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Properties

Label 16.8.361...000.1
Degree $16$
Signature $[8, 4]$
Discriminant $3.612\times 10^{24}$
Root discriminant \(34.27\)
Ramified primes $2,5,29$
Class number $1$ (GRH)
Class group trivial (GRH)
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 - 16*x^14 - 76*x^12 + 528*x^10 + 734*x^8 - 3184*x^6 - 2844*x^4 + 1392*x^2 + 841)
 
Copy content gp:K = bnfinit(y^16 - 16*y^14 - 76*y^12 + 528*y^10 + 734*y^8 - 3184*y^6 - 2844*y^4 + 1392*y^2 + 841, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 16*x^14 - 76*x^12 + 528*x^10 + 734*x^8 - 3184*x^6 - 2844*x^4 + 1392*x^2 + 841);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 16*x^14 - 76*x^12 + 528*x^10 + 734*x^8 - 3184*x^6 - 2844*x^4 + 1392*x^2 + 841)
 

\( x^{16} - 16x^{14} - 76x^{12} + 528x^{10} + 734x^{8} - 3184x^{6} - 2844x^{4} + 1392x^{2} + 841 \) 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:  $[8, 4]$
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:   \(3612067495936000000000000\) \(\medspace = 2^{44}\cdot 5^{12}\cdot 29^{2}\) 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:  \(34.27\)
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:  $2^{23/8}5^{3/4}29^{1/2}\approx 132.0954138186518$
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.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}+\frac{1}{4}a$, $\frac{1}{16}a^{8}-\frac{1}{8}a^{4}-\frac{7}{16}$, $\frac{1}{16}a^{9}-\frac{1}{8}a^{5}-\frac{7}{16}a$, $\frac{1}{16}a^{10}-\frac{1}{8}a^{6}-\frac{7}{16}a^{2}$, $\frac{1}{32}a^{11}-\frac{1}{32}a^{10}-\frac{1}{32}a^{9}-\frac{1}{32}a^{8}-\frac{1}{16}a^{7}+\frac{1}{16}a^{6}+\frac{1}{16}a^{5}+\frac{1}{16}a^{4}-\frac{7}{32}a^{3}+\frac{7}{32}a^{2}+\frac{7}{32}a+\frac{7}{32}$, $\frac{1}{352}a^{12}+\frac{1}{44}a^{10}-\frac{1}{32}a^{8}+\frac{5}{44}a^{6}-\frac{7}{32}a^{4}-\frac{5}{22}a^{2}+\frac{87}{352}$, $\frac{1}{352}a^{13}-\frac{3}{352}a^{11}-\frac{1}{32}a^{10}-\frac{1}{32}a^{8}-\frac{13}{176}a^{7}+\frac{1}{16}a^{6}-\frac{1}{32}a^{5}+\frac{1}{16}a^{4}+\frac{85}{352}a^{3}+\frac{7}{32}a^{2}-\frac{39}{176}a+\frac{7}{32}$, $\frac{1}{579018176}a^{14}-\frac{469903}{579018176}a^{12}+\frac{1283145}{579018176}a^{10}+\frac{9000185}{579018176}a^{8}-\frac{29736765}{579018176}a^{6}+\frac{105307155}{579018176}a^{4}-\frac{112652909}{579018176}a^{2}-\frac{3413185}{19966144}$, $\frac{1}{579018176}a^{15}-\frac{469903}{579018176}a^{13}+\frac{1283145}{579018176}a^{11}+\frac{9000185}{579018176}a^{9}-\frac{29736765}{579018176}a^{7}+\frac{105307155}{579018176}a^{5}-\frac{112652909}{579018176}a^{3}-\frac{3413185}{19966144}a$ 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 $1$ (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:  $C_{2}$, which has order $2$ (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);
 

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:  $11$
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:   $\frac{2027}{4991536}a^{14}-\frac{63475}{9983072}a^{12}-\frac{165839}{4991536}a^{10}+\frac{2053511}{9983072}a^{8}+\frac{1933301}{4991536}a^{6}-\frac{12427565}{9983072}a^{4}-\frac{10142721}{4991536}a^{2}+\frac{10017577}{9983072}$, $\frac{276855}{579018176}a^{14}-\frac{5073335}{579018176}a^{12}-\frac{12150885}{579018176}a^{10}+\frac{212912529}{579018176}a^{8}+\frac{30760141}{579018176}a^{6}-\frac{1557383365}{579018176}a^{4}-\frac{315698015}{579018176}a^{2}+\frac{44289607}{19966144}$, $\frac{797063}{579018176}a^{14}-\frac{12941421}{579018176}a^{12}-\frac{5027819}{52638016}a^{10}+\frac{403028735}{579018176}a^{8}+\frac{24616767}{52638016}a^{6}-\frac{2054845791}{579018176}a^{4}-\frac{44309763}{579018176}a^{2}+\frac{2764275}{1815104}$, $\frac{964919}{579018176}a^{14}-\frac{1440657}{52638016}a^{12}-\frac{66480361}{579018176}a^{10}+\frac{533943865}{579018176}a^{8}+\frac{492053893}{579018176}a^{6}-\frac{3096232345}{579018176}a^{4}-\frac{1672465587}{579018176}a^{2}+\frac{50926295}{19966144}$, $\frac{652215}{289509088}a^{14}-\frac{11031303}{289509088}a^{12}-\frac{39943907}{289509088}a^{10}+\frac{385560801}{289509088}a^{8}+\frac{196422497}{289509088}a^{6}-\frac{2238408093}{289509088}a^{4}-\frac{736354757}{289509088}a^{2}+\frac{33965407}{9983072}$, $\frac{29969}{72377272}a^{14}-\frac{1657625}{289509088}a^{12}-\frac{1627733}{36188636}a^{10}+\frac{41513469}{289509088}a^{8}+\frac{44653177}{72377272}a^{6}-\frac{288505255}{289509088}a^{4}-\frac{19381399}{9047159}a^{2}+\frac{23418063}{9983072}$, $\frac{130753}{579018176}a^{14}-\frac{1147721}{579018176}a^{12}-\frac{25067195}{579018176}a^{10}+\frac{1029539}{579018176}a^{8}+\frac{547734731}{579018176}a^{6}-\frac{61013699}{579018176}a^{4}-\frac{2552867345}{579018176}a^{2}-\frac{16866499}{19966144}$, $\frac{11057}{144754544}a^{15}-\frac{321083}{579018176}a^{14}-\frac{90399}{72377272}a^{13}+\frac{526957}{52638016}a^{12}-\frac{2179377}{289509088}a^{11}+\frac{16509639}{579018176}a^{10}+\frac{19734567}{289509088}a^{9}-\frac{252381663}{579018176}a^{8}+\frac{22951129}{72377272}a^{7}-\frac{214369173}{579018176}a^{6}-\frac{60527737}{144754544}a^{5}+\frac{1799494313}{579018176}a^{4}-\frac{855011933}{289509088}a^{3}+\frac{2025721881}{579018176}a^{2}-\frac{21981573}{9983072}a-\frac{326461}{19966144}$, $\frac{228007}{579018176}a^{15}+\frac{295283}{579018176}a^{14}-\frac{4716443}{579018176}a^{13}-\frac{5492187}{579018176}a^{12}-\frac{985727}{579018176}a^{11}-\frac{822609}{52638016}a^{10}+\frac{209520095}{579018176}a^{9}+\frac{197708603}{579018176}a^{8}-\frac{285811391}{579018176}a^{7}-\frac{25710721}{52638016}a^{6}-\frac{1459060677}{579018176}a^{5}-\frac{1138472893}{579018176}a^{4}+\frac{1458179751}{579018176}a^{3}+\frac{1909779939}{579018176}a^{2}+\frac{61451781}{19966144}a+\frac{3727411}{1815104}$, $\frac{53495}{36188636}a^{15}-\frac{197161}{289509088}a^{14}-\frac{6839849}{289509088}a^{13}+\frac{1407377}{144754544}a^{12}-\frac{16375957}{144754544}a^{11}+\frac{20117497}{289509088}a^{10}+\frac{225973233}{289509088}a^{9}-\frac{37613637}{144754544}a^{8}+\frac{85320401}{72377272}a^{7}-\frac{276813407}{289509088}a^{6}-\frac{1290117767}{289509088}a^{5}+\frac{197492263}{144754544}a^{4}-\frac{746230849}{144754544}a^{3}+\frac{1045831359}{289509088}a^{2}-\frac{6388165}{9983072}a+\frac{6535153}{4991536}$, $\frac{50751}{52638016}a^{15}-\frac{602489}{289509088}a^{14}-\frac{6965947}{579018176}a^{13}+\frac{7689139}{289509088}a^{12}-\frac{67007163}{579018176}a^{11}+\frac{71365917}{289509088}a^{10}+\frac{5294971}{52638016}a^{9}-\frac{97713815}{289509088}a^{8}+\frac{627700575}{579018176}a^{7}-\frac{811309607}{289509088}a^{6}+\frac{59600661}{52638016}a^{5}-\frac{413496323}{289509088}a^{4}+\frac{905993415}{579018176}a^{3}+\frac{804030451}{289509088}a^{2}+\frac{16230151}{19966144}a+\frac{17749923}{9983072}$ 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:  \( 1828598.13777 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 1828598.13777 \cdot 1}{2\cdot\sqrt{3612067495936000000000000}}\cr\approx \mathstrut & 0.191941899210 \end{aligned}\] (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 - 16*x^14 - 76*x^12 + 528*x^10 + 734*x^8 - 3184*x^6 - 2844*x^4 + 1392*x^2 + 841) 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 - 76*x^12 + 528*x^10 + 734*x^8 - 3184*x^6 - 2844*x^4 + 1392*x^2 + 841, 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 - 76*x^12 + 528*x^10 + 734*x^8 - 3184*x^6 - 2844*x^4 + 1392*x^2 + 841); 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 - 16*x^14 - 76*x^12 + 528*x^10 + 734*x^8 - 3184*x^6 - 2844*x^4 + 1392*x^2 + 841); 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

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), 4.4.8000.1, \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{40})^+\)

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: 16.8.11866206109696000000000000.1

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 ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ R ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 2.2.8.44d1.101$x^{16} + 8 x^{15} + 36 x^{14} + 116 x^{13} + 290 x^{12} + 588 x^{11} + 984 x^{10} + 1376 x^{9} + 1613 x^{8} + 1596 x^{7} + 1332 x^{6} + 944 x^{5} + 560 x^{4} + 284 x^{3} + 124 x^{2} + 60 x + 17$$8$$2$$44$16T86$$[2, 2, 3, \frac{7}{2}]^{4}$$
\(5\) 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 29.2.2.2a1.1$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$$$[\ ]^{4}$$
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$$$[\ ]^{4}$$

Spectrum of ring of integers

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