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Properties

Label 16.8.132...000.1
Degree $16$
Signature $[8, 4]$
Discriminant $1.327\times 10^{20}$
Root discriminant \(18.10\)
Ramified primes $2,3,5$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^5:C_4$ (as 16T227)

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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 - 4*x^15 - 6*x^14 + 36*x^13 + 6*x^12 - 112*x^11 - 14*x^10 + 208*x^9 + 22*x^8 - 208*x^7 - 14*x^6 + 112*x^5 + 6*x^4 - 36*x^3 - 6*x^2 + 4*x + 1)
 
Copy content gp:K = bnfinit(y^16 - 4*y^15 - 6*y^14 + 36*y^13 + 6*y^12 - 112*y^11 - 14*y^10 + 208*y^9 + 22*y^8 - 208*y^7 - 14*y^6 + 112*y^5 + 6*y^4 - 36*y^3 - 6*y^2 + 4*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 - 6*x^14 + 36*x^13 + 6*x^12 - 112*x^11 - 14*x^10 + 208*x^9 + 22*x^8 - 208*x^7 - 14*x^6 + 112*x^5 + 6*x^4 - 36*x^3 - 6*x^2 + 4*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 - 6*x^14 + 36*x^13 + 6*x^12 - 112*x^11 - 14*x^10 + 208*x^9 + 22*x^8 - 208*x^7 - 14*x^6 + 112*x^5 + 6*x^4 - 36*x^3 - 6*x^2 + 4*x + 1)
 

\( x^{16} - 4 x^{15} - 6 x^{14} + 36 x^{13} + 6 x^{12} - 112 x^{11} - 14 x^{10} + 208 x^{9} + 22 x^{8} + \cdots + 1 \) 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:  $[8, 4]$
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:   \(132710400000000000000\) \(\medspace = 2^{28}\cdot 3^{4}\cdot 5^{14}\) 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:  \(18.10\)
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:  $2^{15/8}3^{1/2}5^{7/8}\approx 25.977097417310098$
Ramified primes:   \(2\), \(3\), \(5\) 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 not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{6}$, $\frac{1}{362}a^{15}+\frac{15}{362}a^{14}-\frac{83}{362}a^{13}+\frac{44}{181}a^{12}+\frac{49}{362}a^{11}-\frac{43}{181}a^{10}-\frac{19}{362}a^{9}+\frac{14}{181}a^{8}+\frac{11}{362}a^{7}+\frac{1}{362}a^{6}+\frac{5}{362}a^{5}+\frac{13}{181}a^{4}-\frac{43}{362}a^{3}+\frac{26}{181}a^{2}+\frac{77}{362}a-\frac{81}{181}$ 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 $1$ (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:  $C_{2}$, which has order $2$ (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:  $11$
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:   $\frac{6745}{362}a^{15}-\frac{30593}{362}a^{14}-\frac{24075}{362}a^{13}+\frac{255633}{362}a^{12}-\frac{96293}{362}a^{11}-\frac{703331}{362}a^{10}+\frac{280543}{362}a^{9}+\frac{626027}{181}a^{8}-\frac{517313}{362}a^{7}-\frac{1125229}{362}a^{6}+\frac{499619}{362}a^{5}+\frac{488319}{362}a^{4}-\frac{213291}{362}a^{3}-\frac{129815}{362}a^{2}+\frac{26321}{362}a+\frac{6791}{181}$, $a$, $\frac{6845}{181}a^{15}-\frac{30903}{181}a^{14}-\frac{50451}{362}a^{13}+\frac{519273}{362}a^{12}-\frac{184415}{362}a^{11}-\frac{721162}{181}a^{10}+\frac{275023}{181}a^{9}+\frac{2575411}{362}a^{8}-\frac{511326}{181}a^{7}-\frac{1169836}{181}a^{6}+\frac{1010917}{362}a^{5}+\frac{1028355}{362}a^{4}-\frac{446585}{362}a^{3}-\frac{134570}{181}a^{2}+\frac{27686}{181}a+\frac{27163}{362}$, $\frac{2647}{362}a^{15}-\frac{5759}{181}a^{14}-\frac{5866}{181}a^{13}+\frac{49498}{181}a^{12}-\frac{19441}{362}a^{11}-\frac{286105}{362}a^{10}+\frac{31778}{181}a^{9}+\frac{518471}{362}a^{8}-\frac{124371}{362}a^{7}-\frac{245108}{181}a^{6}+\frac{67524}{181}a^{5}+\frac{117671}{181}a^{4}-\frac{66037}{362}a^{3}-\frac{66343}{362}a^{2}+\frac{3717}{181}a+\frac{6491}{362}$, $a^{15}-4a^{14}-6a^{13}+36a^{12}+6a^{11}-112a^{10}-14a^{9}+208a^{8}+22a^{7}-208a^{6}-14a^{5}+112a^{4}+6a^{3}-36a^{2}-5a+3$, $\frac{3467}{362}a^{15}-\frac{16051}{362}a^{14}-\frac{10831}{362}a^{13}+\frac{132241}{362}a^{12}-\frac{30989}{181}a^{11}-\frac{177136}{181}a^{10}+\frac{176305}{362}a^{9}+\frac{623243}{362}a^{8}-\frac{320605}{362}a^{7}-\frac{541343}{362}a^{6}+\frac{300057}{362}a^{5}+\frac{215937}{362}a^{4}-\frac{61056}{181}a^{3}-\frac{25698}{181}a^{2}+\frac{14645}{362}a+\frac{4695}{362}$, $\frac{8980}{181}a^{15}-\frac{41051}{181}a^{14}-\frac{30390}{181}a^{13}+\frac{340455}{181}a^{12}-\frac{140628}{181}a^{11}-\frac{923958}{181}a^{10}+\frac{801051}{362}a^{9}+\frac{3271275}{362}a^{8}-\frac{732372}{181}a^{7}-\frac{1443726}{181}a^{6}+\frac{691251}{181}a^{5}+\frac{606702}{181}a^{4}-\frac{286047}{181}a^{3}-\frac{159300}{181}a^{2}+\frac{69041}{362}a+\frac{32635}{362}$, $\frac{2005}{362}a^{15}-\frac{4058}{181}a^{14}-\frac{6011}{181}a^{13}+\frac{37359}{181}a^{12}+\frac{4325}{181}a^{11}-\frac{119881}{181}a^{10}-\frac{2257}{362}a^{9}+\frac{224998}{181}a^{8}-\frac{13783}{362}a^{7}-\frac{235112}{181}a^{6}+\frac{28633}{181}a^{5}+\frac{127968}{181}a^{4}-\frac{25098}{181}a^{3}-\frac{36922}{181}a^{2}+\frac{7413}{362}a+\frac{3753}{181}$, $\frac{1361}{181}a^{15}-\frac{5649}{181}a^{14}-\frac{14699}{362}a^{13}+\frac{50264}{181}a^{12}+\frac{1429}{362}a^{11}-\frac{153970}{181}a^{10}+\frac{9641}{362}a^{9}+\frac{573423}{362}a^{8}-\frac{15980}{181}a^{7}-\frac{288782}{181}a^{6}+\frac{59403}{362}a^{5}+\frac{156475}{181}a^{4}-\frac{42655}{362}a^{3}-\frac{49593}{181}a^{2}+\frac{4883}{362}a+\frac{10993}{362}$, $\frac{6443}{181}a^{15}-\frac{29150}{181}a^{14}-\frac{46707}{362}a^{13}+\frac{244080}{181}a^{12}-\frac{89371}{181}a^{11}-\frac{673739}{181}a^{10}+\frac{262570}{181}a^{9}+\frac{2398325}{362}a^{8}-\frac{485702}{181}a^{7}-\frac{1078833}{181}a^{6}+\frac{947891}{362}a^{5}+\frac{466168}{181}a^{4}-\frac{204830}{181}a^{3}-\frac{121265}{181}a^{2}+\frac{24425}{181}a+\frac{23833}{362}$, $\frac{33491}{362}a^{15}-\frac{76156}{181}a^{14}-\frac{117605}{362}a^{13}+\frac{1270605}{362}a^{12}-\frac{247461}{181}a^{11}-\frac{3483861}{362}a^{10}+\frac{1442275}{362}a^{9}+\frac{6186205}{362}a^{8}-\frac{2659005}{362}a^{7}-\frac{2763505}{181}a^{6}+\frac{2570773}{362}a^{5}+\frac{2356957}{362}a^{4}-\frac{550188}{181}a^{3}-\frac{608029}{362}a^{2}+\frac{135307}{362}a+\frac{60749}{362}$ 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:  \( 13147.1416221 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 13147.1416221 \cdot 1}{2\cdot\sqrt{132710400000000000000}}\cr\approx \mathstrut & 0.227671309352 \end{aligned}\] (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 - 4*x^15 - 6*x^14 + 36*x^13 + 6*x^12 - 112*x^11 - 14*x^10 + 208*x^9 + 22*x^8 - 208*x^7 - 14*x^6 + 112*x^5 + 6*x^4 - 36*x^3 - 6*x^2 + 4*x + 1) 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 - 4*x^15 - 6*x^14 + 36*x^13 + 6*x^12 - 112*x^11 - 14*x^10 + 208*x^9 + 22*x^8 - 208*x^7 - 14*x^6 + 112*x^5 + 6*x^4 - 36*x^3 - 6*x^2 + 4*x + 1, 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 - 4*x^15 - 6*x^14 + 36*x^13 + 6*x^12 - 112*x^11 - 14*x^10 + 208*x^9 + 22*x^8 - 208*x^7 - 14*x^6 + 112*x^5 + 6*x^4 - 36*x^3 - 6*x^2 + 4*x + 1); 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 - 4*x^15 - 6*x^14 + 36*x^13 + 6*x^12 - 112*x^11 - 14*x^10 + 208*x^9 + 22*x^8 - 208*x^7 - 14*x^6 + 112*x^5 + 6*x^4 - 36*x^3 - 6*x^2 + 4*x + 1); 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

$C_2^5:C_4$ (as 16T227):

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 128
The 26 conjugacy class representatives for $C_2^5:C_4$
Character table for $C_2^5:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.2000.1, \(\Q(\zeta_{20})^+\), 4.2.400.1, 8.4.2880000000.2, 8.4.2880000000.1, 8.4.64000000.1

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: 16.0.8294400000000000000.3

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 R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 2.2.8.28a19.2$x^{16} + 10 x^{15} + 50 x^{14} + 168 x^{13} + 420 x^{12} + 826 x^{11} + 1318 x^{10} + 1742 x^{9} + 1931 x^{8} + 1810 x^{7} + 1438 x^{6} + 966 x^{5} + 542 x^{4} + 248 x^{3} + 88 x^{2} + 26 x + 5$$8$$2$$28$16T172$$[2, 2, 2, 2]^{4}$$
\(3\) Copy content Toggle raw display 3.4.1.0a1.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
3.4.1.0a1.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
3.4.2.4a1.2$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(5\) Copy content Toggle raw display 5.2.8.14a1.5$x^{16} + 32 x^{15} + 464 x^{14} + 4032 x^{13} + 23408 x^{12} + 95872 x^{11} + 285376 x^{10} + 627456 x^{9} + 1027168 x^{8} + 1254912 x^{7} + 1141504 x^{6} + 766976 x^{5} + 374528 x^{4} + 129024 x^{3} + 29696 x^{2} + 4106 x + 266$$8$$2$$14$$C_8: C_2$$$[\ ]_{8}^{2}$$

Spectrum of ring of integers

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