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Properties

Label 16.0.397...264.2
Degree $16$
Signature $[0, 8]$
Discriminant $3.977\times 10^{19}$
Root discriminant \(16.79\)
Ramified primes $2,3,11$
Class number $1$
Class group trivial
Galois group $D_4^2:C_2^2$ (as 16T608)

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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^15 + 26*x^14 - 36*x^13 - 16*x^12 + 140*x^11 - 188*x^10 - 24*x^9 + 337*x^8 - 276*x^7 - 176*x^6 + 340*x^5 + 14*x^4 - 192*x^3 + 2*x^2 + 56*x + 13)
 
Copy content gp:K = bnfinit(y^16 - 8*y^15 + 26*y^14 - 36*y^13 - 16*y^12 + 140*y^11 - 188*y^10 - 24*y^9 + 337*y^8 - 276*y^7 - 176*y^6 + 340*y^5 + 14*y^4 - 192*y^3 + 2*y^2 + 56*y + 13, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 26*x^14 - 36*x^13 - 16*x^12 + 140*x^11 - 188*x^10 - 24*x^9 + 337*x^8 - 276*x^7 - 176*x^6 + 340*x^5 + 14*x^4 - 192*x^3 + 2*x^2 + 56*x + 13);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 8*x^15 + 26*x^14 - 36*x^13 - 16*x^12 + 140*x^11 - 188*x^10 - 24*x^9 + 337*x^8 - 276*x^7 - 176*x^6 + 340*x^5 + 14*x^4 - 192*x^3 + 2*x^2 + 56*x + 13)
 

\( x^{16} - 8 x^{15} + 26 x^{14} - 36 x^{13} - 16 x^{12} + 140 x^{11} - 188 x^{10} - 24 x^{9} + 337 x^{8} + \cdots + 13 \) 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:   \(39770658357865611264\) \(\medspace = 2^{36}\cdot 3^{14}\cdot 11^{2}\) 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:  \(16.79\)
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:  $2^{19/8}3^{7/8}11^{1/2}\approx 44.99079632827353$
Ramified primes:   \(2\), \(3\), \(11\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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:  \(\Q(\zeta_{12})\)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13}a^{14}+\frac{5}{13}a^{13}-\frac{4}{13}a^{12}-\frac{4}{13}a^{11}-\frac{6}{13}a^{8}+\frac{2}{13}a^{7}-\frac{3}{13}a^{6}+\frac{2}{13}a^{5}+\frac{5}{13}a^{4}-\frac{6}{13}a^{3}-\frac{6}{13}a^{2}+\frac{1}{13}a$, $\frac{1}{77160083}a^{15}-\frac{1771901}{77160083}a^{14}-\frac{36669433}{77160083}a^{13}-\frac{15959170}{77160083}a^{12}-\frac{23926333}{77160083}a^{11}+\frac{2790618}{5935391}a^{10}+\frac{271334}{580151}a^{9}-\frac{26687272}{77160083}a^{8}-\frac{32235127}{77160083}a^{7}+\frac{37983485}{77160083}a^{6}+\frac{15341566}{77160083}a^{5}+\frac{14415404}{77160083}a^{4}-\frac{98834}{4061057}a^{3}-\frac{38012314}{77160083}a^{2}+\frac{1478506}{7014553}a-\frac{1983294}{5935391}$ 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 $1$
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:  Trivial group, which has order $1$
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:   \( \frac{42724590}{77160083} a^{15} - \frac{354081765}{77160083} a^{14} + \frac{93458460}{5935391} a^{13} - \frac{1901260190}{77160083} a^{12} - \frac{109739710}{77160083} a^{11} + \frac{463319096}{5935391} a^{10} - \frac{74493066}{580151} a^{9} + \frac{2060298950}{77160083} a^{8} + \frac{13809832898}{77160083} a^{7} - \frac{16275327468}{77160083} a^{6} - \frac{2270264208}{77160083} a^{5} + \frac{15394328289}{77160083} a^{4} - \frac{243082118}{4061057} a^{3} - \frac{6671694319}{77160083} a^{2} + \frac{227232424}{7014553} a + \frac{118241660}{5935391} \)  (order $12$) 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:   $\frac{123599488}{77160083}a^{15}-\frac{1037735346}{77160083}a^{14}+\frac{279414059}{5935391}a^{13}-\frac{5945195670}{77160083}a^{12}+\frac{540986587}{77160083}a^{11}+\frac{1300537508}{5935391}a^{10}-\frac{226387120}{580151}a^{9}+\frac{9781854207}{77160083}a^{8}+\frac{36935121981}{77160083}a^{7}-\frac{49166393572}{77160083}a^{6}-\frac{624986639}{77160083}a^{5}+\frac{41469803026}{77160083}a^{4}-\frac{829919837}{4061057}a^{3}-\frac{16523520628}{77160083}a^{2}+\frac{657431206}{7014553}a+\frac{285303343}{5935391}$, $\frac{63503898}{77160083}a^{15}-\frac{534291696}{77160083}a^{14}+\frac{1872189828}{77160083}a^{13}-\frac{3063473275}{77160083}a^{12}+\frac{265187722}{77160083}a^{11}+\frac{673829605}{5935391}a^{10}-\frac{117183956}{580151}a^{9}+\frac{5054113048}{77160083}a^{8}+\frac{19149064708}{77160083}a^{7}-\frac{25531363785}{77160083}a^{6}-\frac{199961952}{77160083}a^{5}+\frac{21395148147}{77160083}a^{4}-\frac{435499816}{4061057}a^{3}-\frac{8318498753}{77160083}a^{2}+\frac{330710558}{7014553}a+\frac{144396349}{5935391}$, $\frac{1763024}{7014553}a^{15}-\frac{13818230}{7014553}a^{14}+\frac{44641655}{7014553}a^{13}-\frac{63955414}{7014553}a^{12}-\frac{15765052}{7014553}a^{11}+\frac{16698793}{539581}a^{10}-\frac{2378551}{52741}a^{9}+\frac{1321430}{539581}a^{8}+\frac{474743769}{7014553}a^{7}-\frac{454562573}{7014553}a^{6}-\frac{165324571}{7014553}a^{5}+\frac{35693968}{539581}a^{4}-\frac{2604788}{369187}a^{3}-\frac{220300278}{7014553}a^{2}+\frac{55765061}{7014553}a+\frac{3270954}{539581}$, $\frac{2526034}{11022869}a^{15}-\frac{21623707}{11022869}a^{14}+\frac{76901769}{11022869}a^{13}-\frac{127762307}{11022869}a^{12}+\frac{13653399}{11022869}a^{11}+\frac{27979949}{847913}a^{10}-\frac{34358770}{580151}a^{9}+\frac{205522282}{11022869}a^{8}+\frac{829573366}{11022869}a^{7}-\frac{1095499048}{11022869}a^{6}-\frac{42675434}{11022869}a^{5}+\frac{978120521}{11022869}a^{4}-\frac{19292191}{580151}a^{3}-\frac{31570817}{847913}a^{2}+\frac{15238006}{1002079}a+\frac{7230418}{847913}$, $\frac{1626106}{5935391}a^{15}-\frac{175618238}{77160083}a^{14}+\frac{609184678}{77160083}a^{13}-\frac{988407304}{77160083}a^{12}+\frac{84117162}{77160083}a^{11}+\frac{214120597}{5935391}a^{10}-\frac{2832681}{44627}a^{9}+\frac{1515129270}{77160083}a^{8}+\frac{6043815980}{77160083}a^{7}-\frac{7861518097}{77160083}a^{6}-\frac{221815795}{77160083}a^{5}+\frac{6572287655}{77160083}a^{4}-\frac{124120396}{4061057}a^{3}-\frac{2560015154}{77160083}a^{2}+\frac{97475295}{7014553}a+\frac{41793838}{5935391}$, $\frac{50744773}{77160083}a^{15}-\frac{431362424}{77160083}a^{14}+\frac{1521822317}{77160083}a^{13}-\frac{2492309119}{77160083}a^{12}+\frac{159295737}{77160083}a^{11}+\frac{563914781}{5935391}a^{10}-\frac{96926286}{580151}a^{9}+\frac{290915519}{5935391}a^{8}+\frac{16719108096}{77160083}a^{7}-\frac{21623438758}{77160083}a^{6}-\frac{1168409609}{77160083}a^{5}+\frac{1495758531}{5935391}a^{4}-\frac{369896459}{4061057}a^{3}-\frac{8262754410}{77160083}a^{2}+\frac{316777463}{7014553}a+\frac{152015315}{5935391}$, $\frac{6421631}{77160083}a^{15}-\frac{53073981}{77160083}a^{14}+\frac{184939450}{77160083}a^{13}-\frac{305913822}{77160083}a^{12}+\frac{3801395}{5935391}a^{11}+\frac{61907547}{5935391}a^{10}-\frac{11145683}{580151}a^{9}+\frac{565424954}{77160083}a^{8}+\frac{1650312679}{77160083}a^{7}-\frac{2310006098}{77160083}a^{6}+\frac{95513822}{77160083}a^{5}+\frac{1756333765}{77160083}a^{4}-\frac{35045135}{4061057}a^{3}-\frac{679850384}{77160083}a^{2}+\frac{1673358}{539581}a+\frac{13584338}{5935391}$ 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:  \( 12278.7384077 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 12278.7384077 \cdot 1}{12\cdot\sqrt{39770658357865611264}}\cr\approx \mathstrut & 0.394121437104 \end{aligned}\]

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^15 + 26*x^14 - 36*x^13 - 16*x^12 + 140*x^11 - 188*x^10 - 24*x^9 + 337*x^8 - 276*x^7 - 176*x^6 + 340*x^5 + 14*x^4 - 192*x^3 + 2*x^2 + 56*x + 13) 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^15 + 26*x^14 - 36*x^13 - 16*x^12 + 140*x^11 - 188*x^10 - 24*x^9 + 337*x^8 - 276*x^7 - 176*x^6 + 340*x^5 + 14*x^4 - 192*x^3 + 2*x^2 + 56*x + 13, 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^15 + 26*x^14 - 36*x^13 - 16*x^12 + 140*x^11 - 188*x^10 - 24*x^9 + 337*x^8 - 276*x^7 - 176*x^6 + 340*x^5 + 14*x^4 - 192*x^3 + 2*x^2 + 56*x + 13); 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 - 8*x^15 + 26*x^14 - 36*x^13 - 16*x^12 + 140*x^11 - 188*x^10 - 24*x^9 + 337*x^8 - 276*x^7 - 176*x^6 + 340*x^5 + 14*x^4 - 192*x^3 + 2*x^2 + 56*x + 13); 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

$D_4^2:C_2^2$ (as 16T608):

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 256
The 34 conjugacy class representatives for $D_4^2:C_2^2$
Character table for $D_4^2:C_2^2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), 4.2.6912.1 x2, 4.0.1728.1 x2, \(\Q(\zeta_{12})\), 8.0.47775744.1

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: 16.0.300765603831358685184.2

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 ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ R ${\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

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
\(2\) Copy content Toggle raw display 2.2.8.36b16.16$x^{16} + 8 x^{15} + 38 x^{14} + 128 x^{13} + 336 x^{12} + 716 x^{11} + 1274 x^{10} + 1920 x^{9} + 2471 x^{8} + 2720 x^{7} + 2554 x^{6} + 2028 x^{5} + 1342 x^{4} + 720 x^{3} + 300 x^{2} + 96 x + 17$$8$$2$$36$16T102$$[2, 2, 2, 3]^{4}$$
\(3\) Copy content Toggle raw display 3.2.8.14a1.2$x^{16} + 16 x^{15} + 128 x^{14} + 672 x^{13} + 2576 x^{12} + 7616 x^{11} + 17920 x^{10} + 34176 x^{9} + 53344 x^{8} + 68352 x^{7} + 71680 x^{6} + 60928 x^{5} + 41216 x^{4} + 21504 x^{3} + 8192 x^{2} + 2048 x + 259$$8$$2$$14$$QD_{16}$$$[\ ]_{8}^{2}$$
\(11\) Copy content Toggle raw display 11.2.1.0a1.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
11.2.1.0a1.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
11.2.1.0a1.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
11.2.1.0a1.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
11.2.2.2a1.1$x^{4} + 14 x^{3} + 53 x^{2} + 39 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
11.4.1.0a1.1$x^{4} + 8 x^{2} + 10 x + 2$$1$$4$$0$$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)