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Properties

Label 16.0.868...136.2
Degree $16$
Signature $[0, 8]$
Discriminant $8.681\times 10^{24}$
Root discriminant \(36.20\)
Ramified primes $2,7$
Class number $6$ (GRH)
Class group [6] (GRH)
Galois group $D_4:C_4$ (as 16T26)

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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 + 36*x^12 - 104*x^10 + 62*x^8 + 680*x^6 - 252*x^4 + 392*x^2 + 2401)
 
Copy content gp:K = bnfinit(y^16 - 8*y^14 + 36*y^12 - 104*y^10 + 62*y^8 + 680*y^6 - 252*y^4 + 392*y^2 + 2401, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^14 + 36*x^12 - 104*x^10 + 62*x^8 + 680*x^6 - 252*x^4 + 392*x^2 + 2401);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 8*x^14 + 36*x^12 - 104*x^10 + 62*x^8 + 680*x^6 - 252*x^4 + 392*x^2 + 2401)
 

\( x^{16} - 8x^{14} + 36x^{12} - 104x^{10} + 62x^{8} + 680x^{6} - 252x^{4} + 392x^{2} + 2401 \) 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:   \(8680963974111420152283136\) \(\medspace = 2^{66}\cdot 7^{6}\) 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:  \(36.20\)
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:  $2^{67/16}7^{1/2}\approx 48.207224374927556$
Ramified primes:   \(2\), \(7\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2^2$
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 a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{28}a^{12}+\frac{3}{14}a^{10}+\frac{1}{28}a^{8}-\frac{3}{14}a^{6}+\frac{13}{28}a^{4}+\frac{2}{7}a^{2}-\frac{1}{4}$, $\frac{1}{28}a^{13}-\frac{1}{28}a^{11}-\frac{1}{4}a^{10}-\frac{3}{14}a^{9}-\frac{1}{4}a^{8}-\frac{3}{14}a^{7}+\frac{13}{28}a^{5}-\frac{13}{28}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{9060600388}a^{14}-\frac{73796997}{9060600388}a^{12}-\frac{5470569}{9060600388}a^{10}-\frac{1651171479}{9060600388}a^{8}-\frac{845141207}{9060600388}a^{6}-\frac{1989981685}{9060600388}a^{4}-\frac{438610999}{1294371484}a^{2}-\frac{85340587}{184910212}$, $\frac{1}{63424202716}a^{15}-\frac{73796997}{63424202716}a^{13}-\frac{1135310333}{31712101358}a^{11}-\frac{1}{4}a^{10}+\frac{306989309}{31712101358}a^{9}-\frac{1}{4}a^{8}-\frac{14436041789}{63424202716}a^{7}+\frac{20661519285}{63424202716}a^{5}+\frac{1560455291}{4530300194}a^{3}+\frac{1}{4}a^{2}-\frac{112011623}{647185742}a+\frac{1}{4}$ 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:  $C_{6}$, which has order $6$ (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:  $C_{6}$, which has order $6$ (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);
 
Relative class number:   $6$ (assuming GRH)

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:   $\frac{3735}{46227553}a^{14}+\frac{1935}{92455106}a^{12}+\frac{3211}{46227553}a^{10}-\frac{83441}{46227553}a^{8}-\frac{179093}{46227553}a^{6}-\frac{106505}{92455106}a^{4}-\frac{406357}{46227553}a^{2}+\frac{18436987}{46227553}$, $\frac{13636979}{9060600388}a^{14}-\frac{94213741}{9060600388}a^{12}+\frac{360280773}{9060600388}a^{10}-\frac{899139221}{9060600388}a^{8}-\frac{533910999}{9060600388}a^{6}+\frac{8821023889}{9060600388}a^{4}+\frac{1493182665}{1294371484}a^{2}+\frac{7063821}{184910212}$, $\frac{460840}{323592871}a^{14}-\frac{11218631}{1294371484}a^{12}+\frac{8812448}{323592871}a^{10}-\frac{53743713}{1294371484}a^{8}-\frac{66035082}{323592871}a^{6}+\frac{1462962995}{1294371484}a^{4}+\frac{512885094}{323592871}a^{2}-\frac{80972313}{184910212}$, $\frac{22100695}{63424202716}a^{15}-\frac{112010245}{31712101358}a^{13}+\frac{1274632017}{63424202716}a^{11}-\frac{1207909505}{15856050679}a^{9}+\frac{11258214211}{63424202716}a^{7}-\frac{1867394233}{15856050679}a^{5}-\frac{1622151525}{9060600388}a^{3}+\frac{580074149}{647185742}a-1$, $\frac{22994177}{63424202716}a^{15}+\frac{4832057}{9060600388}a^{14}-\frac{258320471}{63424202716}a^{13}-\frac{10478151}{2265150097}a^{12}+\frac{1613610085}{63424202716}a^{11}+\frac{200401557}{9060600388}a^{10}-\frac{6834011895}{63424202716}a^{9}-\frac{165147131}{2265150097}a^{8}+\frac{16829537939}{63424202716}a^{7}+\frac{670105591}{9060600388}a^{6}-\frac{11103164825}{63424202716}a^{5}+\frac{1116411315}{2265150097}a^{4}-\frac{2272406783}{9060600388}a^{3}-\frac{1129436699}{1294371484}a^{2}+\frac{1704958615}{1294371484}a+\frac{10147236}{46227553}$, $\frac{87599441}{63424202716}a^{15}-\frac{32155595}{9060600388}a^{14}-\frac{398647661}{63424202716}a^{13}+\frac{220858949}{9060600388}a^{12}+\frac{898469585}{63424202716}a^{11}-\frac{860977981}{9060600388}a^{10}-\frac{5885153}{63424202716}a^{9}+\frac{2103954367}{9060600388}a^{8}-\frac{16937829509}{63424202716}a^{7}+\frac{1105079419}{9060600388}a^{6}+\frac{51882984229}{63424202716}a^{5}-\frac{21311340025}{9060600388}a^{4}+\frac{30564118117}{9060600388}a^{3}-\frac{3645727485}{1294371484}a^{2}+\frac{2163469097}{1294371484}a-\frac{30111663}{184910212}$, $\frac{15786607}{15856050679}a^{15}+\frac{8577383}{4530300194}a^{14}-\frac{239110957}{31712101358}a^{13}-\frac{153732501}{9060600388}a^{12}+\frac{1044607775}{31712101358}a^{11}+\frac{389497041}{4530300194}a^{10}-\frac{1540496590}{15856050679}a^{9}-\frac{2697751519}{9060600388}a^{8}+\frac{923773014}{15856050679}a^{7}+\frac{2434376781}{4530300194}a^{6}+\frac{20059524899}{31712101358}a^{5}+\frac{3441116153}{9060600388}a^{4}-\frac{1068562895}{4530300194}a^{3}-\frac{261834691}{647185742}a^{2}+\frac{577229293}{323592871}a+\frac{373277263}{184910212}$ 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:  \( 211544.391067 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 211544.391067 \cdot 6}{2\cdot\sqrt{8680963974111420152283136}}\cr\approx \mathstrut & 0.523212052273 \end{aligned}\] (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 - 8*x^14 + 36*x^12 - 104*x^10 + 62*x^8 + 680*x^6 - 252*x^4 + 392*x^2 + 2401) 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 + 36*x^12 - 104*x^10 + 62*x^8 + 680*x^6 - 252*x^4 + 392*x^2 + 2401, 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 + 36*x^12 - 104*x^10 + 62*x^8 + 680*x^6 - 252*x^4 + 392*x^2 + 2401); 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 + 36*x^12 - 104*x^10 + 62*x^8 + 680*x^6 - 252*x^4 + 392*x^2 + 2401); 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

$D_4:C_4$ (as 16T26):

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 14 conjugacy class representatives for $D_4:C_4$
Character table for $D_4:C_4$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.14336.1, 4.0.2048.2, 4.0.7168.1, 8.8.736586891264.1, 8.0.736586891264.1, 8.0.3288334336.2

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(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: deg 16
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 ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{4}$ R ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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 2.1.16.66h1.629$x^{16} + 8 x^{14} + 4 x^{12} + 16 x^{11} + 8 x^{10} + 16 x^{9} + 8 x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 16 x^{3} + 34$$16$$1$$66$16T26$$[2, 3, \frac{7}{2}, 4, 5]$$
\(7\) Copy content Toggle raw display 7.2.1.0a1.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.1.2.1a1.1$x^{2} + 7$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.2.1.0a1.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
7.2.2.2a1.2$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
7.2.2.2a1.2$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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