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Properties

Label 16.8.329...000.2
Degree $16$
Signature $[8, 4]$
Discriminant $3.300\times 10^{25}$
Root discriminant \(39.35\)
Ramified primes $2,5,11,269$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^7.\POPlus(4,3)$ (as 16T1870)

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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^16 - 20*x^14 - 16*x^13 + 124*x^12 + 180*x^11 - 100*x^10 - 584*x^9 - 1360*x^8 + 752*x^7 + 4608*x^6 - 2112*x^5 - 3344*x^4 + 4064*x^3 - 2336*x^2 - 1216*x + 704)
 
Copy content gp:K = bnfinit(y^16 - 20*y^14 - 16*y^13 + 124*y^12 + 180*y^11 - 100*y^10 - 584*y^9 - 1360*y^8 + 752*y^7 + 4608*y^6 - 2112*y^5 - 3344*y^4 + 4064*y^3 - 2336*y^2 - 1216*y + 704, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 20*x^14 - 16*x^13 + 124*x^12 + 180*x^11 - 100*x^10 - 584*x^9 - 1360*x^8 + 752*x^7 + 4608*x^6 - 2112*x^5 - 3344*x^4 + 4064*x^3 - 2336*x^2 - 1216*x + 704);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 20*x^14 - 16*x^13 + 124*x^12 + 180*x^11 - 100*x^10 - 584*x^9 - 1360*x^8 + 752*x^7 + 4608*x^6 - 2112*x^5 - 3344*x^4 + 4064*x^3 - 2336*x^2 - 1216*x + 704)
 

\( x^{16} - 20 x^{14} - 16 x^{13} + 124 x^{12} + 180 x^{11} - 100 x^{10} - 584 x^{9} - 1360 x^{8} + \cdots + 704 \) 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:  $16$
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:  $[8, 4]$
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:   \(32995303182626734400000000\) \(\medspace = 2^{12}\cdot 5^{8}\cdot 11^{4}\cdot 269^{5}\) 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:  \(39.35\)
Copy content comment:Root discriminant
 
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:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{4/3}5^{1/2}11^{1/2}269^{3/4}\approx 1241.2774973494347$
Ramified primes:   \(2\), \(5\), \(11\), \(269\) 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:  \(\Q(\sqrt{269}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{4}a^{6}$, $\frac{1}{4}a^{7}$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{9}$, $\frac{1}{8}a^{10}$, $\frac{1}{16}a^{11}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{12}$, $\frac{1}{16}a^{13}$, $\frac{1}{32}a^{14}-\frac{1}{4}a^{5}$, $\frac{1}{33\cdots 92}a^{15}-\frac{43617176685077}{33\cdots 92}a^{14}-\frac{15895334589273}{849815329867048}a^{13}-\frac{3233686850574}{106226916233381}a^{12}+\frac{25601545031743}{849815329867048}a^{11}+\frac{44758124037515}{849815329867048}a^{10}+\frac{28393783061587}{849815329867048}a^{9}-\frac{33219545570447}{849815329867048}a^{8}+\frac{20825219618511}{424907664933524}a^{7}+\frac{13533296816121}{424907664933524}a^{6}-\frac{45829536751601}{212453832466762}a^{5}+\frac{12451028149914}{106226916233381}a^{4}+\frac{5269628362539}{106226916233381}a^{3}-\frac{33870597786739}{212453832466762}a^{2}-\frac{30784826639185}{106226916233381}a+\frac{21213183006960}{106226916233381}$ 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:  $C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:   $\frac{2919986905}{839531074208}a^{15}+\frac{4652148291}{839531074208}a^{14}-\frac{25959187267}{419765537104}a^{13}-\frac{65402246937}{419765537104}a^{12}+\frac{21335704201}{104941384276}a^{11}+\frac{209348252823}{209882768552}a^{10}+\frac{60984963963}{52470692138}a^{9}-\frac{104062067705}{209882768552}a^{8}-\frac{304171279355}{52470692138}a^{7}-\frac{333395793881}{52470692138}a^{6}+\frac{403905631369}{52470692138}a^{5}+\frac{339625405887}{52470692138}a^{4}-\frac{246667922213}{52470692138}a^{3}+\frac{286228159869}{52470692138}a^{2}+\frac{105221852647}{26235346069}a-\frac{21856637811}{26235346069}$, $\frac{28547028993349}{33\cdots 92}a^{15}+\frac{24356577041219}{16\cdots 96}a^{14}-\frac{245755413883627}{16\cdots 96}a^{13}-\frac{40984734660004}{106226916233381}a^{12}+\frac{84906491830599}{212453832466762}a^{11}+\frac{243562381816029}{106226916233381}a^{10}+\frac{26\cdots 39}{849815329867048}a^{9}-\frac{37889962739983}{424907664933524}a^{8}-\frac{53\cdots 21}{424907664933524}a^{7}-\frac{33\cdots 41}{212453832466762}a^{6}+\frac{29\cdots 49}{212453832466762}a^{5}+\frac{12\cdots 22}{106226916233381}a^{4}-\frac{832962057441438}{106226916233381}a^{3}+\frac{15\cdots 44}{106226916233381}a^{2}+\frac{656023243657136}{106226916233381}a-\frac{76854693326399}{106226916233381}$, $\frac{26742919324707}{16\cdots 96}a^{15}+\frac{73875225114531}{33\cdots 92}a^{14}-\frac{119567476695131}{424907664933524}a^{13}-\frac{10\cdots 71}{16\cdots 96}a^{12}+\frac{860618238753947}{849815329867048}a^{11}+\frac{17\cdots 47}{424907664933524}a^{10}+\frac{460849757855272}{106226916233381}a^{9}-\frac{20\cdots 01}{849815329867048}a^{8}-\frac{10\cdots 45}{424907664933524}a^{7}-\frac{92\cdots 23}{424907664933524}a^{6}+\frac{39\cdots 60}{106226916233381}a^{5}+\frac{16\cdots 45}{106226916233381}a^{4}-\frac{50\cdots 79}{212453832466762}a^{3}+\frac{67\cdots 59}{212453832466762}a^{2}+\frac{388804057816341}{106226916233381}a-\frac{794865613411582}{106226916233381}$, $\frac{68342326434345}{33\cdots 92}a^{15}+\frac{94467694921387}{33\cdots 92}a^{14}-\frac{305811401788537}{849815329867048}a^{13}-\frac{13\cdots 17}{16\cdots 96}a^{12}+\frac{11\cdots 97}{849815329867048}a^{11}+\frac{44\cdots 31}{849815329867048}a^{10}+\frac{47\cdots 99}{849815329867048}a^{9}-\frac{27\cdots 91}{849815329867048}a^{8}-\frac{13\cdots 47}{424907664933524}a^{7}-\frac{59\cdots 03}{212453832466762}a^{6}+\frac{10\cdots 05}{212453832466762}a^{5}+\frac{21\cdots 85}{106226916233381}a^{4}-\frac{66\cdots 83}{212453832466762}a^{3}+\frac{84\cdots 17}{212453832466762}a^{2}+\frac{555570456939717}{106226916233381}a-\frac{967687208197077}{106226916233381}$, $\frac{153145945686467}{33\cdots 92}a^{15}+\frac{221400462586685}{33\cdots 92}a^{14}-\frac{342315561279123}{424907664933524}a^{13}-\frac{15\cdots 65}{849815329867048}a^{12}+\frac{24\cdots 21}{849815329867048}a^{11}+\frac{10\cdots 79}{849815329867048}a^{10}+\frac{11\cdots 91}{849815329867048}a^{9}-\frac{29\cdots 21}{424907664933524}a^{8}-\frac{29\cdots 03}{424907664933524}a^{7}-\frac{14\cdots 67}{212453832466762}a^{6}+\frac{45\cdots 59}{424907664933524}a^{5}+\frac{59\cdots 66}{106226916233381}a^{4}-\frac{14\cdots 81}{212453832466762}a^{3}+\frac{92\cdots 83}{106226916233381}a^{2}+\frac{21\cdots 92}{106226916233381}a-\frac{25\cdots 29}{106226916233381}$, $\frac{6952440427569}{33\cdots 92}a^{15}+\frac{337979520279}{849815329867048}a^{14}-\frac{17761987902373}{424907664933524}a^{13}-\frac{74912593903777}{16\cdots 96}a^{12}+\frac{438674048410095}{16\cdots 96}a^{11}+\frac{411937551532943}{849815329867048}a^{10}-\frac{64164075699695}{849815329867048}a^{9}-\frac{12\cdots 69}{849815329867048}a^{8}-\frac{791569592534479}{212453832466762}a^{7}+\frac{1624021450865}{424907664933524}a^{6}+\frac{42\cdots 59}{424907664933524}a^{5}+\frac{14030533281123}{106226916233381}a^{4}-\frac{633186439094834}{106226916233381}a^{3}+\frac{706763872668044}{106226916233381}a^{2}-\frac{220828556515316}{106226916233381}a-\frac{298859519332167}{106226916233381}$, $\frac{13890314800361}{33\cdots 92}a^{15}+\frac{5284456919879}{424907664933524}a^{14}-\frac{51985177126333}{849815329867048}a^{13}-\frac{481749595472749}{16\cdots 96}a^{12}-\frac{98278592736777}{16\cdots 96}a^{11}+\frac{298061194827917}{212453832466762}a^{10}+\frac{26\cdots 81}{849815329867048}a^{9}+\frac{16\cdots 45}{849815329867048}a^{8}-\frac{27\cdots 51}{424907664933524}a^{7}-\frac{70\cdots 43}{424907664933524}a^{6}-\frac{16\cdots 71}{424907664933524}a^{5}+\frac{17\cdots 53}{106226916233381}a^{4}+\frac{665066814038023}{106226916233381}a^{3}+\frac{148247716264486}{106226916233381}a^{2}+\frac{11\cdots 76}{106226916233381}a+\frac{588284003711489}{106226916233381}$, $\frac{2549632430965}{849815329867048}a^{15}+\frac{8266705640483}{16\cdots 96}a^{14}-\frac{44123346968605}{849815329867048}a^{13}-\frac{224322988259061}{16\cdots 96}a^{12}+\frac{64280178481977}{424907664933524}a^{11}+\frac{82703312640989}{106226916233381}a^{10}+\frac{880487941446425}{849815329867048}a^{9}+\frac{4488991817983}{212453832466762}a^{8}-\frac{18\cdots 89}{424907664933524}a^{7}-\frac{22\cdots 21}{424907664933524}a^{6}+\frac{10\cdots 43}{212453832466762}a^{5}+\frac{251999177711099}{106226916233381}a^{4}-\frac{169629640587041}{106226916233381}a^{3}+\frac{838721401078870}{106226916233381}a^{2}-\frac{42815462809240}{106226916233381}a-\frac{143748646662719}{106226916233381}$, $\frac{638877297593}{33\cdots 92}a^{15}-\frac{121834102677}{16\cdots 96}a^{14}-\frac{3456264114113}{849815329867048}a^{13}-\frac{5437269676977}{16\cdots 96}a^{12}+\frac{3007769957222}{106226916233381}a^{11}+\frac{47216162731097}{849815329867048}a^{10}-\frac{565064435173}{424907664933524}a^{9}-\frac{92134114525911}{424907664933524}a^{8}-\frac{248915292509703}{424907664933524}a^{7}-\frac{43333455767993}{424907664933524}a^{6}+\frac{138775882213804}{106226916233381}a^{5}+\frac{42909588798798}{106226916233381}a^{4}-\frac{41442245258273}{106226916233381}a^{3}+\frac{120088063764022}{106226916233381}a^{2}-\frac{13262893854394}{106226916233381}a-\frac{60157745052701}{106226916233381}$, $\frac{111062848803655}{16\cdots 96}a^{15}+\frac{82279726762911}{849815329867048}a^{14}-\frac{19\cdots 61}{16\cdots 96}a^{13}-\frac{23\cdots 55}{849815329867048}a^{12}+\frac{16\cdots 45}{424907664933524}a^{11}+\frac{73\cdots 31}{424907664933524}a^{10}+\frac{16\cdots 95}{849815329867048}a^{9}-\frac{32\cdots 13}{424907664933524}a^{8}-\frac{10\cdots 29}{106226916233381}a^{7}-\frac{42\cdots 59}{424907664933524}a^{6}+\frac{15\cdots 62}{106226916233381}a^{5}+\frac{80\cdots 77}{106226916233381}a^{4}-\frac{19\cdots 43}{212453832466762}a^{3}+\frac{12\cdots 50}{106226916233381}a^{2}+\frac{32\cdots 00}{106226916233381}a-\frac{31\cdots 67}{106226916233381}$, $\frac{30100690119735}{424907664933524}a^{15}+\frac{446829764107}{424907664933524}a^{14}-\frac{23\cdots 91}{16\cdots 96}a^{13}-\frac{19\cdots 81}{16\cdots 96}a^{12}+\frac{35\cdots 31}{424907664933524}a^{11}+\frac{10\cdots 93}{849815329867048}a^{10}-\frac{17\cdots 41}{424907664933524}a^{9}-\frac{31\cdots 67}{849815329867048}a^{8}-\frac{40\cdots 57}{424907664933524}a^{7}+\frac{44\cdots 20}{106226916233381}a^{6}+\frac{12\cdots 89}{424907664933524}a^{5}-\frac{15\cdots 59}{106226916233381}a^{4}-\frac{15\cdots 27}{106226916233381}a^{3}+\frac{51\cdots 77}{212453832466762}a^{2}-\frac{20\cdots 23}{106226916233381}a-\frac{16\cdots 56}{106226916233381}$ 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:  \( 18200443.0721 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 18200443.0721 \cdot 1}{2\cdot\sqrt{32995303182626734400000000}}\cr\approx \mathstrut & 0.632099347506 \end{aligned}\] (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^16 - 20*x^14 - 16*x^13 + 124*x^12 + 180*x^11 - 100*x^10 - 584*x^9 - 1360*x^8 + 752*x^7 + 4608*x^6 - 2112*x^5 - 3344*x^4 + 4064*x^3 - 2336*x^2 - 1216*x + 704) 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^16 - 20*x^14 - 16*x^13 + 124*x^12 + 180*x^11 - 100*x^10 - 584*x^9 - 1360*x^8 + 752*x^7 + 4608*x^6 - 2112*x^5 - 3344*x^4 + 4064*x^3 - 2336*x^2 - 1216*x + 704, 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^16 - 20*x^14 - 16*x^13 + 124*x^12 + 180*x^11 - 100*x^10 - 584*x^9 - 1360*x^8 + 752*x^7 + 4608*x^6 - 2112*x^5 - 3344*x^4 + 4064*x^3 - 2336*x^2 - 1216*x + 704); 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 = polynomial_ring(QQ); K, a = number_field(x^16 - 20*x^14 - 16*x^13 + 124*x^12 + 180*x^11 - 100*x^10 - 584*x^9 - 1360*x^8 + 752*x^7 + 4608*x^6 - 2112*x^5 - 3344*x^4 + 4064*x^3 - 2336*x^2 - 1216*x + 704); 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

$C_2^7.\POPlus(4,3)$ (as 16T1870):

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); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 73728
The 83 conjugacy class representatives for $C_2^7.\POPlus(4,3)$
Character table for $C_2^7.\POPlus(4,3)$

Intermediate fields

\(\Q(\sqrt{5}) \), 8.8.87556810000.1

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(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.4.0.1}{4} }$ R ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ R ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{5}$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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 2.2.2.4a1.2$x^{4} + 2 x^{3} + 5 x^{2} + 8 x + 5$$2$$2$$4$$C_4$$$[2]^{2}$$
2.2.3.4a1.1$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 5 x + 1$$3$$2$$4$$S_3\times C_3$$$[\ ]_{3}^{6}$$
2.2.3.4a1.1$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 5 x + 1$$3$$2$$4$$S_3\times C_3$$$[\ ]_{3}^{6}$$
\(5\) Copy content Toggle raw display 5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
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$$$[\ ]_{2}^{6}$$
\(11\) Copy content Toggle raw display $\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{11}$$x + 9$$1$$1$$0$Trivial$$[\ ]$$
11.3.1.0a1.1$x^{3} + 2 x + 9$$1$$3$$0$$C_3$$$[\ ]^{3}$$
11.3.1.0a1.1$x^{3} + 2 x + 9$$1$$3$$0$$C_3$$$[\ ]^{3}$$
11.2.2.2a1.2$x^{4} + 14 x^{3} + 53 x^{2} + 28 x + 15$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
11.2.2.2a1.2$x^{4} + 14 x^{3} + 53 x^{2} + 28 x + 15$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(269\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $4$$1$$4$$0$$C_4$$$[\ ]^{4}$$
Deg $4$$4$$1$$3$

Spectrum of ring of integers

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