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Properties

Label 16.0.484...304.261
Degree $16$
Signature $[0, 8]$
Discriminant $4.841\times 10^{23}$
Root discriminant \(30.22\)
Ramified primes $2,3$
Class number $18$
Class group [3, 6]
Galois group $C_2^2 : C_8$ (as 16T24)

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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 - 8*x^14 - 16*x^13 + 16*x^12 + 96*x^11 + 344*x^10 + 784*x^9 + 1758*x^8 + 2512*x^7 + 4256*x^6 + 4032*x^5 + 5296*x^4 + 3104*x^3 + 3280*x^2 + 912*x + 799)
 
Copy content gp:K = bnfinit(y^16 - 8*y^14 - 16*y^13 + 16*y^12 + 96*y^11 + 344*y^10 + 784*y^9 + 1758*y^8 + 2512*y^7 + 4256*y^6 + 4032*y^5 + 5296*y^4 + 3104*y^3 + 3280*y^2 + 912*y + 799, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^14 - 16*x^13 + 16*x^12 + 96*x^11 + 344*x^10 + 784*x^9 + 1758*x^8 + 2512*x^7 + 4256*x^6 + 4032*x^5 + 5296*x^4 + 3104*x^3 + 3280*x^2 + 912*x + 799);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 8*x^14 - 16*x^13 + 16*x^12 + 96*x^11 + 344*x^10 + 784*x^9 + 1758*x^8 + 2512*x^7 + 4256*x^6 + 4032*x^5 + 5296*x^4 + 3104*x^3 + 3280*x^2 + 912*x + 799)
 

\( x^{16} - 8 x^{14} - 16 x^{13} + 16 x^{12} + 96 x^{11} + 344 x^{10} + 784 x^{9} + 1758 x^{8} + 2512 x^{7} + \cdots + 799 \) 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:   \(484116351470433472610304\) \(\medspace = 2^{66}\cdot 3^{8}\) 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:  \(30.22\)
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:  $2^{67/16}3^{1/2}\approx 31.559036391689393$
Ramified primes:   \(2\), \(3\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2\times C_4$
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:  8.0.173946175488.1

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{17}a^{12}-\frac{5}{17}a^{11}+\frac{7}{17}a^{10}-\frac{6}{17}a^{9}+\frac{5}{17}a^{8}+\frac{8}{17}a^{7}+\frac{6}{17}a^{6}-\frac{4}{17}a^{5}-\frac{2}{17}a^{4}-\frac{3}{17}a^{3}-\frac{7}{17}a$, $\frac{1}{17}a^{13}-\frac{1}{17}a^{11}-\frac{5}{17}a^{10}-\frac{8}{17}a^{9}-\frac{1}{17}a^{8}-\frac{5}{17}a^{7}-\frac{8}{17}a^{6}-\frac{5}{17}a^{5}+\frac{4}{17}a^{4}+\frac{2}{17}a^{3}-\frac{7}{17}a^{2}-\frac{1}{17}a$, $\frac{1}{17}a^{14}+\frac{7}{17}a^{11}-\frac{1}{17}a^{10}-\frac{7}{17}a^{9}+\frac{1}{17}a^{6}+\frac{7}{17}a^{3}-\frac{1}{17}a^{2}-\frac{7}{17}a$, $\frac{1}{29\cdots 01}a^{15}-\frac{76\cdots 47}{29\cdots 01}a^{14}+\frac{44\cdots 10}{17\cdots 53}a^{13}-\frac{51\cdots 93}{29\cdots 01}a^{12}+\frac{10\cdots 92}{29\cdots 01}a^{11}-\frac{13\cdots 01}{29\cdots 01}a^{10}+\frac{70\cdots 48}{29\cdots 01}a^{9}-\frac{956251965024231}{30\cdots 33}a^{8}+\frac{13\cdots 50}{29\cdots 01}a^{7}+\frac{46\cdots 06}{29\cdots 01}a^{6}+\frac{11\cdots 68}{30\cdots 33}a^{5}-\frac{22\cdots 89}{29\cdots 01}a^{4}-\frac{30\cdots 71}{29\cdots 01}a^{3}+\frac{68\cdots 58}{29\cdots 01}a^{2}-\frac{70\cdots 48}{29\cdots 01}a-\frac{63\cdots 54}{17\cdots 53}$ 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:  $C_{3}\times C_{6}$, which has order $18$
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:  $C_{3}\times C_{6}$, which has order $18$
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:   $\frac{297123372}{250881362689}a^{15}+\frac{15026297298}{250881362689}a^{14}+\frac{48675980}{14757727217}a^{13}-\frac{128346963756}{250881362689}a^{12}-\frac{229702264904}{250881362689}a^{11}+\frac{350323112166}{250881362689}a^{10}+\frac{1493379623328}{250881362689}a^{9}+\frac{4438412362879}{250881362689}a^{8}+\frac{9403625666428}{250881362689}a^{7}+\frac{20630638582682}{250881362689}a^{6}+\frac{25764541628796}{250881362689}a^{5}+\frac{41541192076864}{250881362689}a^{4}+\frac{31454152325096}{250881362689}a^{3}+\frac{38395166567974}{250881362689}a^{2}+\frac{13488245869800}{250881362689}a+\frac{797829850722}{14757727217}$, $\frac{515757468824918}{30\cdots 33}a^{15}-\frac{23\cdots 60}{30\cdots 33}a^{14}-\frac{252116926048548}{17\cdots 49}a^{13}+\frac{12\cdots 09}{30\cdots 33}a^{12}+\frac{46\cdots 42}{30\cdots 33}a^{11}-\frac{12\cdots 33}{30\cdots 33}a^{10}-\frac{78\cdots 76}{30\cdots 33}a^{9}-\frac{37\cdots 76}{30\cdots 33}a^{8}-\frac{79\cdots 88}{30\cdots 33}a^{7}-\frac{23\cdots 48}{30\cdots 33}a^{6}-\frac{27\cdots 94}{30\cdots 33}a^{5}-\frac{55\cdots 68}{30\cdots 33}a^{4}-\frac{38\cdots 28}{30\cdots 33}a^{3}-\frac{58\cdots 55}{30\cdots 33}a^{2}-\frac{18\cdots 54}{30\cdots 33}a-\frac{13\cdots 54}{17\cdots 49}$, $\frac{480073843217834}{30\cdots 33}a^{15}-\frac{541185905579154}{30\cdots 33}a^{14}-\frac{14486534522264}{104256397975297}a^{13}-\frac{30\cdots 23}{30\cdots 33}a^{12}+\frac{19\cdots 54}{30\cdots 33}a^{11}+\frac{40\cdots 69}{30\cdots 33}a^{10}+\frac{10\cdots 40}{30\cdots 33}a^{9}+\frac{15\cdots 87}{30\cdots 33}a^{8}+\frac{33\cdots 28}{30\cdots 33}a^{7}+\frac{10\cdots 06}{30\cdots 33}a^{6}+\frac{37\cdots 18}{30\cdots 33}a^{5}-\frac{59\cdots 60}{30\cdots 33}a^{4}-\frac{75\cdots 16}{30\cdots 33}a^{3}-\frac{11\cdots 77}{30\cdots 33}a^{2}-\frac{20\cdots 54}{30\cdots 33}a-\frac{38\cdots 69}{17\cdots 49}$, $\frac{15\cdots 36}{29\cdots 01}a^{15}+\frac{49\cdots 49}{29\cdots 01}a^{14}-\frac{82\cdots 12}{17\cdots 53}a^{13}-\frac{29\cdots 33}{29\cdots 01}a^{12}+\frac{30\cdots 58}{29\cdots 01}a^{11}+\frac{18\cdots 54}{29\cdots 01}a^{10}+\frac{55\cdots 54}{29\cdots 01}a^{9}+\frac{12\cdots 76}{30\cdots 33}a^{8}+\frac{25\cdots 76}{29\cdots 01}a^{7}+\frac{36\cdots 98}{29\cdots 01}a^{6}+\frac{55\cdots 96}{30\cdots 33}a^{5}+\frac{48\cdots 58}{29\cdots 01}a^{4}+\frac{49\cdots 23}{29\cdots 01}a^{3}+\frac{25\cdots 15}{29\cdots 01}a^{2}+\frac{16\cdots 11}{29\cdots 01}a+\frac{20\cdots 04}{17\cdots 53}$, $\frac{24\cdots 11}{29\cdots 01}a^{15}-\frac{27\cdots 17}{29\cdots 01}a^{14}-\frac{13\cdots 15}{17\cdots 53}a^{13}-\frac{14\cdots 34}{29\cdots 01}a^{12}+\frac{11\cdots 95}{29\cdots 01}a^{11}+\frac{22\cdots 05}{29\cdots 01}a^{10}+\frac{44\cdots 85}{29\cdots 01}a^{9}+\frac{67\cdots 64}{30\cdots 33}a^{8}+\frac{14\cdots 36}{29\cdots 01}a^{7}+\frac{12\cdots 50}{29\cdots 01}a^{6}+\frac{90\cdots 66}{30\cdots 33}a^{5}-\frac{34\cdots 86}{29\cdots 01}a^{4}-\frac{13\cdots 75}{29\cdots 01}a^{3}-\frac{48\cdots 70}{29\cdots 01}a^{2}-\frac{10\cdots 11}{29\cdots 01}a-\frac{10\cdots 13}{17\cdots 53}$, $\frac{21\cdots 05}{29\cdots 01}a^{15}-\frac{37\cdots 17}{29\cdots 01}a^{14}+\frac{11\cdots 79}{17\cdots 53}a^{13}+\frac{67\cdots 60}{29\cdots 01}a^{12}+\frac{53\cdots 02}{29\cdots 01}a^{11}-\frac{33\cdots 25}{29\cdots 01}a^{10}-\frac{10\cdots 94}{29\cdots 01}a^{9}-\frac{26\cdots 17}{30\cdots 33}a^{8}-\frac{55\cdots 99}{29\cdots 01}a^{7}-\frac{92\cdots 43}{29\cdots 01}a^{6}-\frac{13\cdots 36}{30\cdots 33}a^{5}-\frac{15\cdots 90}{29\cdots 01}a^{4}-\frac{13\cdots 56}{29\cdots 01}a^{3}-\frac{11\cdots 92}{29\cdots 01}a^{2}-\frac{50\cdots 15}{29\cdots 01}a-\frac{18\cdots 77}{17\cdots 53}$, $\frac{49\cdots 85}{29\cdots 01}a^{15}+\frac{87\cdots 30}{29\cdots 01}a^{14}-\frac{27\cdots 02}{17\cdots 53}a^{13}-\frac{15\cdots 09}{29\cdots 01}a^{12}+\frac{819136597885034}{29\cdots 01}a^{11}+\frac{77\cdots 59}{29\cdots 01}a^{10}+\frac{24\cdots 43}{29\cdots 01}a^{9}+\frac{61\cdots 59}{30\cdots 33}a^{8}+\frac{12\cdots 93}{29\cdots 01}a^{7}+\frac{20\cdots 14}{29\cdots 01}a^{6}+\frac{29\cdots 85}{30\cdots 33}a^{5}+\frac{34\cdots 87}{29\cdots 01}a^{4}+\frac{29\cdots 48}{29\cdots 01}a^{3}+\frac{26\cdots 06}{29\cdots 01}a^{2}+\frac{10\cdots 76}{29\cdots 01}a+\frac{43\cdots 99}{17\cdots 53}$ Copy content Toggle raw display
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:  \( 28179.76782913505 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 28179.76782913505 \cdot 18}{2\cdot\sqrt{484116351470433472610304}}\cr\approx \mathstrut & 0.885409120296159 \end{aligned}\]

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 - 8*x^14 - 16*x^13 + 16*x^12 + 96*x^11 + 344*x^10 + 784*x^9 + 1758*x^8 + 2512*x^7 + 4256*x^6 + 4032*x^5 + 5296*x^4 + 3104*x^3 + 3280*x^2 + 912*x + 799) 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 - 8*x^14 - 16*x^13 + 16*x^12 + 96*x^11 + 344*x^10 + 784*x^9 + 1758*x^8 + 2512*x^7 + 4256*x^6 + 4032*x^5 + 5296*x^4 + 3104*x^3 + 3280*x^2 + 912*x + 799, 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 - 8*x^14 - 16*x^13 + 16*x^12 + 96*x^11 + 344*x^10 + 784*x^9 + 1758*x^8 + 2512*x^7 + 4256*x^6 + 4032*x^5 + 5296*x^4 + 3104*x^3 + 3280*x^2 + 912*x + 799); 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 - 8*x^14 - 16*x^13 + 16*x^12 + 96*x^11 + 344*x^10 + 784*x^9 + 1758*x^8 + 2512*x^7 + 4256*x^6 + 4032*x^5 + 5296*x^4 + 3104*x^3 + 3280*x^2 + 912*x + 799); 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

$C_2^2:C_8$ (as 16T24):

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 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.2.9216.1, \(\Q(\zeta_{16})^+\), 4.2.18432.3, 8.4.2147483648.1, 8.0.173946175488.1, 8.4.5435817984.1

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

Galois closure: deg 32
Degree 16 sibling: 16.8.484116351470433472610304.4
Minimal sibling: 16.8.484116351470433472610304.4

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.1.0.1}{1} }^{16}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{8}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{8}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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
\(2\) Copy content Toggle raw display 2.1.16.66h1.520$x^{16} + 16 x^{13} + 4 x^{12} + 8 x^{10} + 8 x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 34$$16$$1$$66$$C_2^2 : C_8$$$[2, 3, \frac{7}{2}, 4, 5]$$
\(3\) Copy content Toggle raw display 3.8.2.8a1.2$x^{16} + 4 x^{13} + 2 x^{12} + 8 x^{10} + 8 x^{9} + 5 x^{8} + 8 x^{7} + 12 x^{6} + 12 x^{5} + 8 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + 7$$2$$8$$8$$C_8\times C_2$$$[\ ]_{2}^{8}$$

Spectrum of ring of integers

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