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Properties

Label 16.0.605...000.7
Degree $16$
Signature $[0, 8]$
Discriminant $6.052\times 10^{23}$
Root discriminant \(30.65\)
Ramified primes $2,3,5,7$
Class number $8$ (GRH)
Class group [2, 2, 2] (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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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 - 19*x^14 + 240*x^12 - 1741*x^10 + 9219*x^8 - 29161*x^6 + 63200*x^4 - 33759*x^2 + 14641)
 
Copy content gp:K = bnfinit(y^16 - 19*y^14 + 240*y^12 - 1741*y^10 + 9219*y^8 - 29161*y^6 + 63200*y^4 - 33759*y^2 + 14641, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 19*x^14 + 240*x^12 - 1741*x^10 + 9219*x^8 - 29161*x^6 + 63200*x^4 - 33759*x^2 + 14641);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 19*x^14 + 240*x^12 - 1741*x^10 + 9219*x^8 - 29161*x^6 + 63200*x^4 - 33759*x^2 + 14641)
 

\( x^{16} - 19x^{14} + 240x^{12} - 1741x^{10} + 9219x^{8} - 29161x^{6} + 63200x^{4} - 33759x^{2} + 14641 \) 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:   \(605165749776000000000000\) \(\medspace = 2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 7^{8}\) 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:  \(30.65\)
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:  $2\cdot 3^{1/2}5^{3/4}7^{1/2}\approx 30.645530678223075$
Ramified primes:   \(2\), \(3\), \(5\), \(7\) 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:  \(\Q\)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_2^2\times C_4$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(420=2^{2}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{420}(1,·)$, $\chi_{420}(83,·)$, $\chi_{420}(71,·)$, $\chi_{420}(13,·)$, $\chi_{420}(211,·)$, $\chi_{420}(281,·)$, $\chi_{420}(29,·)$, $\chi_{420}(223,·)$, $\chi_{420}(97,·)$, $\chi_{420}(293,·)$, $\chi_{420}(167,·)$, $\chi_{420}(169,·)$, $\chi_{420}(239,·)$, $\chi_{420}(307,·)$, $\chi_{420}(377,·)$, $\chi_{420}(379,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{11}a^{9}+\frac{1}{11}a^{7}+\frac{3}{11}a^{5}-\frac{2}{11}a^{3}+\frac{4}{11}a$, $\frac{1}{11}a^{10}+\frac{1}{11}a^{8}+\frac{3}{11}a^{6}-\frac{2}{11}a^{4}+\frac{4}{11}a^{2}$, $\frac{1}{11}a^{11}+\frac{2}{11}a^{7}-\frac{5}{11}a^{5}-\frac{5}{11}a^{3}-\frac{4}{11}a$, $\frac{1}{11}a^{12}+\frac{2}{11}a^{8}-\frac{5}{11}a^{6}-\frac{5}{11}a^{4}-\frac{4}{11}a^{2}$, $\frac{1}{11}a^{13}+\frac{4}{11}a^{7}+\frac{3}{11}a$, $\frac{1}{11137467154979}a^{14}-\frac{399578710000}{11137467154979}a^{12}+\frac{124755454976}{11137467154979}a^{10}+\frac{4072809174926}{11137467154979}a^{8}-\frac{3879007553519}{11137467154979}a^{6}+\frac{494941678642}{1012497014089}a^{4}+\frac{3589762146041}{11137467154979}a^{2}-\frac{10630726872}{92045183099}$, $\frac{1}{122512138704769}a^{15}+\frac{2637912332267}{122512138704769}a^{13}+\frac{124755454976}{122512138704769}a^{11}-\frac{989675895519}{122512138704769}a^{9}-\frac{7928995609875}{122512138704769}a^{7}-\frac{356657010010}{1012497014089}a^{5}+\frac{2577265131952}{122512138704769}a^{3}-\frac{470856642367}{1012497014089}a$ 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:  $C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  $C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:   $4$ (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:   \( -\frac{1913329}{92045183099} a^{15} + \frac{346706790}{1012497014089} a^{13} - \frac{4150184540}{1012497014089} a^{11} + \frac{25395094000}{1012497014089} a^{9} - \frac{115409043000}{1012497014089} a^{7} + \frac{202993944499}{1012497014089} a^{5} - \frac{9911044220}{92045183099} a^{3} - \frac{1313812737840}{1012497014089} a \)  (order $12$) 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:   $\frac{2211}{21019681}a^{14}-\frac{432308}{231216491}a^{12}+\frac{4795860}{231216491}a^{10}-\frac{29346000}{231216491}a^{8}+\frac{116003123}{231216491}a^{6}-\frac{234575241}{231216491}a^{4}+\frac{11452980}{21019681}a^{2}+\frac{8179387}{21019681}$, $\frac{1913329}{92045183099}a^{15}-\frac{346706790}{1012497014089}a^{13}+\frac{4150184540}{1012497014089}a^{11}-\frac{25395094000}{1012497014089}a^{9}+\frac{115409043000}{1012497014089}a^{7}-\frac{202993944499}{1012497014089}a^{5}+\frac{9911044220}{92045183099}a^{3}+\frac{1313812737840}{1012497014089}a-1$, $\frac{1977105}{25359581599}a^{15}+\frac{700972489}{11137467154979}a^{14}-\frac{33840010}{25359581599}a^{13}-\frac{12232808872}{11137467154979}a^{12}+\frac{426869315}{25359581599}a^{11}+\frac{123771447179}{11137467154979}a^{10}-\frac{2974352111}{25359581599}a^{9}-\frac{663232557440}{11137467154979}a^{8}+\frac{16495010200}{25359581599}a^{7}+\frac{1671303830347}{11137467154979}a^{6}-\frac{39480061}{19053029}a^{5}+\frac{31283527881}{1012497014089}a^{4}+\frac{127569956482}{25359581599}a^{3}-\frac{20300710112720}{11137467154979}a^{2}-\frac{392145995}{209583319}a+\frac{152144831913}{92045183099}$, $\frac{1750368606}{11137467154979}a^{14}-\frac{22240823625}{11137467154979}a^{12}+\frac{221921283282}{11137467154979}a^{10}-\frac{875110320306}{11137467154979}a^{8}+\frac{2844398064714}{11137467154979}a^{6}-\frac{58154153858}{1012497014089}a^{4}+\frac{4372605041031}{11137467154979}a^{2}-\frac{16751090454}{92045183099}$, $\frac{4816075}{92045183099}a^{15}-\frac{920857381}{1012497014089}a^{13}+\frac{10446504500}{1012497014089}a^{11}-\frac{63922450000}{1012497014089}a^{9}+\frac{260228515359}{1012497014089}a^{7}-\frac{510959725825}{1012497014089}a^{5}+\frac{24947268500}{92045183099}a^{3}-\frac{608145287463}{1012497014089}a$, $\frac{8554798408}{122512138704769}a^{15}-\frac{7237445019}{11137467154979}a^{14}+\frac{78480172205}{122512138704769}a^{13}+\frac{87580041851}{11137467154979}a^{12}-\frac{1678433057976}{122512138704769}a^{11}-\frac{831521155263}{11137467154979}a^{10}+\frac{23980867099986}{122512138704769}a^{9}+\frac{2754412221479}{11137467154979}a^{8}-\frac{125522026428687}{122512138704769}a^{7}-\frac{6474184370161}{11137467154979}a^{6}+\frac{3892363270514}{1012497014089}a^{5}-\frac{1465975132089}{1012497014089}a^{4}-\frac{578279762728196}{122512138704769}a^{3}+\frac{24180307477910}{11137467154979}a^{2}+\frac{1764287706595}{1012497014089}a-\frac{118312525499}{92045183099}$, $\frac{26282437176}{122512138704769}a^{15}-\frac{5848164616}{11137467154979}a^{14}-\frac{459017646465}{122512138704769}a^{13}+\frac{102997373081}{11137467154979}a^{12}+\frac{5052555576111}{122512138704769}a^{11}-\frac{1147114432008}{11137467154979}a^{10}-\frac{30069572166960}{122512138704769}a^{9}+\frac{7032881266611}{11137467154979}a^{8}+\frac{113533319175414}{122512138704769}a^{7}-\frac{27869094253426}{11137467154979}a^{6}-\frac{150778307485}{92045183099}a^{5}+\frac{5190068062805}{1012497014089}a^{4}+\frac{32131967669810}{122512138704769}a^{3}-\frac{25599188372831}{11137467154979}a^{2}-\frac{542992106839}{1012497014089}a+\frac{70794655219}{92045183099}$ 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:  \( 76884.6053994 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 76884.6053994 \cdot 8}{12\cdot\sqrt{605165749776000000000000}}\cr\approx \mathstrut & 0.160047873165 \end{aligned}\] (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 - 19*x^14 + 240*x^12 - 1741*x^10 + 9219*x^8 - 29161*x^6 + 63200*x^4 - 33759*x^2 + 14641) 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 - 19*x^14 + 240*x^12 - 1741*x^10 + 9219*x^8 - 29161*x^6 + 63200*x^4 - 33759*x^2 + 14641, 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 - 19*x^14 + 240*x^12 - 1741*x^10 + 9219*x^8 - 29161*x^6 + 63200*x^4 - 33759*x^2 + 14641); 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 - 19*x^14 + 240*x^12 - 1741*x^10 + 9219*x^8 - 29161*x^6 + 63200*x^4 - 33759*x^2 + 14641); 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

$C_2^2\times C_4$ (as 16T2):

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)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{15}) \), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{15})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{-5})\), 4.4.882000.1, 4.0.55125.1, 4.0.98000.1, 4.4.6125.1, 8.0.12960000.1, 8.0.777924000000.8, 8.0.9604000000.1, 8.0.777924000000.9, 8.0.3038765625.1, 8.8.777924000000.1, 8.0.777924000000.5

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]
 

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 R R ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 2.4.2.8a1.1$x^{8} + 2 x^{5} + 4 x^{4} + x^{2} + 4 x + 5$$2$$4$$8$$C_4\times C_2$$$[2]^{4}$$
2.4.2.8a1.1$x^{8} + 2 x^{5} + 4 x^{4} + x^{2} + 4 x + 5$$2$$4$$8$$C_4\times C_2$$$[2]^{4}$$
\(3\) Copy content Toggle raw display 3.4.2.4a1.2$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
3.4.2.4a1.2$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(5\) Copy content Toggle raw display 5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
\(7\) Copy content Toggle raw display 7.2.2.2a1.1$x^{4} + 12 x^{3} + 42 x^{2} + 43 x + 9$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
7.2.2.2a1.1$x^{4} + 12 x^{3} + 42 x^{2} + 43 x + 9$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
7.2.2.2a1.1$x^{4} + 12 x^{3} + 42 x^{2} + 43 x + 9$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
7.2.2.2a1.1$x^{4} + 12 x^{3} + 42 x^{2} + 43 x + 9$$2$$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)