^18 - 18*x^17 + x^16 + 46*x^15 - 138*x^14 + 300*x^13 - 521*x^12 + 716*x^11 - 763*x^10 + 598*x^9 - 273*x^8 - 58*x^7 + 250*x^6 - 272*x^5 + 199*x^4 - 110*x^3 + 43*x^2 - 10*x + 1, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 6*x^19 + 15*x^18 - 18*x^17 + x^16 + 46*x^15 - 138*x^14 + 300*x^13 - 521*x^12 + 716*x^11 - 763*x^10 + 598*x^9 - 273*x^8 - 58*x^7 + 250*x^6 - 272*x^5 + 199*x^4 - 110*x^3 + 43*x^2 - 10*x + 1); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 6*x^19 + 15*x^18 - 18*x^17 + x^16 + 46*x^15 - 138*x^14 + 300*x^13 - 521*x^12 + 716*x^11 - 763*x^10 + 598*x^9 - 273*x^8 - 58*x^7 + 250*x^6 - 272*x^5 + 199*x^4 - 110*x^3 + 43*x^2 - 10*x + 1); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2^8.\POPlus(4,5)$ (as 20T1013):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A non-solvable group of order 3686400
The 114 conjugacy class representatives for $C_2^8.\POPlus(4,5)$
Character table for $C_2^8.\POPlus(4,5)$

Intermediate fields

\(\Q(\sqrt{5}) \), 10.2.25347200000.1

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

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 20 sibling: data not computed
Degree 32 sibling: data not computed
Degree 40 siblings: data not computed
Minimal sibling: 20.4.168422420732968960000000000.3

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.10.0.1}{10} }^{2}$ R ${\href{/padicField/7.10.0.1}{10} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.10.0.1}{10} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.5.0.1}{5} }^{4}$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.2.4.16b5.3$x^{8} + 8 x^{7} + 24 x^{6} + 48 x^{5} + 63 x^{4} + 64 x^{3} + 46 x^{2} + 24 x + 9$$4$$2$$16$$D_4\times C_2$$$[2, 2, 3]^{2}$$
2.2.6.12a1.3$x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 126 x^{7} + 141 x^{6} + 126 x^{5} + 90 x^{4} + 50 x^{3} + 23 x^{2} + 8 x + 5$$6$$2$$12$$S_4$$$[\frac{4}{3}, \frac{4}{3}]_{3}^{2}$$
\(5\) Copy content Toggle raw display 5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
5.6.2.6a1.2$x^{12} + 2 x^{10} + 8 x^{9} + 3 x^{8} + 8 x^{7} + 22 x^{6} + 8 x^{5} + 5 x^{4} + 16 x^{3} + 4 x^{2} + 9$$2$$6$$6$$C_6\times C_2$$$[\ ]_{2}^{6}$$
\(89\) Copy content Toggle raw display 89.2.1.0a1.1$x^{2} + 82 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
89.2.1.0a1.1$x^{2} + 82 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
89.3.1.0a1.1$x^{3} + 3 x + 86$$1$$3$$0$$C_3$$$[\ ]^{3}$$
89.3.1.0a1.1$x^{3} + 3 x + 86$$1$$3$$0$$C_3$$$[\ ]^{3}$$
89.4.1.0a1.1$x^{4} + 4 x^{2} + 72 x + 3$$1$$4$$0$$C_4$$$[\ ]^{4}$$
89.2.3.4a1.2$x^{6} + 246 x^{5} + 20181 x^{4} + 552844 x^{3} + 60543 x^{2} + 2214 x + 116$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)