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Properties

Label 16.8.488...625.1
Degree $16$
Signature $[8, 4]$
Discriminant $4.885\times 10^{18}$
Root discriminant \(14.72\)
Ramified primes $5,29$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_4\wr C_2$ (as 16T42)

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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 - 2*x^15 - 3*x^14 + 3*x^13 + x^12 + 5*x^11 - 31*x^10 + 6*x^9 + 89*x^8 + 6*x^7 - 31*x^6 + 5*x^5 + x^4 + 3*x^3 - 3*x^2 - 2*x + 1)
 
Copy content gp:K = bnfinit(y^16 - 2*y^15 - 3*y^14 + 3*y^13 + y^12 + 5*y^11 - 31*y^10 + 6*y^9 + 89*y^8 + 6*y^7 - 31*y^6 + 5*y^5 + y^4 + 3*y^3 - 3*y^2 - 2*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 3*x^14 + 3*x^13 + x^12 + 5*x^11 - 31*x^10 + 6*x^9 + 89*x^8 + 6*x^7 - 31*x^6 + 5*x^5 + x^4 + 3*x^3 - 3*x^2 - 2*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 - 3*x^14 + 3*x^13 + x^12 + 5*x^11 - 31*x^10 + 6*x^9 + 89*x^8 + 6*x^7 - 31*x^6 + 5*x^5 + x^4 + 3*x^3 - 3*x^2 - 2*x + 1)
 

\( x^{16} - 2 x^{15} - 3 x^{14} + 3 x^{13} + x^{12} + 5 x^{11} - 31 x^{10} + 6 x^{9} + 89 x^{8} + 6 x^{7} + \cdots + 1 \) 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:   \(4885218876572265625\) \(\medspace = 5^{10}\cdot 29^{8}\) 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:  \(14.72\)
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:  $5^{3/4}29^{1/2}\approx 18.006383777357115$
Ramified primes:   \(5\), \(29\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2\times C_4$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7}a^{13}+\frac{2}{7}a^{12}+\frac{1}{7}a^{11}-\frac{1}{7}a^{9}+\frac{2}{7}a^{8}+\frac{2}{7}a^{7}-\frac{2}{7}a^{6}-\frac{2}{7}a^{5}+\frac{1}{7}a^{4}-\frac{1}{7}a^{2}-\frac{2}{7}a-\frac{1}{7}$, $\frac{1}{471541}a^{14}-\frac{26170}{471541}a^{13}-\frac{130428}{471541}a^{12}+\frac{52319}{471541}a^{11}-\frac{204366}{471541}a^{10}+\frac{118056}{471541}a^{9}-\frac{222071}{471541}a^{8}+\frac{14487}{67363}a^{7}-\frac{19982}{471541}a^{6}+\frac{50693}{471541}a^{5}-\frac{137003}{471541}a^{4}-\frac{15044}{471541}a^{3}+\frac{139024}{471541}a^{2}-\frac{228259}{471541}a-\frac{67362}{471541}$, $\frac{1}{471541}a^{15}+\frac{15019}{471541}a^{13}-\frac{32594}{471541}a^{12}-\frac{34385}{471541}a^{11}+\frac{77858}{471541}a^{10}-\frac{98457}{471541}a^{9}-\frac{568}{11501}a^{8}+\frac{222889}{471541}a^{7}-\frac{141367}{471541}a^{6}-\frac{148115}{471541}a^{5}+\frac{2564}{15211}a^{4}+\frac{24897}{67363}a^{3}+\frac{225732}{471541}a^{2}+\frac{145448}{471541}a-\frac{15508}{67363}$ 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:  Trivial group, which has order $1$ (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:   $a$, $\frac{296689}{471541}a^{15}-\frac{571091}{471541}a^{14}-\frac{871877}{471541}a^{13}+\frac{711705}{471541}a^{12}+\frac{154254}{471541}a^{11}+\frac{235239}{67363}a^{10}-\frac{9035289}{471541}a^{9}+\frac{1455448}{471541}a^{8}+\frac{3526330}{67363}a^{7}+\frac{3739776}{471541}a^{6}-\frac{3532155}{471541}a^{5}+\frac{2148085}{471541}a^{4}-\frac{150140}{471541}a^{3}+\frac{806343}{471541}a^{2}-\frac{1034461}{471541}a-\frac{277252}{471541}$, $\frac{344595}{471541}a^{15}-\frac{687443}{471541}a^{14}-\frac{28592}{15211}a^{13}+\frac{692534}{471541}a^{12}+\frac{38453}{471541}a^{11}+\frac{2089077}{471541}a^{10}-\frac{10705528}{471541}a^{9}+\frac{2947594}{471541}a^{8}+\frac{25734445}{471541}a^{7}+\frac{4673643}{471541}a^{6}-\frac{7246}{471541}a^{5}+\frac{1738298}{471541}a^{4}-\frac{206016}{471541}a^{3}+\frac{1030524}{471541}a^{2}-\frac{635512}{471541}a-\frac{251651}{471541}$, $\frac{690}{15211}a^{15}-\frac{68381}{471541}a^{14}+\frac{34938}{471541}a^{13}+\frac{24621}{471541}a^{12}-\frac{9984}{67363}a^{11}+\frac{62178}{471541}a^{10}-\frac{906896}{471541}a^{9}+\frac{1364375}{471541}a^{8}+\frac{79865}{471541}a^{7}-\frac{658003}{471541}a^{6}+\frac{233439}{471541}a^{5}+\frac{262064}{67363}a^{4}+\frac{3418294}{471541}a^{3}-\frac{371218}{471541}a^{2}-\frac{196849}{471541}a+\frac{258204}{471541}$, $\frac{6414}{8897}a^{15}-\frac{12630}{8897}a^{14}-\frac{17823}{8897}a^{13}+\frac{14628}{8897}a^{12}+\frac{13}{31}a^{11}+\frac{35409}{8897}a^{10}-\frac{199090}{8897}a^{9}+\frac{44802}{8897}a^{8}+\frac{512825}{8897}a^{7}+\frac{87706}{8897}a^{6}-\frac{86138}{8897}a^{5}+\frac{5327}{1271}a^{4}+\frac{34505}{8897}a^{3}+\frac{20385}{8897}a^{2}-\frac{22230}{8897}a-\frac{87}{287}$, $\frac{355309}{471541}a^{15}-\frac{572510}{471541}a^{14}-\frac{173015}{67363}a^{13}+\frac{429913}{471541}a^{12}+\frac{336694}{471541}a^{11}+\frac{2106474}{471541}a^{10}-\frac{10192369}{471541}a^{9}-\frac{1343165}{471541}a^{8}+\frac{28605061}{471541}a^{7}+\frac{14171033}{471541}a^{6}+\frac{503075}{471541}a^{5}+\frac{2066457}{471541}a^{4}+\frac{748707}{471541}a^{3}+\frac{133025}{67363}a^{2}-\frac{787305}{471541}a-\frac{250870}{471541}$, $\frac{76448}{471541}a^{15}-\frac{92479}{471541}a^{14}-\frac{341709}{471541}a^{13}+\frac{45579}{471541}a^{12}+\frac{182632}{471541}a^{11}+\frac{497916}{471541}a^{10}-\frac{2034146}{471541}a^{9}-\frac{1452721}{471541}a^{8}+\frac{7027123}{471541}a^{7}+\frac{5291639}{471541}a^{6}-\frac{295627}{471541}a^{5}-\frac{1301172}{471541}a^{4}-\frac{844919}{471541}a^{3}+\frac{1492355}{471541}a^{2}+\frac{10685}{8897}a-\frac{129419}{471541}$, $\frac{10569}{471541}a^{15}-\frac{97267}{471541}a^{14}+\frac{37483}{67363}a^{13}-\frac{57755}{471541}a^{12}-\frac{504922}{471541}a^{11}+\frac{299024}{471541}a^{10}-\frac{695922}{471541}a^{9}+\frac{3346235}{471541}a^{8}-\frac{4201340}{471541}a^{7}-\frac{4343160}{471541}a^{6}+\frac{8994081}{471541}a^{5}+\frac{3068536}{471541}a^{4}-\frac{38903}{15211}a^{3}-\frac{158637}{67363}a^{2}-\frac{625669}{471541}a+\frac{14015}{11501}$, $\frac{20268}{471541}a^{15}-\frac{141579}{471541}a^{14}+\frac{216488}{471541}a^{13}+\frac{266638}{471541}a^{12}-\frac{552707}{471541}a^{11}-\frac{43629}{471541}a^{10}-\frac{1118273}{471541}a^{9}+\frac{3838715}{471541}a^{8}-\frac{763416}{471541}a^{7}-\frac{9721119}{471541}a^{6}+\frac{562313}{67363}a^{5}+\frac{8324644}{471541}a^{4}+\frac{3237707}{471541}a^{3}+\frac{228831}{471541}a^{2}-\frac{1863002}{471541}a-\frac{76918}{471541}$, $\frac{53402}{471541}a^{15}-\frac{95691}{471541}a^{14}-\frac{234747}{471541}a^{13}+\frac{248418}{471541}a^{12}+\frac{250430}{471541}a^{11}-\frac{37068}{471541}a^{10}-\frac{1654783}{471541}a^{9}-\frac{3419}{15211}a^{8}+\frac{6460924}{471541}a^{7}+\frac{22450}{15211}a^{6}-\frac{1077146}{67363}a^{5}+\frac{2254019}{471541}a^{4}+\frac{4718582}{471541}a^{3}-\frac{2455094}{471541}a^{2}-\frac{264763}{471541}a+\frac{528118}{471541}$, $\frac{3565}{15211}a^{15}-\frac{167175}{471541}a^{14}-\frac{65532}{67363}a^{13}+\frac{220849}{471541}a^{12}+\frac{39404}{67363}a^{11}+\frac{594220}{471541}a^{10}-\frac{3130812}{471541}a^{9}-\frac{1120670}{471541}a^{8}+\frac{10856977}{471541}a^{7}+\frac{4732223}{471541}a^{6}-\frac{3694702}{471541}a^{5}-\frac{188434}{67363}a^{4}+\frac{132546}{471541}a^{3}-\frac{18431}{67363}a^{2}-\frac{904070}{471541}a-\frac{220030}{471541}$ 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:  \( 1652.33200457 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 1652.33200457 \cdot 1}{2\cdot\sqrt{4885218876572265625}}\cr\approx \mathstrut & 0.149136774060 \end{aligned}\] (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 - 2*x^15 - 3*x^14 + 3*x^13 + x^12 + 5*x^11 - 31*x^10 + 6*x^9 + 89*x^8 + 6*x^7 - 31*x^6 + 5*x^5 + x^4 + 3*x^3 - 3*x^2 - 2*x + 1) 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 - 2*x^15 - 3*x^14 + 3*x^13 + x^12 + 5*x^11 - 31*x^10 + 6*x^9 + 89*x^8 + 6*x^7 - 31*x^6 + 5*x^5 + x^4 + 3*x^3 - 3*x^2 - 2*x + 1, 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 - 2*x^15 - 3*x^14 + 3*x^13 + x^12 + 5*x^11 - 31*x^10 + 6*x^9 + 89*x^8 + 6*x^7 - 31*x^6 + 5*x^5 + x^4 + 3*x^3 - 3*x^2 - 2*x + 1); 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 - 2*x^15 - 3*x^14 + 3*x^13 + x^12 + 5*x^11 - 31*x^10 + 6*x^9 + 89*x^8 + 6*x^7 - 31*x^6 + 5*x^5 + x^4 + 3*x^3 - 3*x^2 - 2*x + 1); 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

$C_4\wr C_2$ (as 16T42):

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 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{145}) \), 4.4.725.1 x2, 4.4.4205.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.4.88410125.1, 8.4.2210253125.1, 8.8.442050625.1

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

Galois closure: deg 32
Degree 8 siblings: 8.4.88410125.1, 8.4.2210253125.1
Degree 16 sibling: 16.0.145220537353515625.1
Minimal sibling: 8.4.88410125.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ R ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ R ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(5\) Copy content Toggle raw display 5.2.4.6a1.3$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 133 x + 31$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
5.4.2.4a1.2$x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(29\) Copy content Toggle raw display 29.4.2.4a1.2$x^{8} + 4 x^{6} + 30 x^{5} + 8 x^{4} + 60 x^{3} + 233 x^{2} + 60 x + 33$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
29.4.2.4a1.2$x^{8} + 4 x^{6} + 30 x^{5} + 8 x^{4} + 60 x^{3} + 233 x^{2} + 60 x + 33$$2$$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)