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Properties

Label 16.4.124470628948573741.1
Degree $16$
Signature $[4, 6]$
Discriminant $1.245\times 10^{17}$
Root discriminant \(11.71\)
Ramified primes $41,83,1289,6469$
Class number $1$
Class group trivial
Galois group $C_2^8.S_8$ (as 16T1948)

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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 - x^15 - 2*x^14 - 4*x^13 + 6*x^12 + 9*x^11 - 4*x^10 - 24*x^9 - 19*x^8 + 23*x^7 + 37*x^6 + 10*x^5 - 24*x^4 - 19*x^3 + 6*x^2 + 5*x - 1)
 
Copy content gp:K = bnfinit(y^16 - y^15 - 2*y^14 - 4*y^13 + 6*y^12 + 9*y^11 - 4*y^10 - 24*y^9 - 19*y^8 + 23*y^7 + 37*y^6 + 10*y^5 - 24*y^4 - 19*y^3 + 6*y^2 + 5*y - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 - 2*x^14 - 4*x^13 + 6*x^12 + 9*x^11 - 4*x^10 - 24*x^9 - 19*x^8 + 23*x^7 + 37*x^6 + 10*x^5 - 24*x^4 - 19*x^3 + 6*x^2 + 5*x - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - x^15 - 2*x^14 - 4*x^13 + 6*x^12 + 9*x^11 - 4*x^10 - 24*x^9 - 19*x^8 + 23*x^7 + 37*x^6 + 10*x^5 - 24*x^4 - 19*x^3 + 6*x^2 + 5*x - 1)
 

\( x^{16} - x^{15} - 2 x^{14} - 4 x^{13} + 6 x^{12} + 9 x^{11} - 4 x^{10} - 24 x^{9} - 19 x^{8} + 23 x^{7} + \cdots - 1 \) 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:  $[4, 6]$
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:   \(124470628948573741\) \(\medspace = 41^{2}\cdot 83^{2}\cdot 1289^{2}\cdot 6469\) 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:  \(11.71\)
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:  $41^{1/2}83^{1/2}1289^{1/2}6469^{1/2}\approx 168451.9368336262$
Ramified primes:   \(41\), \(83\), \(1289\), \(6469\) 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:  \(\Q(\sqrt{6469}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{17807}a^{15}-\frac{8024}{17807}a^{14}+\frac{4245}{17807}a^{13}+\frac{7152}{17807}a^{12}-\frac{6336}{17807}a^{11}-\frac{5248}{17807}a^{10}-\frac{8855}{17807}a^{9}-\frac{6289}{17807}a^{8}-\frac{8410}{17807}a^{7}+\frac{2730}{17807}a^{6}-\frac{143}{17807}a^{5}+\frac{7651}{17807}a^{4}-\frac{3268}{17807}a^{3}+\frac{7241}{17807}a^{2}-\frac{8103}{17807}a-\frac{2983}{17807}$ 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:  $9$
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:   $\frac{2925}{17807}a^{15}-\frac{574}{17807}a^{14}+\frac{5146}{17807}a^{13}-\frac{39239}{17807}a^{12}-\frac{13520}{17807}a^{11}+\frac{34848}{17807}a^{10}+\frac{97345}{17807}a^{9}-\frac{54115}{17807}a^{8}-\frac{274888}{17807}a^{7}-\frac{116935}{17807}a^{6}+\frac{222777}{17807}a^{5}+\frac{351916}{17807}a^{4}-\frac{14348}{17807}a^{3}-\frac{206282}{17807}a^{2}-\frac{35772}{17807}a+\frac{35769}{17807}$, $\frac{1526}{17807}a^{15}-\frac{11215}{17807}a^{14}+\frac{13929}{17807}a^{13}+\frac{16068}{17807}a^{12}+\frac{465}{17807}a^{11}-\frac{66526}{17807}a^{10}-\frac{15024}{17807}a^{9}+\frac{125608}{17807}a^{8}+\frac{94222}{17807}a^{7}-\frac{125507}{17807}a^{6}-\frac{182604}{17807}a^{5}+\frac{47455}{17807}a^{4}+\frac{194869}{17807}a^{3}+\frac{9426}{17807}a^{2}-\frac{78348}{17807}a-\frac{11273}{17807}$, $\frac{8170}{17807}a^{15}-\frac{26320}{17807}a^{14}+\frac{11421}{17807}a^{13}+\frac{7073}{17807}a^{12}+\frac{71057}{17807}a^{11}-\frac{85939}{17807}a^{10}-\frac{102351}{17807}a^{9}+\frac{27679}{17807}a^{8}+\frac{203390}{17807}a^{7}+\frac{80964}{17807}a^{6}-\frac{242346}{17807}a^{5}-\frac{154163}{17807}a^{4}+\frac{64361}{17807}a^{3}+\frac{146572}{17807}a^{2}+\frac{22723}{17807}a-\frac{28941}{17807}$, $\frac{4808}{17807}a^{15}-\frac{9430}{17807}a^{14}-\frac{14669}{17807}a^{13}+\frac{1499}{17807}a^{12}+\frac{75517}{17807}a^{11}+\frac{17942}{17807}a^{10}-\frac{122952}{17807}a^{9}-\frac{143682}{17807}a^{8}+\frac{146873}{17807}a^{7}+\frac{358221}{17807}a^{6}+\frac{24736}{17807}a^{5}-\frac{305973}{17807}a^{4}-\frac{238261}{17807}a^{3}+\frac{126692}{17807}a^{2}+\frac{144948}{17807}a-\frac{43243}{17807}$, $\frac{7459}{17807}a^{15}-\frac{19496}{17807}a^{14}+\frac{20416}{17807}a^{13}-\frac{38618}{17807}a^{12}+\frac{52975}{17807}a^{11}-\frac{58467}{17807}a^{10}+\frac{32332}{17807}a^{9}-\frac{41627}{17807}a^{8}-\frac{49550}{17807}a^{7}-\frac{25945}{17807}a^{6}-\frac{51638}{17807}a^{5}+\frac{122023}{17807}a^{4}+\frac{19578}{17807}a^{3}+\frac{37602}{17807}a^{2}+\frac{14488}{17807}a-\frac{44868}{17807}$, $\frac{12961}{17807}a^{15}-\frac{23991}{17807}a^{14}-\frac{4185}{17807}a^{13}-\frac{23977}{17807}a^{12}+\frac{58409}{17807}a^{11}+\frac{3412}{17807}a^{10}-\frac{56961}{17807}a^{9}-\frac{80318}{17807}a^{8}-\frac{94398}{17807}a^{7}+\frac{18828}{17807}a^{6}+\frac{87540}{17807}a^{5}+\frac{193305}{17807}a^{4}+\frac{77533}{17807}a^{3}-\frac{152552}{17807}a^{2}-\frac{68525}{17807}a+\frac{31948}{17807}$, $\frac{4850}{17807}a^{15}-\frac{8105}{17807}a^{14}-\frac{14449}{17807}a^{13}+\frac{16971}{17807}a^{12}+\frac{23089}{17807}a^{11}+\frac{11210}{17807}a^{10}-\frac{103108}{17807}a^{9}-\frac{16066}{17807}a^{8}+\frac{78565}{17807}a^{7}+\frac{63320}{17807}a^{6}-\frac{34691}{17807}a^{5}-\frac{109280}{17807}a^{4}+\frac{51851}{17807}a^{3}-\frac{14361}{17807}a^{2}-\frac{17308}{17807}a+\frac{45155}{17807}$, $\frac{1750}{17807}a^{15}-\frac{10084}{17807}a^{14}+\frac{3231}{17807}a^{13}+\frac{15486}{17807}a^{12}+\frac{41375}{17807}a^{11}-\frac{49009}{17807}a^{10}-\frac{75388}{17807}a^{9}+\frac{16783}{17807}a^{8}+\frac{204766}{17807}a^{7}+\frac{112066}{17807}a^{6}-\frac{161215}{17807}a^{5}-\frac{215298}{17807}a^{4}+\frac{14854}{17807}a^{3}+\frac{153429}{17807}a^{2}+\frac{29736}{17807}a-\frac{20606}{17807}$, $\frac{16216}{17807}a^{15}-\frac{19242}{17807}a^{14}-\frac{40556}{17807}a^{13}-\frac{17966}{17807}a^{12}+\frac{90849}{17807}a^{11}+\frac{87120}{17807}a^{10}-\frac{157295}{17807}a^{9}-\frac{251033}{17807}a^{8}-\frac{46168}{17807}a^{7}+\frac{232969}{17807}a^{6}+\frac{245320}{17807}a^{5}+\frac{25054}{17807}a^{4}-\frac{256}{17807}a^{3}-\frac{106144}{17807}a^{2}-\frac{89430}{17807}a+\frac{27098}{17807}$ 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:  \( 112.097283421 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 112.097283421 \cdot 1}{2\cdot\sqrt{124470628948573741}}\cr\approx \mathstrut & 0.156397873255 \end{aligned}\]

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 - x^15 - 2*x^14 - 4*x^13 + 6*x^12 + 9*x^11 - 4*x^10 - 24*x^9 - 19*x^8 + 23*x^7 + 37*x^6 + 10*x^5 - 24*x^4 - 19*x^3 + 6*x^2 + 5*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 - x^15 - 2*x^14 - 4*x^13 + 6*x^12 + 9*x^11 - 4*x^10 - 24*x^9 - 19*x^8 + 23*x^7 + 37*x^6 + 10*x^5 - 24*x^4 - 19*x^3 + 6*x^2 + 5*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 - x^15 - 2*x^14 - 4*x^13 + 6*x^12 + 9*x^11 - 4*x^10 - 24*x^9 - 19*x^8 + 23*x^7 + 37*x^6 + 10*x^5 - 24*x^4 - 19*x^3 + 6*x^2 + 5*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 = PolynomialRing(QQ); K, a = NumberField(x^16 - x^15 - 2*x^14 - 4*x^13 + 6*x^12 + 9*x^11 - 4*x^10 - 24*x^9 - 19*x^8 + 23*x^7 + 37*x^6 + 10*x^5 - 24*x^4 - 19*x^3 + 6*x^2 + 5*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

$C_2^8.S_8$ (as 16T1948):

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 non-solvable group of order 10321920
The 185 conjugacy class representatives for $C_2^8.S_8$
Character table for $C_2^8.S_8$

Intermediate fields

8.2.4386467.1

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(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: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $16$ ${\href{/padicField/3.5.0.1}{5} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.3.0.1}{3} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ $16$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ R ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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
\(41\) Copy content Toggle raw display 41.2.1.0a1.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
41.2.1.0a1.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
41.2.1.0a1.1$x^{2} + 38 x + 6$$1$$2$$0$$C_2$$$[\ ]^{2}$$
41.2.2.2a1.2$x^{4} + 76 x^{3} + 1456 x^{2} + 456 x + 77$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
41.6.1.0a1.1$x^{6} + 4 x^{4} + 33 x^{3} + 39 x^{2} + 6 x + 6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
\(83\) Copy content Toggle raw display 83.2.2.2a1.2$x^{4} + 164 x^{3} + 6728 x^{2} + 328 x + 87$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
83.6.1.0a1.1$x^{6} + x^{4} + 76 x^{3} + 32 x^{2} + 17 x + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$
83.6.1.0a1.1$x^{6} + x^{4} + 76 x^{3} + 32 x^{2} + 17 x + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$
\(1289\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
\(6469\) Copy content Toggle raw display $\Q_{6469}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{6469}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$
Deg $6$$1$$6$$0$$C_6$$$[\ ]^{6}$$

Spectrum of ring of integers

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