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Properties

Label 16.4.278...944.4
Degree $16$
Signature $[4, 6]$
Discriminant $2.786\times 10^{24}$
Root discriminant \(33.71\)
Ramified primes $2,3,17$
Class number $1$
Class group trivial
Galois group $C_2^4.D_4^2$ (as 16T1228)

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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 + 26*x^12 - 260*x^10 - 849*x^8 - 36*x^6 + 2202*x^4 + 1972*x^2 + 289)
 
Copy content gp:K = bnfinit(y^16 + 16*y^14 + 26*y^12 - 260*y^10 - 849*y^8 - 36*y^6 + 2202*y^4 + 1972*y^2 + 289, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 16*x^14 + 26*x^12 - 260*x^10 - 849*x^8 - 36*x^6 + 2202*x^4 + 1972*x^2 + 289);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 + 16*x^14 + 26*x^12 - 260*x^10 - 849*x^8 - 36*x^6 + 2202*x^4 + 1972*x^2 + 289)
 

\( x^{16} + 16x^{14} + 26x^{12} - 260x^{10} - 849x^{8} - 36x^{6} + 2202x^{4} + 1972x^{2} + 289 \) 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:  $[4, 6]$
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:   \(2786014518176517344722944\) \(\medspace = 2^{44}\cdot 3^{8}\cdot 17^{6}\) 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:  \(33.71\)
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:  $2^{51/16}3^{1/2}17^{1/2}\approx 65.0606202428235$
Ramified primes:   \(2\), \(3\), \(17\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{20}a^{12}-\frac{1}{10}a^{10}-\frac{1}{4}a^{9}-\frac{1}{20}a^{8}-\frac{1}{2}a^{7}+\frac{1}{5}a^{6}+\frac{1}{4}a^{5}+\frac{7}{20}a^{4}-\frac{1}{5}a^{2}+\frac{1}{4}a+\frac{2}{5}$, $\frac{1}{20}a^{13}-\frac{1}{10}a^{11}+\frac{1}{5}a^{9}-\frac{1}{4}a^{8}-\frac{3}{10}a^{7}-\frac{1}{2}a^{6}+\frac{1}{10}a^{5}+\frac{1}{4}a^{4}-\frac{1}{5}a^{3}+\frac{3}{20}a+\frac{1}{4}$, $\frac{1}{11350220}a^{14}-\frac{4017}{1621460}a^{12}+\frac{146317}{2837555}a^{10}-\frac{2453819}{11350220}a^{8}-\frac{162864}{2837555}a^{6}+\frac{5486867}{11350220}a^{4}+\frac{3611421}{11350220}a^{2}+\frac{112181}{333830}$, $\frac{1}{11350220}a^{15}-\frac{4017}{1621460}a^{13}+\frac{146317}{2837555}a^{11}-\frac{2453819}{11350220}a^{9}-\frac{162864}{2837555}a^{7}+\frac{5486867}{11350220}a^{5}+\frac{3611421}{11350220}a^{3}+\frac{112181}{333830}a$ 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:  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:  $C_{2}\times C_{2}$, which has order $4$
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:  $9$
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:   $\frac{865307}{2270044}a^{14}+\frac{422029}{81073}a^{12}-\frac{5211525}{2270044}a^{10}-\frac{212920751}{2270044}a^{8}-\frac{235142735}{2270044}a^{6}+\frac{523401783}{2270044}a^{4}+\frac{338470295}{1135022}a^{2}+\frac{6143721}{133532}$, $\frac{983359}{11350220}a^{14}+\frac{478107}{405365}a^{12}-\frac{6482817}{11350220}a^{10}-\frac{241492093}{11350220}a^{8}-\frac{256706211}{11350220}a^{6}+\frac{598446337}{11350220}a^{4}+\frac{372910103}{5675110}a^{2}+\frac{7188771}{667660}$, $\frac{835794}{2837555}a^{14}+\frac{1632038}{405365}a^{12}-\frac{4893702}{2837555}a^{10}-\frac{411555831}{5675110}a^{8}-\frac{229751866}{2837555}a^{6}+\frac{1009281289}{5675110}a^{4}+\frac{659720686}{2837555}a^{2}+\frac{11764917}{333830}$, $\frac{407459}{597380}a^{14}+\frac{199147}{21335}a^{12}-\frac{2307927}{597380}a^{10}-\frac{100510373}{597380}a^{8}-\frac{113282081}{597380}a^{6}+\frac{246197717}{597380}a^{4}+\frac{81377294}{149345}a^{2}+\frac{3027571}{35140}$, $\frac{481108}{2837555}a^{14}+\frac{375871}{162146}a^{12}-\frac{1131855}{1135022}a^{10}-\frac{118856254}{2837555}a^{8}-\frac{52654821}{1135022}a^{6}+\frac{59113580}{567511}a^{4}+\frac{152127259}{1135022}a^{2}+\frac{6092283}{333830}$, $\frac{3053}{1135022}a^{14}+\frac{24439}{1621460}a^{12}-\frac{3256861}{11350220}a^{10}-\frac{626762}{2837555}a^{8}+\frac{46007217}{11350220}a^{6}+\frac{5564509}{2837555}a^{4}-\frac{128178947}{11350220}a^{2}-\frac{453123}{667660}$, $\frac{2976167}{5675110}a^{14}+\frac{11603359}{1621460}a^{12}-\frac{36812061}{11350220}a^{10}-\frac{146555651}{1135022}a^{8}-\frac{1600026123}{11350220}a^{6}+\frac{1816884753}{5675110}a^{4}+\frac{4622691793}{11350220}a^{2}+\frac{39284939}{667660}$, $\frac{1233931}{11350220}a^{15}+\frac{116007}{1621460}a^{14}+\frac{245351}{162146}a^{13}+\frac{334771}{324292}a^{12}-\frac{364393}{2270044}a^{11}+\frac{168849}{324292}a^{10}-\frac{142541089}{5675110}a^{9}-\frac{5874574}{405365}a^{8}-\frac{69282357}{2270044}a^{7}-\frac{6753903}{324292}a^{6}+\frac{64636903}{1135022}a^{5}+\frac{4628597}{162146}a^{4}+\frac{87947563}{1135022}a^{3}+\frac{7113865}{162146}a^{2}+\frac{4208779}{333830}a+\frac{711651}{95380}$, $\frac{107019}{85340}a^{15}+\frac{3967767}{5675110}a^{14}+\frac{1698619}{85340}a^{13}+\frac{21374991}{1621460}a^{12}+\frac{750508}{21335}a^{11}+\frac{146843089}{2837555}a^{10}-\frac{10739403}{42670}a^{9}-\frac{1097447539}{11350220}a^{8}-\frac{39490097}{42670}a^{7}-\frac{5419454921}{5675110}a^{6}-\frac{42853281}{42670}a^{5}-\frac{21525722171}{11350220}a^{4}-\frac{33226821}{85340}a^{3}-\frac{3593191302}{2837555}a^{2}-\frac{231637}{5020}a-\frac{11899497}{66766}$ 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:  \( 833924.759268 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 833924.759268 \cdot 1}{2\cdot\sqrt{2786014518176517344722944}}\cr\approx \mathstrut & 0.245925961354 \end{aligned}\]

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 + 26*x^12 - 260*x^10 - 849*x^8 - 36*x^6 + 2202*x^4 + 1972*x^2 + 289) 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 + 26*x^12 - 260*x^10 - 849*x^8 - 36*x^6 + 2202*x^4 + 1972*x^2 + 289, 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 + 26*x^12 - 260*x^10 - 849*x^8 - 36*x^6 + 2202*x^4 + 1972*x^2 + 289); 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 + 26*x^12 - 260*x^10 - 849*x^8 - 36*x^6 + 2202*x^4 + 1972*x^2 + 289); 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

$C_2^4.D_4^2$ (as 16T1228):

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 1024
The 61 conjugacy class representatives for $C_2^4.D_4^2$
Character table for $C_2^4.D_4^2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), 4.4.9792.1, 4.4.4352.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.1534132224.1

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]
 

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.8.2786014518176517344722944.3

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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ R ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{6}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 2.2.8.44d2.60$x^{16} + 12 x^{15} + 68 x^{14} + 256 x^{13} + 714 x^{12} + 1564 x^{11} + 2776 x^{10} + 4068 x^{9} + 4973 x^{8} + 5092 x^{7} + 4364 x^{6} + 3108 x^{5} + 1820 x^{4} + 864 x^{3} + 324 x^{2} + 92 x + 17$$8$$2$$44$16T295$$[2, 2, 3, \frac{7}{2}, \frac{7}{2}]^{4}$$
\(3\) 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$$$[\ ]_{2}^{8}$$
\(17\) Copy content Toggle raw display 17.2.1.0a1.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
17.2.1.0a1.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
17.2.2.2a1.2$x^{4} + 32 x^{3} + 262 x^{2} + 96 x + 26$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
17.4.2.4a1.2$x^{8} + 14 x^{6} + 20 x^{5} + 55 x^{4} + 140 x^{3} + 142 x^{2} + 60 x + 26$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$

Spectrum of ring of integers

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