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Properties

Label 16.0.73263654766724409.1
Degree $16$
Signature $[0, 8]$
Discriminant $7.326\times 10^{16}$
Root discriminant \(11.33\)
Ramified primes $3,13$
Class number $1$
Class group trivial
Galois group $C_4\wr C_2$ (as 16T42)

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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 - 6*x^13 + 13*x^12 + 6*x^11 - 30*x^10 - 9*x^9 + 51*x^8 - 9*x^7 - 30*x^6 + 6*x^5 + 13*x^4 - 6*x^3 + 3*x^2 - 3*x + 1)
 
Copy content gp:K = bnfinit(y^16 - 3*y^15 + 3*y^14 - 6*y^13 + 13*y^12 + 6*y^11 - 30*y^10 - 9*y^9 + 51*y^8 - 9*y^7 - 30*y^6 + 6*y^5 + 13*y^4 - 6*y^3 + 3*y^2 - 3*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 3*x^14 - 6*x^13 + 13*x^12 + 6*x^11 - 30*x^10 - 9*x^9 + 51*x^8 - 9*x^7 - 30*x^6 + 6*x^5 + 13*x^4 - 6*x^3 + 3*x^2 - 3*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 3*x^15 + 3*x^14 - 6*x^13 + 13*x^12 + 6*x^11 - 30*x^10 - 9*x^9 + 51*x^8 - 9*x^7 - 30*x^6 + 6*x^5 + 13*x^4 - 6*x^3 + 3*x^2 - 3*x + 1)
 

\( x^{16} - 3 x^{15} + 3 x^{14} - 6 x^{13} + 13 x^{12} + 6 x^{11} - 30 x^{10} - 9 x^{9} + 51 x^{8} + \cdots + 1 \) 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:  $[0, 8]$
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:   \(73263654766724409\) \(\medspace = 3^{12}\cdot 13^{10}\) 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:  \(11.33\)
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:  $3^{3/4}13^{3/4}\approx 15.606246247497747$
Ramified primes:   \(3\), \(13\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2\times C_4$
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.
Maximal CM subfield:  \(\Q(\sqrt{-3}, \sqrt{13})\)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7}a^{13}-\frac{1}{7}a^{11}-\frac{3}{7}a^{10}+\frac{1}{7}a^{9}+\frac{1}{7}a^{8}+\frac{1}{7}a^{5}+\frac{1}{7}a^{4}-\frac{3}{7}a^{3}-\frac{1}{7}a^{2}+\frac{1}{7}$, $\frac{1}{161}a^{14}+\frac{6}{161}a^{13}-\frac{36}{161}a^{12}-\frac{37}{161}a^{11}-\frac{31}{161}a^{10}+\frac{9}{23}a^{9}+\frac{62}{161}a^{8}+\frac{7}{23}a^{7}-\frac{76}{161}a^{6}+\frac{9}{23}a^{5}+\frac{38}{161}a^{4}+\frac{9}{161}a^{3}-\frac{13}{161}a^{2}+\frac{29}{161}a-\frac{22}{161}$, $\frac{1}{161}a^{15}-\frac{3}{161}a^{13}+\frac{18}{161}a^{12}-\frac{39}{161}a^{11}+\frac{6}{23}a^{10}+\frac{75}{161}a^{9}+\frac{68}{161}a^{8}-\frac{48}{161}a^{7}+\frac{36}{161}a^{6}+\frac{51}{161}a^{5}+\frac{11}{161}a^{4}+\frac{48}{161}a^{3}+\frac{38}{161}a^{2}-\frac{5}{23}a+\frac{40}{161}$ 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:   \( -\frac{30}{23} a^{15} + 3 a^{14} - \frac{48}{23} a^{13} + \frac{173}{23} a^{12} - \frac{325}{23} a^{11} - \frac{294}{23} a^{10} + \frac{487}{23} a^{9} + \frac{812}{23} a^{8} - \frac{998}{23} a^{7} - \frac{528}{23} a^{6} + \frac{563}{23} a^{5} + \frac{406}{23} a^{4} - \frac{382}{23} a^{3} + \frac{33}{23} a^{2} - \frac{77}{23} a + \frac{65}{23} \)  (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:   $\frac{42}{23}a^{15}-\frac{808}{161}a^{14}+\frac{687}{161}a^{13}-\frac{1684}{161}a^{12}+\frac{3641}{161}a^{11}+\frac{2367}{161}a^{10}-\frac{7625}{161}a^{9}-\frac{5494}{161}a^{8}+\frac{1873}{23}a^{7}+\frac{2762}{161}a^{6}-\frac{8080}{161}a^{5}-\frac{2699}{161}a^{4}+\frac{527}{23}a^{3}+\frac{125}{161}a^{2}+\frac{571}{161}a-\frac{594}{161}$, $\frac{94}{161}a^{15}-\frac{16}{7}a^{14}+\frac{431}{161}a^{13}-\frac{677}{161}a^{12}+\frac{1877}{161}a^{11}-\frac{330}{161}a^{10}-\frac{3553}{161}a^{9}-\frac{158}{23}a^{8}+\frac{7402}{161}a^{7}-\frac{687}{161}a^{6}-\frac{4843}{161}a^{5}-\frac{691}{161}a^{4}+\frac{2741}{161}a^{3}-\frac{683}{161}a^{2}+\frac{528}{161}a-\frac{51}{23}$, $\frac{74}{161}a^{15}-\frac{195}{161}a^{14}+\frac{80}{161}a^{13}-\frac{181}{161}a^{12}+\frac{603}{161}a^{11}+\frac{1195}{161}a^{10}-\frac{2848}{161}a^{9}-\frac{269}{23}a^{8}+\frac{5408}{161}a^{7}+\frac{740}{161}a^{6}-\frac{4463}{161}a^{5}-\frac{19}{23}a^{4}+\frac{1889}{161}a^{3}-\frac{311}{161}a^{2}+\frac{288}{161}a-\frac{42}{23}$, $\frac{54}{161}a^{15}-\frac{187}{161}a^{14}+\frac{40}{23}a^{13}-\frac{507}{161}a^{12}+\frac{995}{161}a^{11}-\frac{330}{161}a^{10}-\frac{191}{23}a^{9}+\frac{565}{161}a^{8}+\frac{2091}{161}a^{7}-\frac{153}{23}a^{6}-\frac{1184}{161}a^{5}+\frac{43}{161}a^{4}+\frac{1047}{161}a^{3}+\frac{26}{23}a^{2}-\frac{229}{161}a-\frac{51}{161}$, $\frac{191}{161}a^{15}-\frac{20}{7}a^{14}+\frac{232}{161}a^{13}-\frac{771}{161}a^{12}+\frac{237}{23}a^{11}+\frac{2893}{161}a^{10}-\frac{5455}{161}a^{9}-\frac{4400}{161}a^{8}+\frac{8220}{161}a^{7}+\frac{2874}{161}a^{6}-\frac{5048}{161}a^{5}-\frac{1027}{161}a^{4}+\frac{934}{161}a^{3}+\frac{174}{161}a^{2}+\frac{744}{161}a-\frac{410}{161}$, $\frac{93}{161}a^{15}-\frac{82}{161}a^{14}-\frac{265}{161}a^{13}-\frac{43}{161}a^{12}+\frac{4}{23}a^{11}+\frac{2837}{161}a^{10}-\frac{1066}{161}a^{9}-\frac{6304}{161}a^{8}+\frac{3271}{161}a^{7}+\frac{6038}{161}a^{6}-\frac{3137}{161}a^{5}-\frac{118}{7}a^{4}+\frac{68}{7}a^{3}-\frac{4}{7}a^{2}+\frac{324}{161}a-\frac{88}{161}$, $\frac{113}{161}a^{15}-\frac{555}{161}a^{14}+\frac{110}{23}a^{13}-\frac{848}{161}a^{12}+\frac{2512}{161}a^{11}-\frac{1026}{161}a^{10}-\frac{6756}{161}a^{9}+\frac{116}{7}a^{8}+\frac{11656}{161}a^{7}-\frac{5755}{161}a^{6}-\frac{8341}{161}a^{5}+\frac{2624}{161}a^{4}+\frac{3695}{161}a^{3}-\frac{117}{23}a^{2}+\frac{75}{161}a-\frac{727}{161}$ 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:  \( 140.216883217 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 140.216883217 \cdot 1}{6\cdot\sqrt{73263654766724409}}\cr\approx \mathstrut & 0.209721867299 \end{aligned}\]

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 - 6*x^13 + 13*x^12 + 6*x^11 - 30*x^10 - 9*x^9 + 51*x^8 - 9*x^7 - 30*x^6 + 6*x^5 + 13*x^4 - 6*x^3 + 3*x^2 - 3*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 - 3*x^15 + 3*x^14 - 6*x^13 + 13*x^12 + 6*x^11 - 30*x^10 - 9*x^9 + 51*x^8 - 9*x^7 - 30*x^6 + 6*x^5 + 13*x^4 - 6*x^3 + 3*x^2 - 3*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 - 3*x^15 + 3*x^14 - 6*x^13 + 13*x^12 + 6*x^11 - 30*x^10 - 9*x^9 + 51*x^8 - 9*x^7 - 30*x^6 + 6*x^5 + 13*x^4 - 6*x^3 + 3*x^2 - 3*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 = polynomial_ring(QQ); K, a = number_field(x^16 - 3*x^15 + 3*x^14 - 6*x^13 + 13*x^12 + 6*x^11 - 30*x^10 - 9*x^9 + 51*x^8 - 9*x^7 - 30*x^6 + 6*x^5 + 13*x^4 - 6*x^3 + 3*x^2 - 3*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

$C_4\wr C_2$ (as 16T42):

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 $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-39}) \), 4.2.507.1 x2, 4.0.117.1 x2, \(\Q(\sqrt{-3}, \sqrt{13})\), 8.0.1601613.1, 8.0.270672597.1, 8.0.2313441.1

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(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 8 siblings: 8.0.1601613.1, 8.0.270672597.1
Degree 16 sibling: 16.4.12381557655576425121.1
Minimal sibling: 8.0.1601613.1

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 ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ R ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{8}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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
\(3\) Copy content Toggle raw display 3.2.4.6a1.3$x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 67 x + 19$$4$$2$$6$$Q_8$$$[\ ]_{4}^{2}$$
3.2.4.6a1.3$x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 67 x + 19$$4$$2$$6$$Q_8$$$[\ ]_{4}^{2}$$
\(13\) Copy content Toggle raw display 13.4.2.4a1.2$x^{8} + 6 x^{6} + 24 x^{5} + 13 x^{4} + 72 x^{3} + 156 x^{2} + 48 x + 17$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
13.2.4.6a1.3$x^{8} + 48 x^{7} + 872 x^{6} + 7200 x^{5} + 24216 x^{4} + 14400 x^{3} + 3488 x^{2} + 397 x + 159$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$

Spectrum of ring of integers

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