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Properties

Label 16.0.317...504.1
Degree $16$
Signature $[0, 8]$
Discriminant $3.179\times 10^{24}$
Root discriminant \(33.99\)
Ramified primes $2,3,41$
Class number $34$ (GRH)
Class group [34] (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 + 8*x^14 - 24*x^13 + 88*x^12 - 120*x^11 + 648*x^10 - 880*x^9 + 1766*x^8 - 3760*x^7 + 2888*x^6 - 7824*x^5 + 6372*x^4 - 4512*x^3 + 13480*x^2 + 2720*x + 9406)
 
Copy content gp:K = bnfinit(y^16 + 8*y^14 - 24*y^13 + 88*y^12 - 120*y^11 + 648*y^10 - 880*y^9 + 1766*y^8 - 3760*y^7 + 2888*y^6 - 7824*y^5 + 6372*y^4 - 4512*y^3 + 13480*y^2 + 2720*y + 9406, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 8*x^14 - 24*x^13 + 88*x^12 - 120*x^11 + 648*x^10 - 880*x^9 + 1766*x^8 - 3760*x^7 + 2888*x^6 - 7824*x^5 + 6372*x^4 - 4512*x^3 + 13480*x^2 + 2720*x + 9406);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 + 8*x^14 - 24*x^13 + 88*x^12 - 120*x^11 + 648*x^10 - 880*x^9 + 1766*x^8 - 3760*x^7 + 2888*x^6 - 7824*x^5 + 6372*x^4 - 4512*x^3 + 13480*x^2 + 2720*x + 9406)
 

\( x^{16} + 8 x^{14} - 24 x^{13} + 88 x^{12} - 120 x^{11} + 648 x^{10} - 880 x^{9} + 1766 x^{8} + \cdots + 9406 \) 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:   \(3178904636022651044757504\) \(\medspace = 2^{58}\cdot 3^{8}\cdot 41^{2}\) 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:  \(33.99\)
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:  $2^{61/16}3^{1/2}41^{1/2}\approx 155.82222959264388$
Ramified primes:   \(2\), \(3\), \(41\) 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 a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{17\cdots 21}a^{15}+\frac{24\cdots 54}{17\cdots 21}a^{14}+\frac{26\cdots 12}{17\cdots 21}a^{13}-\frac{66\cdots 18}{17\cdots 21}a^{12}-\frac{59\cdots 87}{17\cdots 21}a^{11}+\frac{10\cdots 45}{17\cdots 21}a^{10}-\frac{76\cdots 97}{17\cdots 21}a^{9}+\frac{36\cdots 16}{17\cdots 21}a^{8}-\frac{42\cdots 05}{17\cdots 21}a^{7}-\frac{70\cdots 75}{17\cdots 21}a^{6}-\frac{57\cdots 22}{17\cdots 21}a^{5}-\frac{51\cdots 99}{17\cdots 21}a^{4}-\frac{77\cdots 42}{17\cdots 21}a^{3}-\frac{52\cdots 05}{17\cdots 21}a^{2}+\frac{65\cdots 81}{17\cdots 21}a+\frac{74\cdots 80}{17\cdots 21}$ 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:  $C_{34}$, which has order $34$ (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:  $C_{34}$, which has order $34$ (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:   $34$ (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:   $\frac{97\cdots 38}{89\cdots 21}a^{15}+\frac{12\cdots 38}{89\cdots 21}a^{14}+\frac{32\cdots 74}{89\cdots 21}a^{13}-\frac{16\cdots 86}{89\cdots 21}a^{12}+\frac{36\cdots 50}{89\cdots 21}a^{11}+\frac{76\cdots 66}{89\cdots 21}a^{10}+\frac{24\cdots 08}{89\cdots 21}a^{9}-\frac{13\cdots 01}{89\cdots 21}a^{8}-\frac{71\cdots 84}{89\cdots 21}a^{7}-\frac{48\cdots 64}{89\cdots 21}a^{6}+\frac{43\cdots 24}{89\cdots 21}a^{5}-\frac{31\cdots 39}{89\cdots 21}a^{4}+\frac{47\cdots 44}{89\cdots 21}a^{3}-\frac{24\cdots 46}{89\cdots 21}a^{2}+\frac{63\cdots 60}{89\cdots 21}a-\frac{11\cdots 85}{89\cdots 21}$, $\frac{11\cdots 81}{17\cdots 21}a^{15}-\frac{71\cdots 50}{17\cdots 21}a^{14}+\frac{91\cdots 58}{17\cdots 21}a^{13}-\frac{31\cdots 07}{17\cdots 21}a^{12}+\frac{11\cdots 95}{17\cdots 21}a^{11}-\frac{17\cdots 96}{17\cdots 21}a^{10}+\frac{74\cdots 20}{17\cdots 21}a^{9}-\frac{11\cdots 62}{17\cdots 21}a^{8}+\frac{20\cdots 10}{17\cdots 21}a^{7}-\frac{37\cdots 83}{17\cdots 21}a^{6}+\frac{22\cdots 14}{17\cdots 21}a^{5}-\frac{50\cdots 56}{17\cdots 21}a^{4}+\frac{34\cdots 26}{17\cdots 21}a^{3}+\frac{10\cdots 81}{17\cdots 21}a^{2}+\frac{50\cdots 98}{17\cdots 21}a+\frac{68\cdots 51}{17\cdots 21}$, $\frac{40\cdots 04}{17\cdots 21}a^{15}+\frac{25\cdots 92}{17\cdots 21}a^{14}+\frac{26\cdots 80}{17\cdots 21}a^{13}-\frac{77\cdots 13}{17\cdots 21}a^{12}+\frac{26\cdots 25}{17\cdots 21}a^{11}-\frac{14\cdots 01}{17\cdots 21}a^{10}+\frac{19\cdots 58}{17\cdots 21}a^{9}-\frac{15\cdots 94}{17\cdots 21}a^{8}+\frac{23\cdots 72}{17\cdots 21}a^{7}-\frac{80\cdots 62}{17\cdots 21}a^{6}-\frac{17\cdots 48}{17\cdots 21}a^{5}-\frac{16\cdots 59}{17\cdots 21}a^{4}+\frac{11\cdots 64}{17\cdots 21}a^{3}+\frac{14\cdots 27}{17\cdots 21}a^{2}+\frac{32\cdots 14}{17\cdots 21}a+\frac{31\cdots 45}{17\cdots 21}$, $\frac{47\cdots 73}{10\cdots 29}a^{15}-\frac{58\cdots 58}{10\cdots 29}a^{14}+\frac{39\cdots 22}{10\cdots 29}a^{13}-\frac{14\cdots 06}{10\cdots 29}a^{12}+\frac{54\cdots 30}{10\cdots 29}a^{11}-\frac{98\cdots 55}{10\cdots 29}a^{10}+\frac{33\cdots 38}{10\cdots 29}a^{9}-\frac{62\cdots 32}{10\cdots 29}a^{8}+\frac{10\cdots 62}{10\cdots 29}a^{7}-\frac{17\cdots 29}{10\cdots 29}a^{6}+\frac{14\cdots 38}{10\cdots 29}a^{5}-\frac{20\cdots 53}{10\cdots 29}a^{4}+\frac{13\cdots 38}{10\cdots 29}a^{3}-\frac{25\cdots 54}{10\cdots 29}a^{2}+\frac{11\cdots 16}{10\cdots 29}a+\frac{12\cdots 65}{10\cdots 29}$, $\frac{98\cdots 97}{10\cdots 29}a^{15}-\frac{35\cdots 89}{10\cdots 29}a^{14}+\frac{66\cdots 64}{10\cdots 29}a^{13}-\frac{25\cdots 03}{10\cdots 29}a^{12}+\frac{86\cdots 14}{10\cdots 29}a^{11}-\frac{11\cdots 21}{10\cdots 29}a^{10}+\frac{55\cdots 82}{10\cdots 29}a^{9}-\frac{87\cdots 14}{10\cdots 29}a^{8}+\frac{12\cdots 52}{10\cdots 29}a^{7}-\frac{27\cdots 25}{10\cdots 29}a^{6}+\frac{16\cdots 46}{10\cdots 29}a^{5}-\frac{37\cdots 39}{10\cdots 29}a^{4}+\frac{36\cdots 90}{10\cdots 29}a^{3}+\frac{97\cdots 78}{10\cdots 29}a^{2}+\frac{51\cdots 60}{10\cdots 29}a+\frac{23\cdots 23}{10\cdots 29}$, $\frac{13\cdots 95}{17\cdots 21}a^{15}-\frac{55\cdots 98}{17\cdots 21}a^{14}+\frac{86\cdots 74}{17\cdots 21}a^{13}-\frac{35\cdots 24}{17\cdots 21}a^{12}+\frac{11\cdots 45}{17\cdots 21}a^{11}-\frac{15\cdots 14}{17\cdots 21}a^{10}+\frac{73\cdots 84}{17\cdots 21}a^{9}-\frac{12\cdots 71}{17\cdots 21}a^{8}+\frac{16\cdots 06}{17\cdots 21}a^{7}-\frac{36\cdots 45}{17\cdots 21}a^{6}+\frac{26\cdots 06}{17\cdots 21}a^{5}-\frac{49\cdots 98}{17\cdots 21}a^{4}+\frac{56\cdots 26}{17\cdots 21}a^{3}+\frac{79\cdots 35}{17\cdots 21}a^{2}+\frac{69\cdots 74}{17\cdots 21}a-\frac{88\cdots 25}{17\cdots 21}$, $\frac{36\cdots 13}{17\cdots 21}a^{15}-\frac{34\cdots 72}{17\cdots 21}a^{14}+\frac{23\cdots 96}{17\cdots 21}a^{13}-\frac{84\cdots 00}{17\cdots 21}a^{12}+\frac{28\cdots 63}{17\cdots 21}a^{11}-\frac{28\cdots 61}{17\cdots 21}a^{10}+\frac{18\cdots 22}{17\cdots 21}a^{9}-\frac{25\cdots 97}{17\cdots 21}a^{8}+\frac{33\cdots 54}{17\cdots 21}a^{7}-\frac{86\cdots 97}{17\cdots 21}a^{6}+\frac{41\cdots 78}{17\cdots 21}a^{5}-\frac{13\cdots 65}{17\cdots 21}a^{4}+\frac{13\cdots 56}{17\cdots 21}a^{3}+\frac{55\cdots 57}{17\cdots 21}a^{2}+\frac{22\cdots 02}{17\cdots 21}a+\frac{70\cdots 93}{17\cdots 21}$ 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:  \( 11964.3106427 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 11964.3106427 \cdot 34}{2\cdot\sqrt{3178904636022651044757504}}\cr\approx \mathstrut & 0.277100174304 \end{aligned}\] (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 - 24*x^13 + 88*x^12 - 120*x^11 + 648*x^10 - 880*x^9 + 1766*x^8 - 3760*x^7 + 2888*x^6 - 7824*x^5 + 6372*x^4 - 4512*x^3 + 13480*x^2 + 2720*x + 9406) 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 - 24*x^13 + 88*x^12 - 120*x^11 + 648*x^10 - 880*x^9 + 1766*x^8 - 3760*x^7 + 2888*x^6 - 7824*x^5 + 6372*x^4 - 4512*x^3 + 13480*x^2 + 2720*x + 9406, 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 - 24*x^13 + 88*x^12 - 120*x^11 + 648*x^10 - 880*x^9 + 1766*x^8 - 3760*x^7 + 2888*x^6 - 7824*x^5 + 6372*x^4 - 4512*x^3 + 13480*x^2 + 2720*x + 9406); 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 + 8*x^14 - 24*x^13 + 88*x^12 - 120*x^11 + 648*x^10 - 880*x^9 + 1766*x^8 - 3760*x^7 + 2888*x^6 - 7824*x^5 + 6372*x^4 - 4512*x^3 + 13480*x^2 + 2720*x + 9406); 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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), \(\Q(\zeta_{16})^+\), 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: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.0.16493020657882951871102976.1

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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ R ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(2\) Copy content Toggle raw display 2.1.16.58n1.725$x^{16} + 8 x^{15} + 4 x^{12} + 8 x^{11} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 30$$16$$1$$58$16T112$$[2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$$
\(3\) Copy content Toggle raw display 3.8.2.8a1.2$x^{16} + 4 x^{13} + 2 x^{12} + 8 x^{10} + 8 x^{9} + 5 x^{8} + 8 x^{7} + 12 x^{6} + 12 x^{5} + 8 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + 7$$2$$8$$8$$C_8\times C_2$$$[\ ]_{2}^{8}$$
\(41\) Copy content Toggle raw display 41.4.1.0a1.1$x^{4} + 23 x + 6$$1$$4$$0$$C_4$$$[\ ]^{4}$$
41.4.1.0a1.1$x^{4} + 23 x + 6$$1$$4$$0$$C_4$$$[\ ]^{4}$$
41.4.1.0a1.1$x^{4} + 23 x + 6$$1$$4$$0$$C_4$$$[\ ]^{4}$$
41.2.2.2a1.1$x^{4} + 76 x^{3} + 1456 x^{2} + 497 x + 36$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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