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Properties

Label 16.0.344...000.1
Degree $16$
Signature $[0, 8]$
Discriminant $3.445\times 10^{18}$
Root discriminant \(14.41\)
Ramified primes $2,5,29$
Class number $1$
Class group trivial
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^15 + 26*x^14 - 42*x^13 + 33*x^12 - 16*x^11 + 21*x^10 - 6*x^9 - 59*x^8 + 96*x^7 - 69*x^6 + 40*x^5 - 10*x^4 - 22*x^3 + 57*x^2 - 42*x + 31)
 
Copy content gp:K = bnfinit(y^16 - 8*y^15 + 26*y^14 - 42*y^13 + 33*y^12 - 16*y^11 + 21*y^10 - 6*y^9 - 59*y^8 + 96*y^7 - 69*y^6 + 40*y^5 - 10*y^4 - 22*y^3 + 57*y^2 - 42*y + 31, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 26*x^14 - 42*x^13 + 33*x^12 - 16*x^11 + 21*x^10 - 6*x^9 - 59*x^8 + 96*x^7 - 69*x^6 + 40*x^5 - 10*x^4 - 22*x^3 + 57*x^2 - 42*x + 31);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 8*x^15 + 26*x^14 - 42*x^13 + 33*x^12 - 16*x^11 + 21*x^10 - 6*x^9 - 59*x^8 + 96*x^7 - 69*x^6 + 40*x^5 - 10*x^4 - 22*x^3 + 57*x^2 - 42*x + 31)
 

\( x^{16} - 8 x^{15} + 26 x^{14} - 42 x^{13} + 33 x^{12} - 16 x^{11} + 21 x^{10} - 6 x^{9} - 59 x^{8} + \cdots + 31 \) 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:   \(3444736000000000000\) \(\medspace = 2^{24}\cdot 5^{12}\cdot 29^{2}\) 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:  \(14.41\)
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:  $2^{3/2}5^{3/4}29^{1/2}\approx 50.92974429446663$
Ramified primes:   \(2\), \(5\), \(29\) 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 not a CM field.
Maximal CM subfield:  8.0.64000000.2

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{279589}a^{14}-\frac{7}{279589}a^{13}-\frac{64064}{279589}a^{12}+\frac{104886}{279589}a^{11}+\frac{33355}{279589}a^{10}-\frac{56639}{279589}a^{9}-\frac{87178}{279589}a^{8}-\frac{3695}{9019}a^{7}+\frac{45361}{279589}a^{6}+\frac{103086}{279589}a^{5}+\frac{120887}{279589}a^{4}-\frac{89908}{279589}a^{3}-\frac{1363}{9641}a^{2}+\frac{44292}{279589}a-\frac{750}{9019}$, $\frac{1}{11463149}a^{15}+\frac{13}{11463149}a^{14}+\frac{215385}{11463149}a^{13}+\frac{3017441}{11463149}a^{12}+\frac{5486143}{11463149}a^{11}+\frac{610461}{11463149}a^{10}-\frac{381191}{11463149}a^{9}-\frac{4933584}{11463149}a^{8}+\frac{2507474}{11463149}a^{7}+\frac{4085785}{11463149}a^{6}-\frac{4527529}{11463149}a^{5}-\frac{747647}{11463149}a^{4}-\frac{1558098}{11463149}a^{3}+\frac{2049642}{11463149}a^{2}+\frac{1142179}{11463149}a+\frac{57152}{369779}$ 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:   \( \frac{2548}{279589} a^{14} - \frac{17836}{279589} a^{13} + \frac{44904}{279589} a^{12} - \frac{37556}{279589} a^{11} - \frac{6516}{279589} a^{10} - \frac{48248}{279589} a^{9} + \frac{143711}{279589} a^{8} + \frac{976}{9019} a^{7} - \frac{170018}{279589} a^{6} - \frac{150532}{279589} a^{5} + \frac{192587}{279589} a^{4} + \frac{177396}{279589} a^{3} - \frac{2164}{9641} a^{2} - \frac{97940}{279589} a + \frac{1028}{9019} \)  (order $10$) 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:   $\frac{890}{279589}a^{14}-\frac{6230}{279589}a^{13}+\frac{19196}{279589}a^{12}-\frac{34186}{279589}a^{11}+\frac{49516}{279589}a^{10}-\frac{82690}{279589}a^{9}+\frac{137322}{279589}a^{8}-\frac{5634}{9019}a^{7}+\frac{110474}{279589}a^{6}+\frac{41348}{279589}a^{5}-\frac{52335}{279589}a^{4}-\frac{55666}{279589}a^{3}-\frac{7945}{9641}a^{2}+\frac{277420}{279589}a-\frac{9113}{9019}$, $\frac{231383}{11463149}a^{15}-\frac{1572500}{11463149}a^{14}+\frac{3792658}{11463149}a^{13}-\frac{2809835}{11463149}a^{12}-\frac{2050848}{11463149}a^{11}+\frac{348412}{11463149}a^{10}+\frac{7576315}{11463149}a^{9}-\frac{3033809}{11463149}a^{8}-\frac{6329619}{11463149}a^{7}-\frac{1702818}{11463149}a^{6}+\frac{7793619}{11463149}a^{5}+\frac{4623330}{11463149}a^{4}-\frac{4738526}{11463149}a^{3}+\frac{2532585}{11463149}a^{2}-\frac{5861504}{11463149}a+\frac{76958}{369779}$, $\frac{201276}{11463149}a^{15}-\frac{1261110}{11463149}a^{14}+\frac{2419330}{11463149}a^{13}+\frac{837091}{11463149}a^{12}-\frac{8467461}{11463149}a^{11}+\frac{232601}{369779}a^{10}+\frac{4721872}{11463149}a^{9}-\frac{3615676}{11463149}a^{8}-\frac{8367390}{11463149}a^{7}-\frac{1548322}{11463149}a^{6}+\frac{21048098}{11463149}a^{5}-\frac{18829867}{11463149}a^{4}+\frac{8418541}{11463149}a^{3}+\frac{7940547}{11463149}a^{2}-\frac{10334289}{11463149}a+\frac{202485}{369779}$, $\frac{458778}{11463149}a^{15}-\frac{3346125}{11463149}a^{14}+\frac{9411078}{11463149}a^{13}-\frac{11059359}{11463149}a^{12}+\frac{1486029}{11463149}a^{11}+\frac{3568004}{11463149}a^{10}+\frac{6066118}{11463149}a^{9}+\frac{1646593}{11463149}a^{8}-\frac{31688934}{11463149}a^{7}+\frac{29193578}{11463149}a^{6}+\frac{214858}{11463149}a^{5}-\frac{143266}{395281}a^{4}+\frac{862832}{11463149}a^{3}-\frac{10036686}{11463149}a^{2}+\frac{20809661}{11463149}a-\frac{66683}{369779}$, $\frac{313883}{11463149}a^{15}-\frac{2422285}{11463149}a^{14}+\frac{7167054}{11463149}a^{13}-\frac{8477635}{11463149}a^{12}-\frac{1685213}{11463149}a^{11}+\frac{12806686}{11463149}a^{10}-\frac{9415714}{11463149}a^{9}+\frac{10967382}{11463149}a^{8}-\frac{34131587}{11463149}a^{7}+\frac{1360598}{395281}a^{6}-\frac{6779761}{11463149}a^{5}-\frac{14285966}{11463149}a^{4}+\frac{10294816}{11463149}a^{3}-\frac{13431989}{11463149}a^{2}+\frac{26171066}{11463149}a-\frac{359421}{369779}$, $\frac{318991}{11463149}a^{15}-\frac{2577937}{11463149}a^{14}+\frac{8445722}{11463149}a^{13}-\frac{428927}{369779}a^{12}+\frac{8150227}{11463149}a^{11}+\frac{118171}{11463149}a^{10}+\frac{5614468}{11463149}a^{9}-\frac{8102030}{11463149}a^{8}-\frac{20151790}{11463149}a^{7}+\frac{38678548}{11463149}a^{6}-\frac{11182623}{11463149}a^{5}-\frac{1563790}{11463149}a^{4}-\frac{8702221}{11463149}a^{3}-\frac{1817163}{11463149}a^{2}+\frac{14292898}{11463149}a-\frac{410965}{369779}$, $\frac{113778}{11463149}a^{15}-\frac{1101180}{11463149}a^{14}+\frac{4460877}{11463149}a^{13}-\frac{9027365}{11463149}a^{12}+\frac{7757163}{11463149}a^{11}+\frac{1428189}{11463149}a^{10}-\frac{4471666}{11463149}a^{9}-\frac{2342615}{11463149}a^{8}-\frac{5533126}{11463149}a^{7}+\frac{24325787}{11463149}a^{6}-\frac{14945837}{11463149}a^{5}-\frac{9468125}{11463149}a^{4}+\frac{9651680}{11463149}a^{3}-\frac{9954644}{11463149}a^{2}+\frac{9798880}{11463149}a+\frac{243534}{369779}$ 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:  \( 1674.03195575 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1674.03195575 \cdot 1}{10\cdot\sqrt{3444736000000000000}}\cr\approx \mathstrut & 0.219091091504 \end{aligned}\]

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 - 8*x^15 + 26*x^14 - 42*x^13 + 33*x^12 - 16*x^11 + 21*x^10 - 6*x^9 - 59*x^8 + 96*x^7 - 69*x^6 + 40*x^5 - 10*x^4 - 22*x^3 + 57*x^2 - 42*x + 31) 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^15 + 26*x^14 - 42*x^13 + 33*x^12 - 16*x^11 + 21*x^10 - 6*x^9 - 59*x^8 + 96*x^7 - 69*x^6 + 40*x^5 - 10*x^4 - 22*x^3 + 57*x^2 - 42*x + 31, 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^15 + 26*x^14 - 42*x^13 + 33*x^12 - 16*x^11 + 21*x^10 - 6*x^9 - 59*x^8 + 96*x^7 - 69*x^6 + 40*x^5 - 10*x^4 - 22*x^3 + 57*x^2 - 42*x + 31); 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^15 + 26*x^14 - 42*x^13 + 33*x^12 - 16*x^11 + 21*x^10 - 6*x^9 - 59*x^8 + 96*x^7 - 69*x^6 + 40*x^5 - 10*x^4 - 22*x^3 + 57*x^2 - 42*x + 31); 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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), \(\Q(\zeta_{5})\), 4.0.8000.2, \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.64000000.2

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: 16.0.707281000000000000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }^{2}$ R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ R ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{12}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 2.8.2.24a1.41$x^{16} + 2 x^{12} + 2 x^{11} + 2 x^{10} + 7 x^{8} + 2 x^{7} + 3 x^{6} + 2 x^{5} + 7 x^{4} + 6 x^{3} + 6 x^{2} + 7$$2$$8$$24$$C_8\times C_2$$$[3]^{8}$$
\(5\) Copy content Toggle raw display 5.4.4.12a1.4$x^{16} + 16 x^{14} + 16 x^{13} + 104 x^{12} + 192 x^{11} + 448 x^{10} + 864 x^{9} + 1432 x^{8} + 2048 x^{7} + 2624 x^{6} + 2752 x^{5} + 2208 x^{4} + 1280 x^{3} + 512 x^{2} + 128 x + 21$$4$$4$$12$$C_4^2$$$[\ ]_{4}^{4}$$
\(29\) Copy content Toggle raw display 29.4.1.0a1.1$x^{4} + 2 x^{2} + 15 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
29.4.1.0a1.1$x^{4} + 2 x^{2} + 15 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
29.2.2.2a1.1$x^{4} + 48 x^{3} + 580 x^{2} + 125 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
29.4.1.0a1.1$x^{4} + 2 x^{2} + 15 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$

Spectrum of ring of integers

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