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Properties

Label 16.12.245...664.4
Degree $16$
Signature $[12, 2]$
Discriminant $2.451\times 10^{24}$
Root discriminant \(33.45\)
Ramified primes $2,3$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $(C_2^3\times C_4):C_4$ (as 16T292)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^16 - 72*x^12 + 216*x^10 + 198*x^8 - 1296*x^6 + 1512*x^4 - 648*x^2 + 81)
 
Copy content gp:K = bnfinit(y^16 - 72*y^12 + 216*y^10 + 198*y^8 - 1296*y^6 + 1512*y^4 - 648*y^2 + 81, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 72*x^12 + 216*x^10 + 198*x^8 - 1296*x^6 + 1512*x^4 - 648*x^2 + 81);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 72*x^12 + 216*x^10 + 198*x^8 - 1296*x^6 + 1512*x^4 - 648*x^2 + 81)
 

\( x^{16} - 72x^{12} + 216x^{10} + 198x^{8} - 1296x^{6} + 1512x^{4} - 648x^{2} + 81 \) 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:  $[12, 2]$
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:   \(2450839029319069455089664\) \(\medspace = 2^{62}\cdot 3^{12}\) 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:  \(33.45\)
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:  $2^{137/32}3^{3/4}\approx 44.32263892833827$
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:  \(\Q\)
$\Aut(K/\Q)$:   $C_4$
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.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{6}a^{6}-\frac{1}{6}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{6}a^{7}-\frac{1}{6}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{18}a^{8}-\frac{1}{2}$, $\frac{1}{18}a^{9}-\frac{1}{2}a$, $\frac{1}{18}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{36}a^{11}-\frac{1}{36}a^{10}-\frac{1}{36}a^{9}-\frac{1}{36}a^{8}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{108}a^{12}-\frac{1}{36}a^{8}-\frac{1}{12}a^{4}+\frac{1}{4}$, $\frac{1}{108}a^{13}-\frac{1}{36}a^{9}-\frac{1}{12}a^{5}+\frac{1}{4}a$, $\frac{1}{756}a^{14}-\frac{1}{252}a^{12}-\frac{1}{36}a^{10}-\frac{5}{252}a^{8}-\frac{1}{84}a^{6}-\frac{1}{84}a^{4}-\frac{13}{28}a^{2}+\frac{1}{28}$, $\frac{1}{756}a^{15}-\frac{1}{252}a^{13}-\frac{1}{36}a^{10}+\frac{1}{126}a^{9}-\frac{1}{36}a^{8}-\frac{1}{84}a^{7}-\frac{1}{84}a^{5}+\frac{2}{7}a^{3}+\frac{1}{4}a^{2}-\frac{3}{14}a+\frac{1}{4}$ 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:  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:  $13$
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:   $\frac{131}{378}a^{14}-\frac{187}{756}a^{12}+\frac{223}{9}a^{10}-\frac{14405}{252}a^{8}-\frac{4601}{42}a^{6}+\frac{31139}{84}a^{4}-\frac{1805}{7}a^{2}+\frac{1145}{28}$, $\frac{127}{378}a^{14}+\frac{197}{756}a^{12}-24a^{10}+\frac{13591}{252}a^{8}+\frac{764}{7}a^{6}-\frac{9845}{28}a^{4}+\frac{3235}{14}a^{2}-\frac{901}{28}$, $\frac{61}{756}a^{14}+\frac{55}{756}a^{12}-\frac{23}{4}a^{10}+\frac{3083}{252}a^{8}+\frac{2305}{84}a^{6}-\frac{2251}{28}a^{4}+\frac{1307}{28}a^{2}-\frac{149}{28}$, $\frac{491}{756}a^{14}+\frac{191}{378}a^{12}-\frac{1669}{36}a^{10}+\frac{13133}{126}a^{8}+\frac{17569}{84}a^{6}-\frac{28487}{42}a^{4}+\frac{12811}{28}a^{2}-\frac{969}{14}$, $\frac{193}{756}a^{14}+\frac{71}{378}a^{12}-\frac{73}{4}a^{10}+\frac{2627}{63}a^{8}+\frac{6863}{84}a^{6}-\frac{3797}{14}a^{4}+\frac{5163}{28}a^{2}-\frac{195}{7}$, $\frac{2}{189}a^{14}+\frac{5}{378}a^{12}+\frac{7}{9}a^{10}-\frac{407}{126}a^{8}-\frac{17}{42}a^{6}+\frac{401}{21}a^{4}-\frac{375}{14}a^{2}+\frac{68}{7}$, $\frac{193}{189}a^{14}+\frac{589}{756}a^{12}-\frac{1313}{18}a^{10}+\frac{41549}{252}a^{8}+\frac{13817}{42}a^{6}-\frac{90127}{84}a^{4}+\frac{5037}{7}a^{2}-\frac{2987}{28}$, $\frac{571}{378}a^{15}+\frac{11}{18}a^{14}+\frac{443}{378}a^{13}+\frac{1}{2}a^{12}-\frac{3883}{36}a^{11}-\frac{523}{12}a^{10}+\frac{61133}{252}a^{9}+\frac{3469}{36}a^{8}+\frac{10246}{21}a^{7}+\frac{598}{3}a^{6}-\frac{33182}{21}a^{5}-\frac{1886}{3}a^{4}+\frac{29527}{28}a^{3}+\frac{1651}{4}a^{2}-\frac{4339}{28}a-\frac{243}{4}$, $\frac{28}{27}a^{15}+\frac{41}{189}a^{14}+\frac{83}{108}a^{13}+\frac{23}{126}a^{12}-\frac{889}{12}a^{11}-\frac{557}{36}a^{10}+\frac{1522}{9}a^{9}+\frac{8525}{252}a^{8}+\frac{989}{3}a^{7}+\frac{3019}{42}a^{6}-\frac{13171}{12}a^{5}-\frac{4654}{21}a^{4}+\frac{3039}{4}a^{3}+\frac{3909}{28}a^{2}-123a-\frac{459}{28}$, $\frac{163}{378}a^{15}-\frac{221}{252}a^{14}+\frac{127}{378}a^{13}-\frac{85}{126}a^{12}-\frac{1109}{36}a^{11}+\frac{1127}{18}a^{10}+\frac{17417}{252}a^{9}-\frac{35605}{252}a^{8}+\frac{2953}{21}a^{7}-\frac{23641}{84}a^{6}-\frac{3162}{7}a^{5}+\frac{38597}{42}a^{4}+\frac{8201}{28}a^{3}-\frac{8735}{14}a^{2}-\frac{983}{28}a+\frac{2697}{28}$, $\frac{2}{189}a^{15}-\frac{103}{378}a^{14}-\frac{5}{378}a^{13}-\frac{53}{252}a^{12}-\frac{7}{9}a^{11}+\frac{175}{9}a^{10}+\frac{407}{126}a^{9}-\frac{11059}{252}a^{8}+\frac{17}{42}a^{7}-\frac{1828}{21}a^{6}-\frac{401}{21}a^{5}+\frac{7995}{28}a^{4}+\frac{375}{14}a^{3}-\frac{2735}{14}a^{2}-\frac{68}{7}a+\frac{865}{28}$, $\frac{179}{189}a^{15}+\frac{173}{378}a^{14}-\frac{575}{756}a^{13}+\frac{22}{63}a^{12}+\frac{811}{12}a^{11}-\frac{1177}{36}a^{10}-\frac{18937}{126}a^{9}+\frac{18619}{252}a^{8}-\frac{6478}{21}a^{7}+\frac{6197}{42}a^{6}+\frac{82343}{84}a^{5}-\frac{10093}{21}a^{4}-\frac{18027}{28}a^{3}+\frac{9019}{28}a^{2}+\frac{1287}{14}a-\frac{1383}{28}$, $\frac{619}{756}a^{15}+\frac{619}{756}a^{14}-\frac{151}{252}a^{13}+\frac{151}{252}a^{12}+\frac{2107}{36}a^{11}-\frac{2107}{36}a^{10}-\frac{33767}{252}a^{9}+\frac{33767}{252}a^{8}-\frac{21935}{84}a^{7}+\frac{21935}{84}a^{6}+\frac{24389}{28}a^{5}-\frac{24389}{28}a^{4}-\frac{16663}{28}a^{3}+\frac{16663}{28}a^{2}+\frac{2461}{28}a-\frac{2489}{28}$ 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:  \( 6828348.410317181 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{2}\cdot 6828348.410317181 \cdot 1}{2\cdot\sqrt{2450839029319069455089664}}\cr\approx \mathstrut & 0.352653312539926 \end{aligned}\] (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 - 72*x^12 + 216*x^10 + 198*x^8 - 1296*x^6 + 1512*x^4 - 648*x^2 + 81) 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 - 72*x^12 + 216*x^10 + 198*x^8 - 1296*x^6 + 1512*x^4 - 648*x^2 + 81, 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 - 72*x^12 + 216*x^10 + 198*x^8 - 1296*x^6 + 1512*x^4 - 648*x^2 + 81); 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 - 72*x^12 + 216*x^10 + 198*x^8 - 1296*x^6 + 1512*x^4 - 648*x^2 + 81); 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

$(C_2^3\times C_4):C_4$ (as 16T292):

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 $(C_2^3\times C_4):C_4$
Character table for $(C_2^3\times C_4):C_4$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\zeta_{16})^+\), 4.4.18432.1, \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\zeta_{48})^+\)

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: 16.8.612709757329767363772416.21, 16.0.612709757329767363772416.95, 16.8.9803356117276277820358656.110, 16.0.9803356117276277820358656.257, 16.4.2450839029319069455089664.110
Degree 32 siblings: deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32
Minimal sibling: 16.8.9803356117276277820358656.110

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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{12}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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 2.1.16.62h1.2057$x^{16} + 8 x^{15} + 20 x^{12} + 16 x^{9} + 2 x^{8} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 30$$16$$1$$62$16T292$$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$$
\(3\) Copy content Toggle raw display 3.4.4.12a1.1$x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 24 x^{12} + 48 x^{11} + 96 x^{10} + 64 x^{9} + 24 x^{8} + 96 x^{7} + 96 x^{6} + 32 x^{4} + 64 x^{3} + 3 x^{2} + 16$$4$$4$$12$$C_8: C_2$$$[\ ]_{4}^{4}$$

Spectrum of ring of integers

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