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Properties

Label 16.0.419...000.2
Degree $16$
Signature $[0, 8]$
Discriminant $4.194\times 10^{20}$
Root discriminant \(19.45\)
Ramified primes $2,5$
Class number $2$
Class group [2]
Galois group $C_4\wr C_2$ (as 16T42)

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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 - 10*x^12 + 95*x^8 - 250*x^4 + 625)
 
Copy content gp:K = bnfinit(y^16 - 10*y^12 + 95*y^8 - 250*y^4 + 625, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 10*x^12 + 95*x^8 - 250*x^4 + 625);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 10*x^12 + 95*x^8 - 250*x^4 + 625)
 

\( x^{16} - 10x^{12} + 95x^{8} - 250x^{4} + 625 \) 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:   \(419430400000000000000\) \(\medspace = 2^{36}\cdot 5^{14}\) 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:  \(19.45\)
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:  $2^{9/4}5^{7/8}\approx 19.449849449321462$
Ramified primes:   \(2\), \(5\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2\times C_4$
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:  \(\Q(i, \sqrt{5})\)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{10}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{10}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{10}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{50}a^{11}-\frac{1}{5}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{2}{5}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2100}a^{12}-\frac{1}{20}a^{10}-\frac{1}{42}a^{8}-\frac{1}{4}a^{6}-\frac{53}{210}a^{4}-\frac{1}{4}a^{2}-\frac{23}{84}$, $\frac{1}{2100}a^{13}-\frac{1}{100}a^{11}-\frac{1}{42}a^{9}-\frac{3}{20}a^{7}+\frac{26}{105}a^{5}-\frac{1}{2}a^{4}+\frac{1}{20}a^{3}-\frac{1}{2}a^{2}-\frac{23}{84}a-\frac{1}{2}$, $\frac{1}{10500}a^{14}+\frac{53}{2100}a^{10}-\frac{1}{20}a^{8}+\frac{419}{2100}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{2}{21}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{10500}a^{15}+\frac{11}{2100}a^{11}-\frac{1}{20}a^{9}-\frac{211}{2100}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{32}{105}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{2}$ 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:  No
Index:  Not computed
Inessential primes:  $3$

Class group and class number

Ideal class group:  $C_{2}$, which has order $2$
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 $2$
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:   \( -\frac{1}{210} a^{12} + \frac{4}{105} a^{8} - \frac{10}{21} a^{4} + \frac{31}{42} \)  (order $4$) 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:   $\frac{4}{525}a^{12}-\frac{17}{210}a^{8}+\frac{97}{210}a^{4}-\frac{37}{42}$, $\frac{4}{875}a^{14}-\frac{1}{420}a^{12}-\frac{27}{700}a^{10}+\frac{2}{105}a^{8}+\frac{229}{700}a^{6}-\frac{5}{21}a^{4}-\frac{19}{28}a^{2}-\frac{11}{84}$, $\frac{1}{750}a^{14}-\frac{1}{420}a^{12}+\frac{1}{300}a^{10}+\frac{2}{105}a^{8}+\frac{13}{300}a^{6}-\frac{5}{21}a^{4}+\frac{7}{12}a^{2}-\frac{11}{84}$, $\frac{1}{2100}a^{15}-\frac{17}{10500}a^{14}-\frac{13}{2100}a^{13}-\frac{17}{2100}a^{12}-\frac{2}{525}a^{11}+\frac{11}{525}a^{10}+\frac{5}{84}a^{9}+\frac{23}{420}a^{8}+\frac{1}{21}a^{7}-\frac{149}{1050}a^{6}-\frac{197}{420}a^{5}-\frac{193}{420}a^{4}+\frac{53}{420}a^{3}+\frac{53}{84}a^{2}+\frac{17}{21}a+\frac{17}{42}$, $\frac{1}{1050}a^{15}-\frac{17}{10500}a^{14}-\frac{1}{525}a^{13}+\frac{17}{2100}a^{12}-\frac{4}{525}a^{11}+\frac{11}{525}a^{10}-\frac{1}{210}a^{9}-\frac{23}{420}a^{8}+\frac{2}{21}a^{7}-\frac{149}{1050}a^{6}+\frac{1}{105}a^{5}+\frac{193}{420}a^{4}-\frac{26}{105}a^{3}+\frac{53}{84}a^{2}-\frac{19}{21}a-\frac{17}{42}$, $\frac{1}{2625}a^{15}-\frac{1}{750}a^{14}+\frac{1}{700}a^{13}-\frac{19}{2100}a^{12}+\frac{23}{2100}a^{11}-\frac{1}{300}a^{10}+\frac{1}{35}a^{9}+\frac{11}{210}a^{8}-\frac{109}{2100}a^{7}-\frac{13}{300}a^{6}-\frac{9}{35}a^{5}-\frac{43}{210}a^{4}+\frac{139}{420}a^{3}+\frac{5}{12}a^{2}+\frac{19}{28}a+\frac{17}{84}$, $\frac{1}{2625}a^{15}-\frac{17}{10500}a^{14}-\frac{1}{2100}a^{13}+\frac{1}{84}a^{12}+\frac{23}{2100}a^{11}+\frac{11}{525}a^{10}+\frac{1}{42}a^{9}-\frac{19}{420}a^{8}-\frac{109}{2100}a^{7}-\frac{149}{1050}a^{6}-\frac{26}{105}a^{5}+\frac{37}{84}a^{4}+\frac{349}{420}a^{3}+\frac{53}{84}a^{2}+\frac{149}{84}a+\frac{59}{42}$ 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:  \( 8166.23968032 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 8166.23968032 \cdot 2}{4\cdot\sqrt{419430400000000000000}}\cr\approx \mathstrut & 0.484285115571 \end{aligned}\]

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 - 10*x^12 + 95*x^8 - 250*x^4 + 625) 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 - 10*x^12 + 95*x^8 - 250*x^4 + 625, 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 - 10*x^12 + 95*x^8 - 250*x^4 + 625); 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 - 10*x^12 + 95*x^8 - 250*x^4 + 625); 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

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), 4.0.8000.1 x2, 4.2.2000.1 x2, \(\Q(i, \sqrt{5})\), 8.0.20480000000.2, 8.0.20480000000.3, 8.0.64000000.3

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: 8.0.20480000000.2, 8.0.20480000000.3
Degree 16 sibling: 16.4.419430400000000000000.4
Minimal sibling: 8.0.20480000000.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{4}$ R ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 2.2.8.36b1.31$x^{16} + 8 x^{15} + 36 x^{14} + 112 x^{13} + 268 x^{12} + 516 x^{11} + 830 x^{10} + 1136 x^{9} + 1347 x^{8} + 1388 x^{7} + 1250 x^{6} + 972 x^{5} + 650 x^{4} + 360 x^{3} + 162 x^{2} + 52 x + 17$$8$$2$$36$16T42$$[2, 2, 3]^{4}$$
\(5\) Copy content Toggle raw display 5.1.8.7a1.4$x^{8} + 20$$8$$1$$7$$C_8:C_2$$$[\ ]_{8}^{2}$$
5.1.8.7a1.4$x^{8} + 20$$8$$1$$7$$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)