Properties

Label 16.0.957...944.1
Degree $16$
Signature $(0, 8)$
Discriminant $9.574\times 10^{21}$
Root discriminant \(23.65\)
Ramified primes $2,3$
Class number $2$
Class group [2]
Galois group $C_4:C_4$ (as 16T8)

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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 + 16*x^14 - 8*x^13 + 100*x^12 - 48*x^11 + 280*x^10 - 144*x^9 + 388*x^8 + 32*x^7 + 160*x^6 - 224*x^5 + 272*x^4 - 32*x^3 - 320*x^2 - 128*x + 184)
 
Copy content gp:K = bnfinit(y^16 + 16*y^14 - 8*y^13 + 100*y^12 - 48*y^11 + 280*y^10 - 144*y^9 + 388*y^8 + 32*y^7 + 160*y^6 - 224*y^5 + 272*y^4 - 32*y^3 - 320*y^2 - 128*y + 184, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 16*x^14 - 8*x^13 + 100*x^12 - 48*x^11 + 280*x^10 - 144*x^9 + 388*x^8 + 32*x^7 + 160*x^6 - 224*x^5 + 272*x^4 - 32*x^3 - 320*x^2 - 128*x + 184);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 + 16*x^14 - 8*x^13 + 100*x^12 - 48*x^11 + 280*x^10 - 144*x^9 + 388*x^8 + 32*x^7 + 160*x^6 - 224*x^5 + 272*x^4 - 32*x^3 - 320*x^2 - 128*x + 184)
 

\( x^{16} + 16 x^{14} - 8 x^{13} + 100 x^{12} - 48 x^{11} + 280 x^{10} - 144 x^{9} + 388 x^{8} + 32 x^{7} + \cdots + 184 \) 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:   \(9573589958277615058944\) \(\medspace = 2^{54}\cdot 3^{12}\) 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:  \(23.65\)
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:  $2^{27/8}3^{3/4}\approx 23.64923933252054$
Ramified primes:   \(2\), \(3\) 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:  \(\Q\)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_4:C_4$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{28}a^{12}-\frac{1}{14}a^{11}-\frac{1}{14}a^{10}-\frac{1}{14}a^{9}+\frac{3}{14}a^{8}+\frac{1}{14}a^{7}-\frac{3}{14}a^{6}-\frac{1}{7}a^{5}+\frac{3}{7}a^{4}+\frac{2}{7}a^{3}+\frac{1}{7}a^{2}-\frac{2}{7}a+\frac{2}{7}$, $\frac{1}{28}a^{13}+\frac{1}{28}a^{11}-\frac{3}{14}a^{10}+\frac{1}{14}a^{9}-\frac{1}{14}a^{7}-\frac{1}{14}a^{6}+\frac{1}{7}a^{5}+\frac{1}{7}a^{4}-\frac{2}{7}a^{3}-\frac{2}{7}a-\frac{3}{7}$, $\frac{1}{231196}a^{14}+\frac{3837}{231196}a^{13}-\frac{1020}{57799}a^{12}+\frac{3245}{57799}a^{11}-\frac{473}{2513}a^{10}+\frac{8278}{57799}a^{9}-\frac{1831}{57799}a^{8}-\frac{1163}{57799}a^{7}+\frac{5195}{115598}a^{6}+\frac{25433}{57799}a^{5}+\frac{14692}{57799}a^{4}-\frac{17978}{57799}a^{3}+\frac{9119}{57799}a^{2}+\frac{14317}{57799}a-\frac{1020}{2513}$, $\frac{1}{79957055836}a^{15}-\frac{44963}{39978527918}a^{14}+\frac{3689138}{19989263959}a^{13}+\frac{805910667}{79957055836}a^{12}-\frac{420580511}{11422436548}a^{11}+\frac{1499864657}{39978527918}a^{10}+\frac{5513955425}{39978527918}a^{9}-\frac{4899746393}{39978527918}a^{8}-\frac{1023245829}{19989263959}a^{7}-\frac{1478288125}{19989263959}a^{6}-\frac{4442972904}{19989263959}a^{5}+\frac{1390677775}{2855609137}a^{4}-\frac{4700696317}{19989263959}a^{3}-\frac{3065797459}{19989263959}a^{2}-\frac{7331940533}{19989263959}a+\frac{424030279}{869098433}$ 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:  $C_{2}$, 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);
 
Relative class number:   $2$

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:   $\frac{168342565}{39978527918}a^{15}-\frac{192588412}{19989263959}a^{14}+\frac{2115957285}{39978527918}a^{13}-\frac{15683190015}{79957055836}a^{12}+\frac{5624121312}{19989263959}a^{11}-\frac{45797770353}{39978527918}a^{10}+\frac{10307122472}{19989263959}a^{9}-\frac{59740283647}{19989263959}a^{8}+\frac{2282632184}{2855609137}a^{7}-\frac{30320501013}{19989263959}a^{6}-\frac{1417188166}{2855609137}a^{5}-\frac{38908573222}{19989263959}a^{4}+\frac{42216283462}{19989263959}a^{3}+\frac{52110999298}{19989263959}a^{2}-\frac{6183601184}{19989263959}a-\frac{1272116677}{869098433}$, $\frac{1001966379}{79957055836}a^{15}+\frac{1107071771}{79957055836}a^{14}+\frac{16066206255}{79957055836}a^{13}+\frac{8785760469}{79957055836}a^{12}+\frac{45328543285}{39978527918}a^{11}+\frac{1712539033}{2855609137}a^{10}+\frac{110381091497}{39978527918}a^{9}+\frac{22375957062}{19989263959}a^{8}+\frac{49221187422}{19989263959}a^{7}+\frac{156039458597}{39978527918}a^{6}+\frac{50748487620}{19989263959}a^{5}-\frac{6933642022}{19989263959}a^{4}-\frac{184406120}{2855609137}a^{3}+\frac{69804638011}{19989263959}a^{2}-\frac{21661204094}{19989263959}a-\frac{1955494519}{869098433}$, $\frac{2135732833}{79957055836}a^{15}+\frac{1377977651}{79957055836}a^{14}+\frac{97168485}{222721604}a^{13}+\frac{5638326137}{79957055836}a^{12}+\frac{106993222067}{39978527918}a^{11}+\frac{20363130499}{39978527918}a^{10}+\frac{299633340199}{39978527918}a^{9}+\frac{4181416915}{2855609137}a^{8}+\frac{203200638024}{19989263959}a^{7}+\frac{361164348501}{39978527918}a^{6}+\frac{160007629294}{19989263959}a^{5}+\frac{31291636573}{19989263959}a^{4}+\frac{146628770562}{19989263959}a^{3}+\frac{87046416123}{19989263959}a^{2}-\frac{92812878556}{19989263959}a-\frac{5024526363}{869098433}$, $\frac{7116027}{563077858}a^{15}+\frac{5894879}{1126155716}a^{14}+\frac{31384889}{160879388}a^{13}-\frac{26713501}{1126155716}a^{12}+\frac{626658383}{563077858}a^{11}-\frac{32374108}{281538929}a^{10}+\frac{764328129}{281538929}a^{9}-\frac{139723188}{281538929}a^{8}+\frac{833662524}{281538929}a^{7}+\frac{1330148873}{563077858}a^{6}+\frac{433242666}{281538929}a^{5}-\frac{1147229464}{281538929}a^{4}+\frac{278550880}{281538929}a^{3}+\frac{72577019}{281538929}a^{2}-\frac{1303230484}{281538929}a-\frac{101298825}{12240823}$, $\frac{311663215}{39978527918}a^{15}+\frac{46735621}{11422436548}a^{14}+\frac{2622692157}{19989263959}a^{13}+\frac{1161601625}{79957055836}a^{12}+\frac{34563881067}{39978527918}a^{11}+\frac{6676220407}{39978527918}a^{10}+\frac{54989330576}{19989263959}a^{9}+\frac{14725449870}{19989263959}a^{8}+\frac{96298832964}{19989263959}a^{7}+\frac{67066439061}{19989263959}a^{6}+\frac{90837599906}{19989263959}a^{5}+\frac{3325835253}{2855609137}a^{4}+\frac{84844943888}{19989263959}a^{3}+\frac{41700115715}{19989263959}a^{2}-\frac{46450434900}{19989263959}a-\frac{2618911625}{869098433}$, $\frac{1704334635}{79957055836}a^{15}+\frac{949461389}{39978527918}a^{14}+\frac{26456867607}{79957055836}a^{13}+\frac{12399721485}{79957055836}a^{12}+\frac{34308601217}{19989263959}a^{11}+\frac{11354518796}{19989263959}a^{10}+\frac{67016754598}{19989263959}a^{9}-\frac{25731019967}{39978527918}a^{8}+\frac{192729652}{2855609137}a^{7}+\frac{35099820913}{39978527918}a^{6}+\frac{151629914}{2855609137}a^{5}-\frac{5676864594}{869098433}a^{4}-\frac{105354230866}{19989263959}a^{3}+\frac{68491473240}{19989263959}a^{2}+\frac{56896105682}{19989263959}a-\frac{457935203}{869098433}$, $\frac{1236732753}{39978527918}a^{15}+\frac{2048600735}{79957055836}a^{14}+\frac{2814223289}{5711218274}a^{13}+\frac{10591893043}{79957055836}a^{12}+\frac{112683562729}{39978527918}a^{11}+\frac{24803350073}{39978527918}a^{10}+\frac{139266075645}{19989263959}a^{9}+\frac{11230564525}{39978527918}a^{8}+\frac{134888051834}{19989263959}a^{7}+\frac{5080176611}{869098433}a^{6}+\frac{108729226780}{19989263959}a^{5}-\frac{83957039153}{19989263959}a^{4}+\frac{29962511972}{19989263959}a^{3}+\frac{121856691911}{19989263959}a^{2}-\frac{17637949766}{19989263959}a-\frac{3981796453}{869098433}$ 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:  \( 22843.7922823 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 22843.7922823 \cdot 2}{2\cdot\sqrt{9573589958277615058944}}\cr\approx \mathstrut & 0.56711317671 \end{aligned}\]

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 + 16*x^14 - 8*x^13 + 100*x^12 - 48*x^11 + 280*x^10 - 144*x^9 + 388*x^8 + 32*x^7 + 160*x^6 - 224*x^5 + 272*x^4 - 32*x^3 - 320*x^2 - 128*x + 184) 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 + 16*x^14 - 8*x^13 + 100*x^12 - 48*x^11 + 280*x^10 - 144*x^9 + 388*x^8 + 32*x^7 + 160*x^6 - 224*x^5 + 272*x^4 - 32*x^3 - 320*x^2 - 128*x + 184, 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 + 16*x^14 - 8*x^13 + 100*x^12 - 48*x^11 + 280*x^10 - 144*x^9 + 388*x^8 + 32*x^7 + 160*x^6 - 224*x^5 + 272*x^4 - 32*x^3 - 320*x^2 - 128*x + 184); 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 + 16*x^14 - 8*x^13 + 100*x^12 - 48*x^11 + 280*x^10 - 144*x^9 + 388*x^8 + 32*x^7 + 160*x^6 - 224*x^5 + 272*x^4 - 32*x^3 - 320*x^2 - 128*x + 184); 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

$C_4:C_4$ (as 16T8):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma: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 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-6 +3 \sqrt{2}})\), \(\Q(\zeta_{24})^+\), \(\Q(\sqrt{-2 + \sqrt{2}})\), \(\Q(\sqrt{-3 + \sqrt{6}})\) x2, \(\Q(\sqrt{-3 + \sqrt{3}})\) x2, 8.0.1358954496.3, 8.0.3057647616.7, 8.8.12230590464.1

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]
 

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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.1.0.1}{1} }^{16}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 2.1.16.54o1.297$x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 14$$16$$1$$54$$C_4:C_4$$$[2, 3, \frac{7}{2}, 4]$$
\(3\) Copy content Toggle raw display 3.4.4.12a1.3$x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 24 x^{12} + 48 x^{11} + 96 x^{10} + 64 x^{9} + 24 x^{8} + 96 x^{7} + 96 x^{6} + 32 x^{4} + 64 x^{3} + 19$$4$$4$$12$$C_4:C_4$$$[\ ]_{4}^{4}$$

Spectrum of ring of integers

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