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Properties

Label 16.0.100...625.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.004\times 10^{19}$
Root discriminant \(15.40\)
Ramified primes $3,5,7,13$
Class number $1$
Class group trivial
Galois group $C_4.D_4^2$ (as 16T511)

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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 - 3*x^15 + 3*x^14 - 12*x^13 + 33*x^12 - 18*x^11 + 45*x^10 - 105*x^9 + 15*x^8 - 63*x^7 + 144*x^6 + 45*x^5 + 81*x^4 - 45*x^3 - 9*x^2 - 18*x + 9)
 
Copy content gp:K = bnfinit(y^16 - 3*y^15 + 3*y^14 - 12*y^13 + 33*y^12 - 18*y^11 + 45*y^10 - 105*y^9 + 15*y^8 - 63*y^7 + 144*y^6 + 45*y^5 + 81*y^4 - 45*y^3 - 9*y^2 - 18*y + 9, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 3*x^14 - 12*x^13 + 33*x^12 - 18*x^11 + 45*x^10 - 105*x^9 + 15*x^8 - 63*x^7 + 144*x^6 + 45*x^5 + 81*x^4 - 45*x^3 - 9*x^2 - 18*x + 9);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 3*x^15 + 3*x^14 - 12*x^13 + 33*x^12 - 18*x^11 + 45*x^10 - 105*x^9 + 15*x^8 - 63*x^7 + 144*x^6 + 45*x^5 + 81*x^4 - 45*x^3 - 9*x^2 - 18*x + 9)
 

\( x^{16} - 3 x^{15} + 3 x^{14} - 12 x^{13} + 33 x^{12} - 18 x^{11} + 45 x^{10} - 105 x^{9} + 15 x^{8} + \cdots + 9 \) 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:   \(10044752324575775625\) \(\medspace = 3^{14}\cdot 5^{4}\cdot 7^{6}\cdot 13^{4}\) 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:  \(15.40\)
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:  $3^{7/8}5^{1/2}7^{3/4}13^{1/2}\approx 90.73217175218808$
Ramified primes:   \(3\), \(5\), \(7\), \(13\) 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.
Maximal CM subfield:  \(\Q(\sqrt{-3}) \)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}$, $\frac{1}{3}a^{9}$, $\frac{1}{3}a^{10}$, $\frac{1}{3}a^{11}$, $\frac{1}{3}a^{12}$, $\frac{1}{39}a^{13}-\frac{1}{13}a^{12}+\frac{4}{39}a^{11}+\frac{5}{39}a^{10}+\frac{1}{13}a^{9}+\frac{2}{39}a^{8}+\frac{6}{13}a^{7}+\frac{3}{13}a^{6}-\frac{6}{13}a^{5}-\frac{2}{13}a^{4}-\frac{2}{13}a^{3}-\frac{3}{13}a^{2}-\frac{2}{13}a-\frac{6}{13}$, $\frac{1}{39}a^{14}-\frac{5}{39}a^{12}+\frac{4}{39}a^{11}+\frac{5}{39}a^{10}-\frac{2}{39}a^{9}-\frac{2}{39}a^{8}-\frac{5}{13}a^{7}+\frac{3}{13}a^{6}+\frac{6}{13}a^{5}+\frac{5}{13}a^{4}+\frac{4}{13}a^{3}+\frac{2}{13}a^{2}+\frac{1}{13}a-\frac{5}{13}$, $\frac{1}{701883}a^{15}+\frac{115}{233961}a^{14}-\frac{1972}{233961}a^{13}+\frac{21622}{233961}a^{12}-\frac{1426}{77987}a^{11}+\frac{1667}{77987}a^{10}+\frac{653}{233961}a^{9}+\frac{5748}{77987}a^{8}+\frac{13915}{233961}a^{7}-\frac{892}{5999}a^{6}-\frac{10063}{77987}a^{5}+\frac{31492}{77987}a^{4}-\frac{24944}{77987}a^{3}+\frac{30031}{77987}a^{2}-\frac{23467}{77987}a-\frac{31874}{77987}$ 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:   \( \frac{7604}{33423} a^{15} - \frac{13609}{33423} a^{14} + \frac{698}{11141} a^{13} - \frac{88415}{33423} a^{12} + \frac{4490}{857} a^{11} + \frac{23939}{11141} a^{10} + \frac{146501}{11141} a^{9} - \frac{212731}{11141} a^{8} - \frac{158390}{11141} a^{7} - \frac{364558}{11141} a^{6} + \frac{313383}{11141} a^{5} + \frac{397476}{11141} a^{4} + \frac{608232}{11141} a^{3} + \frac{114207}{11141} a^{2} - \frac{726}{11141} a - \frac{98591}{11141} \)  (order $6$) 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:   $\frac{144217}{701883}a^{15}-\frac{122260}{233961}a^{14}+\frac{100652}{233961}a^{13}-\frac{584132}{233961}a^{12}+\frac{1404596}{233961}a^{11}-\frac{132164}{77987}a^{10}+\frac{2496983}{233961}a^{9}-\frac{1512249}{77987}a^{8}-\frac{803792}{233961}a^{7}-\frac{1522848}{77987}a^{6}+\frac{1924092}{77987}a^{5}+\frac{1578574}{77987}a^{4}+\frac{2225123}{77987}a^{3}+\frac{248636}{77987}a^{2}-\frac{112471}{77987}a-\frac{362854}{77987}$, $\frac{129767}{233961}a^{15}-\frac{29143}{17997}a^{14}+\frac{112800}{77987}a^{13}-\frac{1410398}{233961}a^{12}+\frac{301850}{17997}a^{11}-\frac{1616893}{233961}a^{10}+\frac{4481182}{233961}a^{9}-\frac{3710221}{77987}a^{8}-\frac{201160}{77987}a^{7}-\frac{1682630}{77987}a^{6}+\frac{4478259}{77987}a^{5}+\frac{202485}{5999}a^{4}+\frac{3000632}{77987}a^{3}-\frac{226790}{77987}a^{2}-\frac{59432}{77987}a-\frac{398449}{77987}$, $\frac{86378}{233961}a^{15}-\frac{14507}{17997}a^{14}+\frac{4189}{77987}a^{13}-\frac{237515}{77987}a^{12}+\frac{643211}{77987}a^{11}+\frac{1014989}{233961}a^{10}+\frac{1632437}{233961}a^{9}-\frac{5649487}{233961}a^{8}-\frac{1947447}{77987}a^{7}-\frac{744643}{77987}a^{6}+\frac{2841401}{77987}a^{5}+\frac{3747638}{77987}a^{4}+\frac{2567590}{77987}a^{3}+\frac{186348}{77987}a^{2}-\frac{445189}{77987}a-\frac{29376}{5999}$, $\frac{7109}{77987}a^{15}-\frac{50950}{233961}a^{14}+\frac{18433}{233961}a^{13}-\frac{188182}{233961}a^{12}+\frac{515570}{233961}a^{11}+\frac{96302}{233961}a^{10}+\frac{464480}{233961}a^{9}-\frac{461831}{77987}a^{8}-\frac{329271}{77987}a^{7}-\frac{171322}{77987}a^{6}+\frac{529784}{77987}a^{5}+\frac{707405}{77987}a^{4}+\frac{595802}{77987}a^{3}+\frac{263656}{77987}a^{2}+\frac{107573}{77987}a-\frac{6338}{77987}$, $\frac{1493}{77987}a^{15}+\frac{2116}{17997}a^{14}-\frac{102164}{233961}a^{13}+\frac{9080}{77987}a^{12}-\frac{241028}{233961}a^{11}+\frac{1065461}{233961}a^{10}-\frac{14174}{233961}a^{9}+\frac{386188}{233961}a^{8}-\frac{994160}{77987}a^{7}-\frac{35796}{5999}a^{6}+\frac{204895}{77987}a^{5}+\frac{1308324}{77987}a^{4}+\frac{1203400}{77987}a^{3}+\frac{387748}{77987}a^{2}-\frac{132620}{77987}a-\frac{178302}{77987}$, $\frac{274679}{701883}a^{15}-\frac{224612}{233961}a^{14}+\frac{40233}{77987}a^{13}-\frac{304831}{77987}a^{12}+\frac{2364965}{233961}a^{11}+\frac{18659}{233961}a^{10}+\frac{994263}{77987}a^{9}-\frac{2301538}{77987}a^{8}-\frac{3446008}{233961}a^{7}-\frac{1616366}{77987}a^{6}+\frac{3003963}{77987}a^{5}+\frac{3076494}{77987}a^{4}+\frac{3308350}{77987}a^{3}+\frac{488011}{77987}a^{2}-\frac{128566}{77987}a-\frac{477796}{77987}$, $\frac{5149}{701883}a^{15}+\frac{13409}{77987}a^{14}-\frac{63163}{77987}a^{13}+\frac{284189}{233961}a^{12}-\frac{598994}{233961}a^{11}+\frac{2000108}{233961}a^{10}-\frac{188518}{17997}a^{9}+\frac{759259}{77987}a^{8}-\frac{5792756}{233961}a^{7}+\frac{1829938}{77987}a^{6}-\frac{660914}{77987}a^{5}+\frac{2050996}{77987}a^{4}-\frac{1383835}{77987}a^{3}+\frac{101378}{77987}a^{2}-\frac{701608}{77987}a+\frac{337360}{77987}$ 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:  \( 1854.603895 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1854.603895 \cdot 1}{6\cdot\sqrt{10044752324575775625}}\cr\approx \mathstrut & 0.2369022839 \end{aligned}\]

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 - 3*x^15 + 3*x^14 - 12*x^13 + 33*x^12 - 18*x^11 + 45*x^10 - 105*x^9 + 15*x^8 - 63*x^7 + 144*x^6 + 45*x^5 + 81*x^4 - 45*x^3 - 9*x^2 - 18*x + 9) 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 - 3*x^15 + 3*x^14 - 12*x^13 + 33*x^12 - 18*x^11 + 45*x^10 - 105*x^9 + 15*x^8 - 63*x^7 + 144*x^6 + 45*x^5 + 81*x^4 - 45*x^3 - 9*x^2 - 18*x + 9, 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 - 3*x^15 + 3*x^14 - 12*x^13 + 33*x^12 - 18*x^11 + 45*x^10 - 105*x^9 + 15*x^8 - 63*x^7 + 144*x^6 + 45*x^5 + 81*x^4 - 45*x^3 - 9*x^2 - 18*x + 9); 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 - 3*x^15 + 3*x^14 - 12*x^13 + 33*x^12 - 18*x^11 + 45*x^10 - 105*x^9 + 15*x^8 - 63*x^7 + 144*x^6 + 45*x^5 + 81*x^4 - 45*x^3 - 9*x^2 - 18*x + 9); 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

$C_4.D_4^2$ (as 16T511):

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 46 conjugacy class representatives for $C_4.D_4^2$
Character table for $C_4.D_4^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.189.1, 4.0.117.1, 4.0.2457.1, 8.0.18753525.1, 8.0.3169345725.2, 8.0.6036849.2

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(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: 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 ${\href{/padicField/2.8.0.1}{8} }^{2}$ R R R ${\href{/padicField/11.8.0.1}{8} }^{2}$ R ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(3\) 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}$$$[\ ]_{8}^{2}$$
\(5\) Copy content Toggle raw display 5.4.1.0a1.1$x^{4} + 4 x^{2} + 4 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
5.4.1.0a1.1$x^{4} + 4 x^{2} + 4 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(7\) Copy content Toggle raw display 7.2.1.0a1.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
7.2.1.0a1.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
7.2.1.0a1.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
7.2.1.0a1.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
7.1.4.3a1.2$x^{4} + 21$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$
7.1.4.3a1.2$x^{4} + 21$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$
\(13\) Copy content Toggle raw display 13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.2.2.2a1.2$x^{4} + 24 x^{3} + 148 x^{2} + 48 x + 17$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
13.2.2.2a1.2$x^{4} + 24 x^{3} + 148 x^{2} + 48 x + 17$$2$$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)