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Properties

Label 16.0.143...481.6
Degree $16$
Signature $[0, 8]$
Discriminant $1.439\times 10^{38}$
Root discriminant \(242.59\)
Ramified primes $17,89$
Class number $112$ (GRH)
Class group [4, 28] (GRH)
Galois group $C_2^3.D_4$ (as 16T153)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 49*x^14 + 160*x^13 + 7474*x^12 - 11638*x^11 - 276102*x^10 - 188070*x^9 + 2944004*x^8 + 8159304*x^7 - 157760522*x^6 + 561923534*x^5 + 2309963333*x^4 - 13952938768*x^3 + 80015826933*x^2 - 89895915272*x + 34103781328)
 
Copy content gp:K = bnfinit(y^16 - 2*y^15 - 49*y^14 + 160*y^13 + 7474*y^12 - 11638*y^11 - 276102*y^10 - 188070*y^9 + 2944004*y^8 + 8159304*y^7 - 157760522*y^6 + 561923534*y^5 + 2309963333*y^4 - 13952938768*y^3 + 80015826933*y^2 - 89895915272*y + 34103781328, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 49*x^14 + 160*x^13 + 7474*x^12 - 11638*x^11 - 276102*x^10 - 188070*x^9 + 2944004*x^8 + 8159304*x^7 - 157760522*x^6 + 561923534*x^5 + 2309963333*x^4 - 13952938768*x^3 + 80015826933*x^2 - 89895915272*x + 34103781328);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 - 49*x^14 + 160*x^13 + 7474*x^12 - 11638*x^11 - 276102*x^10 - 188070*x^9 + 2944004*x^8 + 8159304*x^7 - 157760522*x^6 + 561923534*x^5 + 2309963333*x^4 - 13952938768*x^3 + 80015826933*x^2 - 89895915272*x + 34103781328)
 

\( x^{16} - 2 x^{15} - 49 x^{14} + 160 x^{13} + 7474 x^{12} - 11638 x^{11} - 276102 x^{10} + \cdots + 34103781328 \) 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:  $[0, 8]$
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:   \(143902101499474565788603458614754288481\) \(\medspace = 17^{12}\cdot 89^{12}\) 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:  \(242.59\)
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:  $17^{3/4}89^{3/4}\approx 242.5935199570227$
Ramified primes:   \(17\), \(89\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_4$
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.
Maximal CM subfield:  8.0.1514445171352129.1

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{7}-\frac{1}{4}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{3}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{7}-\frac{1}{8}a^{4}-\frac{1}{2}a^{3}+\frac{1}{8}a-\frac{1}{2}$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{10}-\frac{1}{16}a^{8}+\frac{1}{16}a^{7}-\frac{1}{4}a^{6}+\frac{3}{16}a^{5}-\frac{3}{16}a^{4}-\frac{3}{16}a^{2}+\frac{3}{16}a+\frac{1}{4}$, $\frac{1}{16}a^{12}-\frac{1}{16}a^{10}-\frac{1}{16}a^{9}+\frac{1}{16}a^{7}-\frac{1}{16}a^{6}-\frac{3}{16}a^{4}-\frac{3}{16}a^{3}+\frac{3}{16}a+\frac{1}{4}$, $\frac{1}{32}a^{13}-\frac{1}{8}a^{9}+\frac{1}{16}a^{7}+\frac{1}{8}a^{6}-\frac{3}{8}a^{3}-\frac{3}{32}a+\frac{3}{8}$, $\frac{1}{160}a^{14}+\frac{1}{160}a^{13}-\frac{1}{40}a^{11}-\frac{1}{20}a^{10}+\frac{1}{40}a^{9}-\frac{9}{80}a^{8}+\frac{1}{16}a^{7}+\frac{9}{40}a^{6}-\frac{7}{40}a^{5}-\frac{1}{4}a^{4}-\frac{1}{40}a^{3}-\frac{79}{160}a^{2}-\frac{59}{160}a-\frac{17}{40}$, $\frac{1}{21\cdots 20}a^{15}-\frac{15\cdots 19}{21\cdots 20}a^{14}-\frac{33\cdots 79}{21\cdots 72}a^{13}-\frac{69\cdots 97}{10\cdots 60}a^{12}+\frac{65\cdots 73}{54\cdots 80}a^{11}-\frac{59\cdots 13}{10\cdots 60}a^{10}-\frac{52\cdots 87}{54\cdots 80}a^{9}+\frac{22\cdots 75}{21\cdots 72}a^{8}+\frac{11\cdots 23}{10\cdots 60}a^{7}+\frac{37\cdots 81}{10\cdots 60}a^{6}+\frac{25\cdots 03}{10\cdots 36}a^{5}+\frac{24\cdots 53}{10\cdots 60}a^{4}+\frac{60\cdots 11}{21\cdots 20}a^{3}-\frac{76\cdots 79}{21\cdots 20}a^{2}+\frac{58\cdots 27}{13\cdots 20}a-\frac{68\cdots 77}{27\cdots 84}$ 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:  $C_{4}\times C_{28}$, which has order $112$ (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:  $C_{4}\times C_{28}$, which has order $112$ (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:  $7$
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:   $\frac{25\cdots 13}{58\cdots 44}a^{15}+\frac{38\cdots 41}{58\cdots 44}a^{14}-\frac{51\cdots 83}{14\cdots 36}a^{13}-\frac{20\cdots 63}{73\cdots 18}a^{12}+\frac{10\cdots 04}{36\cdots 09}a^{11}+\frac{73\cdots 19}{14\cdots 36}a^{10}+\frac{14\cdots 33}{29\cdots 72}a^{9}-\frac{50\cdots 51}{29\cdots 72}a^{8}-\frac{14\cdots 03}{14\cdots 36}a^{7}-\frac{14\cdots 29}{73\cdots 18}a^{6}+\frac{19\cdots 60}{36\cdots 09}a^{5}+\frac{27\cdots 57}{14\cdots 36}a^{4}+\frac{83\cdots 65}{58\cdots 44}a^{3}+\frac{52\cdots 65}{58\cdots 44}a^{2}-\frac{98\cdots 71}{73\cdots 18}a+\frac{30\cdots 52}{36\cdots 09}$, $\frac{72\cdots 97}{27\cdots 20}a^{15}+\frac{43\cdots 33}{55\cdots 44}a^{14}-\frac{21\cdots 41}{13\cdots 60}a^{13}-\frac{26\cdots 09}{13\cdots 60}a^{12}+\frac{14\cdots 93}{68\cdots 80}a^{11}+\frac{39\cdots 67}{13\cdots 60}a^{10}-\frac{63\cdots 21}{68\cdots 80}a^{9}-\frac{63\cdots 77}{13\cdots 60}a^{8}+\frac{10\cdots 51}{13\cdots 60}a^{7}+\frac{17\cdots 81}{13\cdots 60}a^{6}-\frac{33\cdots 61}{68\cdots 80}a^{5}+\frac{87\cdots 61}{13\cdots 60}a^{4}+\frac{32\cdots 59}{55\cdots 44}a^{3}-\frac{19\cdots 71}{55\cdots 44}a^{2}+\frac{20\cdots 13}{68\cdots 80}a-\frac{68\cdots 97}{86\cdots 60}$, $\frac{52\cdots 43}{43\cdots 80}a^{15}+\frac{55\cdots 81}{43\cdots 80}a^{14}+\frac{39\cdots 19}{21\cdots 40}a^{13}-\frac{12\cdots 71}{21\cdots 40}a^{12}+\frac{94\cdots 41}{10\cdots 20}a^{11}+\frac{20\cdots 29}{21\cdots 40}a^{10}+\frac{23\cdots 57}{10\cdots 20}a^{9}-\frac{65\cdots 97}{21\cdots 40}a^{8}-\frac{41\cdots 11}{21\cdots 40}a^{7}-\frac{84\cdots 73}{21\cdots 40}a^{6}+\frac{28\cdots 39}{10\cdots 20}a^{5}+\frac{10\cdots 99}{21\cdots 40}a^{4}+\frac{15\cdots 21}{43\cdots 80}a^{3}+\frac{11\cdots 41}{43\cdots 80}a^{2}-\frac{20\cdots 47}{53\cdots 60}a+\frac{12\cdots 01}{26\cdots 80}$, $\frac{36\cdots 69}{21\cdots 20}a^{15}-\frac{40\cdots 31}{21\cdots 20}a^{14}-\frac{18\cdots 47}{21\cdots 72}a^{13}+\frac{21\cdots 87}{10\cdots 60}a^{12}+\frac{69\cdots 57}{54\cdots 80}a^{11}-\frac{88\cdots 97}{10\cdots 60}a^{10}-\frac{25\cdots 03}{54\cdots 80}a^{9}-\frac{16\cdots 41}{21\cdots 72}a^{8}+\frac{47\cdots 07}{10\cdots 60}a^{7}+\frac{19\cdots 89}{10\cdots 60}a^{6}-\frac{27\cdots 53}{10\cdots 36}a^{5}+\frac{78\cdots 17}{10\cdots 60}a^{4}+\frac{99\cdots 79}{21\cdots 20}a^{3}-\frac{42\cdots 71}{21\cdots 20}a^{2}+\frac{19\cdots 56}{17\cdots 65}a-\frac{11\cdots 09}{27\cdots 84}$, $\frac{20\cdots 71}{10\cdots 36}a^{15}-\frac{25\cdots 93}{54\cdots 80}a^{14}-\frac{10\cdots 43}{54\cdots 80}a^{13}+\frac{23\cdots 45}{27\cdots 84}a^{12}+\frac{51\cdots 51}{27\cdots 40}a^{11}-\frac{14\cdots 43}{27\cdots 40}a^{10}-\frac{28\cdots 01}{27\cdots 40}a^{9}+\frac{13\cdots 19}{34\cdots 30}a^{8}+\frac{44\cdots 24}{34\cdots 73}a^{7}-\frac{11\cdots 61}{68\cdots 60}a^{6}+\frac{13\cdots 37}{27\cdots 40}a^{5}+\frac{49\cdots 27}{54\cdots 68}a^{4}-\frac{49\cdots 93}{54\cdots 80}a^{3}+\frac{22\cdots 97}{54\cdots 80}a^{2}-\frac{24\cdots 53}{54\cdots 80}a+\frac{22\cdots 21}{13\cdots 20}$, $\frac{13\cdots 71}{54\cdots 80}a^{15}+\frac{24\cdots 47}{17\cdots 65}a^{14}-\frac{22\cdots 37}{54\cdots 80}a^{13}-\frac{17\cdots 62}{17\cdots 65}a^{12}+\frac{49\cdots 23}{27\cdots 84}a^{11}+\frac{31\cdots 03}{27\cdots 84}a^{10}-\frac{58\cdots 63}{27\cdots 40}a^{9}-\frac{27\cdots 21}{13\cdots 20}a^{8}-\frac{20\cdots 87}{27\cdots 40}a^{7}-\frac{29\cdots 93}{27\cdots 84}a^{6}-\frac{53\cdots 11}{13\cdots 20}a^{5}-\frac{21\cdots 31}{13\cdots 20}a^{4}-\frac{91\cdots 61}{54\cdots 80}a^{3}-\frac{39\cdots 79}{13\cdots 20}a^{2}-\frac{23\cdots 53}{10\cdots 36}a-\frac{30\cdots 31}{13\cdots 20}$, $\frac{21\cdots 61}{21\cdots 20}a^{15}-\frac{36\cdots 47}{21\cdots 20}a^{14}-\frac{71\cdots 79}{10\cdots 60}a^{13}+\frac{90\cdots 23}{10\cdots 60}a^{12}+\frac{45\cdots 61}{54\cdots 80}a^{11}-\frac{67\cdots 81}{10\cdots 60}a^{10}-\frac{43\cdots 99}{10\cdots 36}a^{9}-\frac{90\cdots 93}{10\cdots 60}a^{8}+\frac{74\cdots 83}{10\cdots 60}a^{7}+\frac{49\cdots 97}{10\cdots 60}a^{6}-\frac{46\cdots 69}{54\cdots 80}a^{5}+\frac{48\cdots 93}{10\cdots 60}a^{4}+\frac{29\cdots 43}{21\cdots 20}a^{3}-\frac{41\cdots 67}{21\cdots 20}a^{2}+\frac{62\cdots 33}{27\cdots 40}a-\frac{12\cdots 31}{13\cdots 20}$ 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:  \( 207783740805 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 207783740805 \cdot 112}{2\cdot\sqrt{143902101499474565788603458614754288481}}\cr\approx \mathstrut & 2.35616101830918 \end{aligned}\] (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 - 2*x^15 - 49*x^14 + 160*x^13 + 7474*x^12 - 11638*x^11 - 276102*x^10 - 188070*x^9 + 2944004*x^8 + 8159304*x^7 - 157760522*x^6 + 561923534*x^5 + 2309963333*x^4 - 13952938768*x^3 + 80015826933*x^2 - 89895915272*x + 34103781328) 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 - 2*x^15 - 49*x^14 + 160*x^13 + 7474*x^12 - 11638*x^11 - 276102*x^10 - 188070*x^9 + 2944004*x^8 + 8159304*x^7 - 157760522*x^6 + 561923534*x^5 + 2309963333*x^4 - 13952938768*x^3 + 80015826933*x^2 - 89895915272*x + 34103781328, 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 - 2*x^15 - 49*x^14 + 160*x^13 + 7474*x^12 - 11638*x^11 - 276102*x^10 - 188070*x^9 + 2944004*x^8 + 8159304*x^7 - 157760522*x^6 + 561923534*x^5 + 2309963333*x^4 - 13952938768*x^3 + 80015826933*x^2 - 89895915272*x + 34103781328); 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^16 - 2*x^15 - 49*x^14 + 160*x^13 + 7474*x^12 - 11638*x^11 - 276102*x^10 - 188070*x^9 + 2944004*x^8 + 8159304*x^7 - 157760522*x^6 + 561923534*x^5 + 2309963333*x^4 - 13952938768*x^3 + 80015826933*x^2 - 89895915272*x + 34103781328); 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

$C_2^3.D_4$ (as 16T153):

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 solvable group of order 64
The 13 conjugacy class representatives for $C_2^3.D_4$
Character table for $C_2^3.D_4$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.38915873.1, 4.0.437257.1, 4.0.25721.1, 8.0.1514445171352129.1

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 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.8.143902101499474565788603458614754288481.3

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.2.0.1}{2} }^{8}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ R ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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
\(17\) Copy content Toggle raw display 17.1.4.3a1.1$x^{4} + 17$$4$$1$$3$$C_4$$$[\ ]_{4}$$
17.1.4.3a1.1$x^{4} + 17$$4$$1$$3$$C_4$$$[\ ]_{4}$$
17.1.4.3a1.1$x^{4} + 17$$4$$1$$3$$C_4$$$[\ ]_{4}$$
17.1.4.3a1.1$x^{4} + 17$$4$$1$$3$$C_4$$$[\ ]_{4}$$
\(89\) Copy content Toggle raw display 89.2.4.6a1.2$x^{8} + 328 x^{7} + 40356 x^{6} + 2208424 x^{5} + 45454294 x^{4} + 6625272 x^{3} + 363204 x^{2} + 8856 x + 170$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
89.2.4.6a1.2$x^{8} + 328 x^{7} + 40356 x^{6} + 2208424 x^{5} + 45454294 x^{4} + 6625272 x^{3} + 363204 x^{2} + 8856 x + 170$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$

Spectrum of ring of integers

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