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Properties

Label 16.0.159...081.31
Degree $16$
Signature $[0, 8]$
Discriminant $1.599\times 10^{36}$
Root discriminant \(183.12\)
Ramified primes $37,149$
Class number $96$ (GRH)
Class group [2, 2, 24] (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 - x^15 + 47*x^14 - 527*x^13 + 680*x^12 - 22632*x^11 + 160488*x^10 - 449724*x^9 + 4759005*x^8 - 26059486*x^7 + 41859250*x^6 + 360175*x^5 + 197316058*x^4 - 1248920470*x^3 + 2495253754*x^2 - 2126532772*x + 669593281)
 
Copy content gp:K = bnfinit(y^16 - y^15 + 47*y^14 - 527*y^13 + 680*y^12 - 22632*y^11 + 160488*y^10 - 449724*y^9 + 4759005*y^8 - 26059486*y^7 + 41859250*y^6 + 360175*y^5 + 197316058*y^4 - 1248920470*y^3 + 2495253754*y^2 - 2126532772*y + 669593281, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 + 47*x^14 - 527*x^13 + 680*x^12 - 22632*x^11 + 160488*x^10 - 449724*x^9 + 4759005*x^8 - 26059486*x^7 + 41859250*x^6 + 360175*x^5 + 197316058*x^4 - 1248920470*x^3 + 2495253754*x^2 - 2126532772*x + 669593281);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - x^15 + 47*x^14 - 527*x^13 + 680*x^12 - 22632*x^11 + 160488*x^10 - 449724*x^9 + 4759005*x^8 - 26059486*x^7 + 41859250*x^6 + 360175*x^5 + 197316058*x^4 - 1248920470*x^3 + 2495253754*x^2 - 2126532772*x + 669593281)
 

\( x^{16} - x^{15} + 47 x^{14} - 527 x^{13} + 680 x^{12} - 22632 x^{11} + 160488 x^{10} - 449724 x^{9} + \cdots + 669593281 \) 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:   \(1599229661124692101046354619853228081\) \(\medspace = 37^{12}\cdot 149^{8}\) 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:  \(183.12\)
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:  $37^{3/4}149^{3/4}\approx 639.7952248237154$
Ramified primes:   \(37\), \(149\) 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:  8.0.56961692006209.2

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{9}a^{12}-\frac{1}{9}a^{11}-\frac{1}{9}a^{10}-\frac{1}{9}a^{9}-\frac{1}{9}a^{8}+\frac{4}{9}a^{7}+\frac{1}{9}a^{6}+\frac{1}{9}a^{5}+\frac{1}{3}a^{4}-\frac{2}{9}a^{3}-\frac{4}{9}a-\frac{4}{9}$, $\frac{1}{3339}a^{13}+\frac{118}{3339}a^{12}-\frac{10}{1113}a^{11}-\frac{38}{1113}a^{10}+\frac{39}{371}a^{9}-\frac{526}{3339}a^{8}-\frac{137}{371}a^{7}+\frac{221}{1113}a^{6}-\frac{883}{3339}a^{5}+\frac{1570}{3339}a^{4}-\frac{22}{3339}a^{3}+\frac{137}{3339}a^{2}-\frac{235}{1113}a-\frac{131}{477}$, $\frac{1}{1238769}a^{14}-\frac{169}{1238769}a^{13}+\frac{21383}{1238769}a^{12}-\frac{198151}{1238769}a^{11}+\frac{94655}{1238769}a^{10}-\frac{99779}{1238769}a^{9}-\frac{58031}{1238769}a^{8}-\frac{53195}{1238769}a^{7}+\frac{109783}{412923}a^{6}-\frac{597938}{1238769}a^{5}+\frac{19309}{137641}a^{4}+\frac{10532}{1238769}a^{3}-\frac{271528}{1238769}a^{2}-\frac{4906}{19663}a+\frac{3292}{8427}$, $\frac{1}{93\cdots 39}a^{15}-\frac{28\cdots 97}{31\cdots 13}a^{14}+\frac{13\cdots 26}{93\cdots 39}a^{13}-\frac{35\cdots 30}{93\cdots 39}a^{12}-\frac{13\cdots 24}{93\cdots 39}a^{11}-\frac{47\cdots 12}{10\cdots 71}a^{10}-\frac{21\cdots 22}{93\cdots 39}a^{9}-\frac{11\cdots 76}{93\cdots 39}a^{8}+\frac{39\cdots 59}{93\cdots 39}a^{7}+\frac{43\cdots 24}{93\cdots 39}a^{6}+\frac{45\cdots 01}{31\cdots 13}a^{5}-\frac{39\cdots 50}{31\cdots 13}a^{4}+\frac{15\cdots 18}{31\cdots 13}a^{3}-\frac{37\cdots 08}{13\cdots 77}a^{2}-\frac{15\cdots 13}{63\cdots 37}a-\frac{39\cdots 96}{27\cdots 73}$ 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:  $C_{2}\times C_{2}\times C_{24}$, which has order $96$ (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:  $C_{2}\times C_{2}\times C_{24}$, which has order $96$ (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:   $\frac{44\cdots 42}{55\cdots 83}a^{15}+\frac{23\cdots 81}{61\cdots 87}a^{14}+\frac{21\cdots 64}{55\cdots 83}a^{13}-\frac{28\cdots 10}{78\cdots 69}a^{12}-\frac{55\cdots 52}{55\cdots 83}a^{11}-\frac{33\cdots 94}{18\cdots 61}a^{10}+\frac{80\cdots 76}{78\cdots 69}a^{9}-\frac{11\cdots 63}{55\cdots 83}a^{8}+\frac{19\cdots 18}{55\cdots 83}a^{7}-\frac{86\cdots 54}{55\cdots 83}a^{6}+\frac{59\cdots 39}{61\cdots 87}a^{5}+\frac{10\cdots 35}{61\cdots 87}a^{4}+\frac{34\cdots 08}{18\cdots 61}a^{3}-\frac{40\cdots 85}{55\cdots 83}a^{2}+\frac{74\cdots 31}{87\cdots 41}a-\frac{34\cdots 18}{11\cdots 67}$, $\frac{26\cdots 16}{26\cdots 97}a^{15}+\frac{42\cdots 58}{86\cdots 99}a^{14}+\frac{14\cdots 74}{28\cdots 33}a^{13}-\frac{40\cdots 00}{86\cdots 99}a^{12}-\frac{29\cdots 88}{26\cdots 97}a^{11}-\frac{20\cdots 26}{86\cdots 99}a^{10}+\frac{34\cdots 32}{26\cdots 97}a^{9}-\frac{22\cdots 20}{86\cdots 99}a^{8}+\frac{11\cdots 80}{26\cdots 97}a^{7}-\frac{52\cdots 10}{26\cdots 97}a^{6}+\frac{30\cdots 28}{26\cdots 97}a^{5}+\frac{62\cdots 44}{26\cdots 97}a^{4}+\frac{61\cdots 08}{26\cdots 97}a^{3}-\frac{11\cdots 34}{12\cdots 57}a^{2}+\frac{19\cdots 36}{17\cdots 51}a-\frac{22\cdots 39}{75\cdots 79}$, $\frac{36\cdots 05}{38\cdots 81}a^{15}+\frac{16\cdots 46}{38\cdots 81}a^{14}+\frac{17\cdots 52}{38\cdots 81}a^{13}-\frac{16\cdots 32}{38\cdots 81}a^{12}-\frac{16\cdots 90}{42\cdots 09}a^{11}-\frac{82\cdots 19}{38\cdots 81}a^{10}+\frac{46\cdots 99}{38\cdots 81}a^{9}-\frac{93\cdots 87}{38\cdots 81}a^{8}+\frac{15\cdots 58}{38\cdots 81}a^{7}-\frac{23\cdots 10}{12\cdots 27}a^{6}+\frac{43\cdots 10}{38\cdots 81}a^{5}+\frac{80\cdots 95}{38\cdots 81}a^{4}+\frac{84\cdots 61}{38\cdots 81}a^{3}-\frac{15\cdots 49}{18\cdots 61}a^{2}+\frac{78\cdots 87}{78\cdots 69}a-\frac{45\cdots 86}{11\cdots 67}$, $\frac{11\cdots 23}{93\cdots 39}a^{15}+\frac{18\cdots 22}{93\cdots 39}a^{14}+\frac{58\cdots 01}{93\cdots 39}a^{13}-\frac{45\cdots 26}{93\cdots 39}a^{12}-\frac{13\cdots 63}{31\cdots 13}a^{11}-\frac{27\cdots 64}{93\cdots 39}a^{10}+\frac{11\cdots 56}{93\cdots 39}a^{9}-\frac{21\cdots 37}{93\cdots 39}a^{8}+\frac{49\cdots 94}{93\cdots 39}a^{7}-\frac{56\cdots 93}{31\cdots 13}a^{6}+\frac{30\cdots 23}{93\cdots 39}a^{5}+\frac{83\cdots 99}{93\cdots 39}a^{4}+\frac{25\cdots 76}{93\cdots 39}a^{3}-\frac{36\cdots 53}{44\cdots 59}a^{2}+\frac{16\cdots 57}{19\cdots 11}a-\frac{84\cdots 91}{27\cdots 73}$, $\frac{28\cdots 18}{31\cdots 13}a^{15}-\frac{26\cdots 42}{93\cdots 39}a^{14}+\frac{38\cdots 12}{93\cdots 39}a^{13}-\frac{43\cdots 69}{93\cdots 39}a^{12}+\frac{16\cdots 87}{93\cdots 39}a^{11}-\frac{18\cdots 83}{93\cdots 39}a^{10}+\frac{12\cdots 23}{93\cdots 39}a^{9}-\frac{24\cdots 73}{93\cdots 39}a^{8}+\frac{37\cdots 81}{93\cdots 39}a^{7}-\frac{66\cdots 42}{31\cdots 13}a^{6}+\frac{13\cdots 29}{93\cdots 39}a^{5}+\frac{11\cdots 51}{31\cdots 13}a^{4}+\frac{23\cdots 27}{93\cdots 39}a^{3}-\frac{13\cdots 48}{13\cdots 77}a^{2}+\frac{10\cdots 33}{90\cdots 91}a-\frac{39\cdots 12}{90\cdots 91}$, $\frac{85\cdots 78}{93\cdots 39}a^{15}+\frac{20\cdots 74}{93\cdots 39}a^{14}+\frac{45\cdots 14}{93\cdots 39}a^{13}-\frac{98\cdots 95}{31\cdots 13}a^{12}-\frac{58\cdots 86}{10\cdots 71}a^{11}-\frac{20\cdots 76}{93\cdots 39}a^{10}+\frac{66\cdots 11}{93\cdots 39}a^{9}-\frac{12\cdots 44}{10\cdots 71}a^{8}+\frac{35\cdots 31}{93\cdots 39}a^{7}-\frac{32\cdots 68}{31\cdots 13}a^{6}-\frac{14\cdots 77}{31\cdots 13}a^{5}+\frac{72\cdots 84}{93\cdots 39}a^{4}+\frac{18\cdots 73}{93\cdots 39}a^{3}-\frac{61\cdots 85}{13\cdots 77}a^{2}+\frac{89\cdots 86}{27\cdots 73}a-\frac{54\cdots 94}{90\cdots 91}$, $\frac{38\cdots 69}{93\cdots 39}a^{15}+\frac{59\cdots 38}{93\cdots 39}a^{14}+\frac{65\cdots 74}{31\cdots 13}a^{13}-\frac{15\cdots 76}{93\cdots 39}a^{12}-\frac{12\cdots 68}{93\cdots 39}a^{11}-\frac{90\cdots 01}{93\cdots 39}a^{10}+\frac{38\cdots 24}{93\cdots 39}a^{9}-\frac{74\cdots 00}{93\cdots 39}a^{8}+\frac{54\cdots 71}{31\cdots 13}a^{7}-\frac{58\cdots 78}{93\cdots 39}a^{6}+\frac{38\cdots 91}{31\cdots 13}a^{5}+\frac{30\cdots 92}{93\cdots 39}a^{4}+\frac{84\cdots 19}{93\cdots 39}a^{3}-\frac{37\cdots 53}{13\cdots 77}a^{2}+\frac{57\cdots 68}{19\cdots 11}a-\frac{29\cdots 97}{27\cdots 73}$ 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:  \( 2477297760.12 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2477297760.12 \cdot 96}{2\cdot\sqrt{1599229661124692101046354619853228081}}\cr\approx \mathstrut & 0.228403616502 \end{aligned}\] (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 - x^15 + 47*x^14 - 527*x^13 + 680*x^12 - 22632*x^11 + 160488*x^10 - 449724*x^9 + 4759005*x^8 - 26059486*x^7 + 41859250*x^6 + 360175*x^5 + 197316058*x^4 - 1248920470*x^3 + 2495253754*x^2 - 2126532772*x + 669593281) 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 - x^15 + 47*x^14 - 527*x^13 + 680*x^12 - 22632*x^11 + 160488*x^10 - 449724*x^9 + 4759005*x^8 - 26059486*x^7 + 41859250*x^6 + 360175*x^5 + 197316058*x^4 - 1248920470*x^3 + 2495253754*x^2 - 2126532772*x + 669593281, 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 - x^15 + 47*x^14 - 527*x^13 + 680*x^12 - 22632*x^11 + 160488*x^10 - 449724*x^9 + 4759005*x^8 - 26059486*x^7 + 41859250*x^6 + 360175*x^5 + 197316058*x^4 - 1248920470*x^3 + 2495253754*x^2 - 2126532772*x + 669593281); 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 - x^15 + 47*x^14 - 527*x^13 + 680*x^12 - 22632*x^11 + 160488*x^10 - 449724*x^9 + 4759005*x^8 - 26059486*x^7 + 41859250*x^6 + 360175*x^5 + 197316058*x^4 - 1248920470*x^3 + 2495253754*x^2 - 2126532772*x + 669593281); 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

\(\Q(\sqrt{37}) \), 4.4.203981.1, 4.0.50653.1, 4.0.7547297.1, 8.0.56961692006209.2

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: 16.0.35504497706629289335330118915361516626281.45

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ ${\href{/padicField/3.4.0.1}{4} }^{4}$ ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.2.0.1}{2} }^{8}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ R ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.1.0.1}{1} }^{16}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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
\(37\) Copy content Toggle raw display 37.2.4.6a1.2$x^{8} + 132 x^{7} + 6542 x^{6} + 144540 x^{5} + 1212081 x^{4} + 289080 x^{3} + 26168 x^{2} + 1056 x + 53$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
37.2.4.6a1.2$x^{8} + 132 x^{7} + 6542 x^{6} + 144540 x^{5} + 1212081 x^{4} + 289080 x^{3} + 26168 x^{2} + 1056 x + 53$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
\(149\) Copy content Toggle raw display $\Q_{149}$$x + 147$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{149}$$x + 147$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{149}$$x + 147$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{149}$$x + 147$$1$$1$$0$Trivial$$[\ ]$$
149.1.2.1a1.1$x^{2} + 149$$2$$1$$1$$C_2$$$[\ ]_{2}$$
149.1.2.1a1.1$x^{2} + 149$$2$$1$$1$$C_2$$$[\ ]_{2}$$
149.1.4.3a1.1$x^{4} + 149$$4$$1$$3$$C_4$$$[\ ]_{4}$$
149.1.4.3a1.1$x^{4} + 149$$4$$1$$3$$C_4$$$[\ ]_{4}$$

Spectrum of ring of integers

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