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Properties

Label 16.16.414...448.1
Degree $16$
Signature $[16, 0]$
Discriminant $4.147\times 10^{29}$
Root discriminant \(70.98\)
Ramified primes $2,449,1889$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2\wr \OD_{16}$ (as 16T1584)

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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 - 88*x^14 + 2928*x^12 - 47936*x^10 + 418966*x^8 - 1996208*x^6 + 4988280*x^4 - 5617888*x^2 + 1612808)
 
Copy content gp:K = bnfinit(y^16 - 88*y^14 + 2928*y^12 - 47936*y^10 + 418966*y^8 - 1996208*y^6 + 4988280*y^4 - 5617888*y^2 + 1612808, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 88*x^14 + 2928*x^12 - 47936*x^10 + 418966*x^8 - 1996208*x^6 + 4988280*x^4 - 5617888*x^2 + 1612808);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 88*x^14 + 2928*x^12 - 47936*x^10 + 418966*x^8 - 1996208*x^6 + 4988280*x^4 - 5617888*x^2 + 1612808)
 

\( x^{16} - 88 x^{14} + 2928 x^{12} - 47936 x^{10} + 418966 x^{8} - 1996208 x^{6} + 4988280 x^{4} + \cdots + 1612808 \) 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:  $[16, 0]$
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:   \(414692653834021167975731560448\) \(\medspace = 2^{59}\cdot 449^{2}\cdot 1889^{2}\) 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:  \(70.98\)
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:  $2^{141/32}449^{1/2}1889^{1/2}\approx 19527.766249117198$
Ramified primes:   \(2\), \(449\), \(1889\) 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:  \(\Q(\sqrt{2}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{2}a^{11}$, $\frac{1}{16}a^{12}-\frac{1}{4}a^{11}-\frac{1}{8}a^{8}-\frac{3}{8}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}$, $\frac{1}{16}a^{13}-\frac{1}{8}a^{9}-\frac{3}{8}a^{5}+\frac{1}{4}a$, $\frac{1}{90\cdots 92}a^{14}+\frac{32\cdots 75}{11\cdots 74}a^{12}+\frac{95\cdots 91}{45\cdots 96}a^{10}-\frac{16\cdots 49}{56\cdots 37}a^{8}+\frac{39\cdots 93}{26\cdots 88}a^{6}-\frac{11\cdots 19}{33\cdots 61}a^{4}-\frac{10\cdots 27}{22\cdots 48}a^{2}+\frac{621525644649394}{12\cdots 13}$, $\frac{1}{90\cdots 92}a^{15}+\frac{32\cdots 75}{11\cdots 74}a^{13}+\frac{95\cdots 91}{45\cdots 96}a^{11}-\frac{16\cdots 49}{56\cdots 37}a^{9}+\frac{39\cdots 93}{26\cdots 88}a^{7}-\frac{11\cdots 19}{33\cdots 61}a^{5}-\frac{10\cdots 27}{22\cdots 48}a^{3}+\frac{621525644649394}{12\cdots 13}a$ 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:  Not computed
Index:  $1$
Inessential primes:  None

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:  $15$
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:   $\frac{23692341}{281782655624}a^{14}-\frac{498828351}{70445663906}a^{12}+\frac{30922965701}{140891327812}a^{10}-\frac{112925453832}{35222831953}a^{8}+\frac{3275151267145}{140891327812}a^{6}-\frac{2870856143884}{35222831953}a^{4}+\frac{8341361814311}{70445663906}a^{2}-\frac{1215435148253}{35222831953}$, $\frac{17897419618225}{53\cdots 76}a^{14}-\frac{377921725907145}{13\cdots 44}a^{12}+\frac{23\cdots 43}{26\cdots 88}a^{10}-\frac{86\cdots 29}{66\cdots 22}a^{8}+\frac{25\cdots 81}{26\cdots 88}a^{6}-\frac{22\cdots 33}{66\cdots 22}a^{4}+\frac{69\cdots 89}{13\cdots 44}a^{2}-\frac{14\cdots 50}{74425843916689}$, $\frac{41738502630549}{53\cdots 76}a^{14}-\frac{17\cdots 45}{26\cdots 88}a^{12}+\frac{54\cdots 15}{26\cdots 88}a^{10}-\frac{39\cdots 47}{13\cdots 44}a^{8}+\frac{58\cdots 97}{26\cdots 88}a^{6}-\frac{10\cdots 01}{13\cdots 44}a^{4}+\frac{14\cdots 17}{13\cdots 44}a^{2}-\frac{54\cdots 35}{148851687833378}$, $\frac{33006715021406}{56\cdots 37}a^{14}-\frac{21\cdots 45}{45\cdots 96}a^{12}+\frac{16\cdots 27}{11\cdots 74}a^{10}-\frac{46\cdots 87}{22\cdots 48}a^{8}+\frac{46\cdots 16}{33\cdots 61}a^{6}-\frac{58\cdots 81}{13\cdots 44}a^{4}+\frac{31\cdots 76}{56\cdots 37}a^{2}-\frac{32\cdots 55}{25\cdots 26}$, $\frac{119793975730411}{45\cdots 96}a^{14}-\frac{10\cdots 07}{45\cdots 96}a^{12}+\frac{15\cdots 79}{22\cdots 48}a^{10}-\frac{23\cdots 13}{22\cdots 48}a^{8}+\frac{10\cdots 51}{13\cdots 44}a^{6}-\frac{38\cdots 39}{13\cdots 44}a^{4}+\frac{51\cdots 23}{11\cdots 74}a^{2}-\frac{36\cdots 97}{25\cdots 26}$, $\frac{130963131678681}{22\cdots 48}a^{14}-\frac{21\cdots 21}{45\cdots 96}a^{12}+\frac{16\cdots 55}{11\cdots 74}a^{10}-\frac{46\cdots 67}{22\cdots 48}a^{8}+\frac{94\cdots 31}{66\cdots 22}a^{6}-\frac{60\cdots 45}{13\cdots 44}a^{4}+\frac{34\cdots 74}{56\cdots 37}a^{2}-\frac{48\cdots 75}{25\cdots 26}$, $\frac{242449249134715}{45\cdots 96}a^{14}-\frac{10\cdots 81}{22\cdots 48}a^{12}+\frac{32\cdots 99}{22\cdots 48}a^{10}-\frac{24\cdots 83}{11\cdots 74}a^{8}+\frac{21\cdots 23}{13\cdots 44}a^{6}-\frac{39\cdots 19}{66\cdots 22}a^{4}+\frac{10\cdots 51}{11\cdots 74}a^{2}-\frac{49\cdots 54}{12\cdots 13}$, $\frac{23692341}{281782655624}a^{15}-\frac{171469595401279}{90\cdots 92}a^{14}-\frac{498828351}{70445663906}a^{13}+\frac{19\cdots 87}{11\cdots 74}a^{12}+\frac{30922965701}{140891327812}a^{11}-\frac{25\cdots 77}{45\cdots 96}a^{10}-\frac{112925453832}{35222831953}a^{9}+\frac{52\cdots 13}{56\cdots 37}a^{8}+\frac{3275151267145}{140891327812}a^{7}-\frac{20\cdots 23}{26\cdots 88}a^{6}-\frac{2870856143884}{35222831953}a^{5}+\frac{10\cdots 56}{33\cdots 61}a^{4}+\frac{8341361814311}{70445663906}a^{3}-\frac{12\cdots 67}{22\cdots 48}a^{2}-\frac{1215435148253}{35222831953}a+\frac{19\cdots 71}{12\cdots 13}$, $\frac{100399905779759}{22\cdots 48}a^{15}+\frac{7711099361389}{13\cdots 44}a^{14}-\frac{33\cdots 75}{90\cdots 92}a^{13}-\frac{635954894548485}{13\cdots 44}a^{12}+\frac{63\cdots 80}{56\cdots 37}a^{11}+\frac{47\cdots 90}{33\cdots 61}a^{10}-\frac{70\cdots 13}{45\cdots 96}a^{9}-\frac{65\cdots 39}{33\cdots 61}a^{8}+\frac{70\cdots 01}{66\cdots 22}a^{7}+\frac{85\cdots 17}{66\cdots 22}a^{6}-\frac{89\cdots 39}{26\cdots 88}a^{5}-\frac{25\cdots 31}{66\cdots 22}a^{4}+\frac{23\cdots 79}{56\cdots 37}a^{3}+\frac{15\cdots 92}{33\cdots 61}a^{2}-\frac{39\cdots 19}{50\cdots 52}a-\frac{622169599192311}{74425843916689}$, $\frac{113064699311427}{45\cdots 96}a^{15}-\frac{209564723536521}{90\cdots 92}a^{14}-\frac{91\cdots 55}{45\cdots 96}a^{13}+\frac{976222388875363}{56\cdots 37}a^{12}+\frac{13\cdots 33}{22\cdots 48}a^{11}-\frac{19\cdots 23}{45\cdots 96}a^{10}-\frac{17\cdots 45}{22\cdots 48}a^{9}+\frac{24\cdots 49}{56\cdots 37}a^{8}+\frac{64\cdots 87}{13\cdots 44}a^{7}-\frac{36\cdots 93}{26\cdots 88}a^{6}-\frac{19\cdots 83}{13\cdots 44}a^{5}-\frac{31\cdots 50}{33\cdots 61}a^{4}+\frac{23\cdots 17}{11\cdots 74}a^{3}+\frac{19\cdots 15}{22\cdots 48}a^{2}-\frac{26\cdots 61}{25\cdots 26}a-\frac{69\cdots 75}{12\cdots 13}$, $\frac{197822731513747}{90\cdots 92}a^{15}+\frac{1063856750327}{20\cdots 08}a^{14}-\frac{83\cdots 95}{45\cdots 96}a^{13}-\frac{21948490610591}{50\cdots 52}a^{12}+\frac{25\cdots 13}{45\cdots 96}a^{11}+\frac{13\cdots 57}{10\cdots 04}a^{10}-\frac{18\cdots 63}{22\cdots 48}a^{9}-\frac{45\cdots 03}{25\cdots 26}a^{8}+\frac{16\cdots 71}{26\cdots 88}a^{7}+\frac{71\cdots 51}{595406751333512}a^{6}-\frac{29\cdots 75}{13\cdots 44}a^{5}-\frac{54\cdots 57}{148851687833378}a^{4}+\frac{84\cdots 67}{22\cdots 48}a^{3}+\frac{22\cdots 87}{50\cdots 52}a^{2}-\frac{53\cdots 09}{25\cdots 26}a-\frac{90\cdots 76}{12\cdots 13}$, $\frac{802221882420075}{90\cdots 92}a^{15}-\frac{595641643339173}{90\cdots 92}a^{14}-\frac{34\cdots 83}{45\cdots 96}a^{13}+\frac{13\cdots 09}{22\cdots 48}a^{12}+\frac{10\cdots 53}{45\cdots 96}a^{11}-\frac{91\cdots 47}{45\cdots 96}a^{10}-\frac{81\cdots 59}{22\cdots 48}a^{9}+\frac{38\cdots 89}{11\cdots 74}a^{8}+\frac{72\cdots 99}{26\cdots 88}a^{7}-\frac{76\cdots 29}{26\cdots 88}a^{6}-\frac{13\cdots 43}{13\cdots 44}a^{5}+\frac{79\cdots 91}{66\cdots 22}a^{4}+\frac{36\cdots 75}{22\cdots 48}a^{3}-\frac{46\cdots 17}{22\cdots 48}a^{2}-\frac{12\cdots 77}{25\cdots 26}a+\frac{83\cdots 44}{12\cdots 13}$, $\frac{86727310918653}{90\cdots 92}a^{15}+\frac{293608478544617}{90\cdots 92}a^{14}-\frac{71\cdots 95}{90\cdots 92}a^{13}-\frac{24\cdots 55}{90\cdots 92}a^{12}+\frac{11\cdots 61}{45\cdots 96}a^{11}+\frac{38\cdots 55}{45\cdots 96}a^{10}-\frac{16\cdots 45}{45\cdots 96}a^{9}-\frac{55\cdots 41}{45\cdots 96}a^{8}+\frac{73\cdots 89}{26\cdots 88}a^{7}+\frac{22\cdots 21}{26\cdots 88}a^{6}-\frac{28\cdots 91}{26\cdots 88}a^{5}-\frac{71\cdots 47}{26\cdots 88}a^{4}+\frac{39\cdots 43}{22\cdots 48}a^{3}+\frac{80\cdots 81}{22\cdots 48}a^{2}-\frac{33\cdots 71}{50\cdots 52}a-\frac{55\cdots 91}{50\cdots 52}$, $\frac{25354105168519}{53\cdots 76}a^{15}+\frac{678964931110745}{45\cdots 96}a^{14}-\frac{10\cdots 05}{26\cdots 88}a^{13}-\frac{11\cdots 93}{90\cdots 92}a^{12}+\frac{32\cdots 51}{26\cdots 88}a^{11}+\frac{89\cdots 99}{22\cdots 48}a^{10}-\frac{22\cdots 71}{13\cdots 44}a^{9}-\frac{26\cdots 07}{45\cdots 96}a^{8}+\frac{31\cdots 71}{26\cdots 88}a^{7}+\frac{55\cdots 45}{13\cdots 44}a^{6}-\frac{52\cdots 01}{13\cdots 44}a^{5}-\frac{38\cdots 81}{26\cdots 88}a^{4}+\frac{71\cdots 61}{13\cdots 44}a^{3}+\frac{23\cdots 27}{11\cdots 74}a^{2}-\frac{25\cdots 19}{148851687833378}a-\frac{32\cdots 25}{50\cdots 52}$, $\frac{878310307820011}{45\cdots 96}a^{15}-\frac{22\cdots 97}{90\cdots 92}a^{14}-\frac{92\cdots 78}{56\cdots 37}a^{13}+\frac{95\cdots 85}{45\cdots 96}a^{12}+\frac{11\cdots 55}{22\cdots 48}a^{11}-\frac{29\cdots 71}{45\cdots 96}a^{10}-\frac{85\cdots 47}{11\cdots 74}a^{9}+\frac{21\cdots 79}{22\cdots 48}a^{8}+\frac{73\cdots 99}{13\cdots 44}a^{7}-\frac{18\cdots 85}{26\cdots 88}a^{6}-\frac{65\cdots 90}{33\cdots 61}a^{5}+\frac{32\cdots 09}{13\cdots 44}a^{4}+\frac{33\cdots 77}{11\cdots 74}a^{3}-\frac{80\cdots 37}{22\cdots 48}a^{2}-\frac{12\cdots 83}{12\cdots 13}a+\frac{28\cdots 95}{25\cdots 26}$ 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:  \( 10926714830.0 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 10926714830.0 \cdot 1}{2\cdot\sqrt{414692653834021167975731560448}}\cr\approx \mathstrut & 0.556002024302 \end{aligned}\] (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 - 88*x^14 + 2928*x^12 - 47936*x^10 + 418966*x^8 - 1996208*x^6 + 4988280*x^4 - 5617888*x^2 + 1612808) 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 - 88*x^14 + 2928*x^12 - 47936*x^10 + 418966*x^8 - 1996208*x^6 + 4988280*x^4 - 5617888*x^2 + 1612808, 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 - 88*x^14 + 2928*x^12 - 47936*x^10 + 418966*x^8 - 1996208*x^6 + 4988280*x^4 - 5617888*x^2 + 1612808); 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 - 88*x^14 + 2928*x^12 - 47936*x^10 + 418966*x^8 - 1996208*x^6 + 4988280*x^4 - 5617888*x^2 + 1612808); 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

$C_2\wr \OD_{16}$ (as 16T1584):

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 4096
The 73 conjugacy class representatives for $C_2\wr \OD_{16}$
Character table for $C_2\wr \OD_{16}$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.7923040256.1

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 $16$ ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ $16$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ $16$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ $16$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ $16$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}$

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 2.1.8.31a1.171$x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 2$$8$$1$$31$$(C_8:C_2):C_2$$$[2, 3, \frac{7}{2}, 4, 5]$$
2.1.8.28b1.43$x^{8} + 8 x^{7} + 8 x^{5} + 4 x^{4} + 8 x^{2} + 2$$8$$1$$28$$C_2^3: C_4$$$[2, 3, \frac{7}{2}, 4, \frac{17}{4}]$$
\(449\) Copy content Toggle raw display $\Q_{449}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{449}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{449}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{449}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{449}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{449}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{449}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{449}$$x$$1$$1$$0$Trivial$$[\ ]$$
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$$$[\ ]_{2}$$
\(1889\) Copy content Toggle raw display $\Q_{1889}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1889}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1889}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1889}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1889}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1889}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1889}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1889}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1889}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1889}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1889}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{1889}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $4$$2$$2$$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)