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Properties

Label 16.0.138...464.3
Degree $16$
Signature $[0, 8]$
Discriminant $1.385\times 10^{23}$
Root discriminant \(27.95\)
Ramified primes $2,17$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_4^2:D_4$ (as 16T305)

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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 - 14*x^14 + 105*x^12 - 460*x^10 + 1260*x^8 - 1768*x^6 + 2788*x^4 - 4624*x^2 + 4624)
 
Copy content gp:K = bnfinit(y^16 - 14*y^14 + 105*y^12 - 460*y^10 + 1260*y^8 - 1768*y^6 + 2788*y^4 - 4624*y^2 + 4624, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 14*x^14 + 105*x^12 - 460*x^10 + 1260*x^8 - 1768*x^6 + 2788*x^4 - 4624*x^2 + 4624);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 14*x^14 + 105*x^12 - 460*x^10 + 1260*x^8 - 1768*x^6 + 2788*x^4 - 4624*x^2 + 4624)
 

\( x^{16} - 14x^{14} + 105x^{12} - 460x^{10} + 1260x^{8} - 1768x^{6} + 2788x^{4} - 4624x^{2} + 4624 \) 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:   \(138538045941822955454464\) \(\medspace = 2^{36}\cdot 17^{10}\) 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:  \(27.95\)
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:  $2^{3}17^{3/4}\approx 66.97715222874152$
Ramified primes:   \(2\), \(17\) 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.
Maximal CM subfield:  8.0.342102016.2

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{10}-\frac{1}{4}a^{7}+\frac{3}{16}a^{6}+\frac{1}{4}a^{5}+\frac{3}{8}a^{4}-\frac{1}{2}a^{3}-\frac{3}{8}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{16}a^{11}-\frac{1}{4}a^{8}+\frac{3}{16}a^{7}-\frac{1}{4}a^{6}+\frac{3}{8}a^{5}-\frac{3}{8}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a$, $\frac{1}{272}a^{12}+\frac{3}{272}a^{10}-\frac{65}{272}a^{8}-\frac{1}{4}a^{7}+\frac{67}{272}a^{6}+\frac{1}{4}a^{5}+\frac{9}{68}a^{4}-\frac{1}{2}a^{3}-\frac{3}{8}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{272}a^{13}+\frac{3}{272}a^{11}+\frac{3}{272}a^{9}-\frac{1}{272}a^{7}-\frac{25}{68}a^{5}+\frac{1}{8}a^{3}-\frac{1}{4}a$, $\frac{1}{2913959392}a^{14}-\frac{2542047}{1456979696}a^{12}-\frac{68335961}{2913959392}a^{10}-\frac{20614977}{364244924}a^{8}+\frac{11854335}{1456979696}a^{6}-\frac{6310629}{91061231}a^{4}+\frac{9901373}{21426172}a^{2}-\frac{1542078}{5356543}$, $\frac{1}{5827918784}a^{15}+\frac{87953}{91061231}a^{13}-\frac{36196703}{5827918784}a^{11}-\frac{66390279}{2913959392}a^{9}+\frac{203056}{91061231}a^{7}-\frac{1}{4}a^{6}-\frac{9362123}{42852344}a^{5}+\frac{1}{4}a^{4}+\frac{25159289}{85704688}a^{3}-\frac{1}{2}a^{2}-\frac{11524855}{42852344}a-\frac{1}{2}$ 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:  No
Index:  Not computed
Inessential primes:  $2$

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:  $7$
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:   $\frac{2379}{85704688}a^{14}+\frac{3617}{21426172}a^{12}-\frac{171169}{85704688}a^{10}+\frac{553157}{42852344}a^{8}-\frac{367965}{21426172}a^{6}+\frac{258009}{10713086}a^{4}-\frac{1070167}{21426172}a^{2}+\frac{5234923}{10713086}$, $\frac{244115}{1456979696}a^{14}-\frac{826699}{364244924}a^{12}+\frac{27319213}{1456979696}a^{10}-\frac{74318027}{728489848}a^{8}+\frac{291451527}{728489848}a^{6}-\frac{414028553}{364244924}a^{4}+\frac{12351188}{5356543}a^{2}-\frac{10553661}{5356543}$, $\frac{2690979}{5827918784}a^{15}+\frac{618355}{2913959392}a^{14}-\frac{2062415}{364244924}a^{13}-\frac{571599}{182122462}a^{12}+\frac{224397115}{5827918784}a^{11}+\frac{74051311}{2913959392}a^{10}-\frac{424793061}{2913959392}a^{9}-\frac{172566701}{1456979696}a^{8}+\frac{239600673}{728489848}a^{7}+\frac{251391451}{728489848}a^{6}-\frac{153390805}{728489848}a^{5}-\frac{10294243}{21426172}a^{4}+\frac{35856999}{85704688}a^{3}+\frac{32209029}{42852344}a^{2}-\frac{1531021}{42852344}a+\frac{224535}{21426172}$, $\frac{5668007}{5827918784}a^{15}+\frac{1071865}{1456979696}a^{14}-\frac{9411309}{728489848}a^{13}-\frac{7766759}{728489848}a^{12}+\frac{530012095}{5827918784}a^{11}+\frac{115386151}{1456979696}a^{10}-\frac{1042048861}{2913959392}a^{9}-\frac{31439615}{91061231}a^{8}+\frac{583309451}{728489848}a^{7}+\frac{666058893}{728489848}a^{6}-\frac{374338125}{728489848}a^{5}-\frac{27548363}{21426172}a^{4}+\frac{88538595}{85704688}a^{3}+\frac{10940644}{5356543}a^{2}-\frac{152616877}{42852344}a-\frac{54212781}{10713086}$, $\frac{342247}{5827918784}a^{15}-\frac{209203}{1456979696}a^{14}-\frac{900523}{728489848}a^{13}+\frac{456479}{364244924}a^{12}+\frac{56820007}{5827918784}a^{11}-\frac{10989789}{1456979696}a^{10}-\frac{125474589}{2913959392}a^{9}+\frac{22340593}{728489848}a^{8}+\frac{10231036}{91061231}a^{7}-\frac{86584639}{728489848}a^{6}-\frac{148179595}{728489848}a^{5}+\frac{882910}{5356543}a^{4}+\frac{26883711}{85704688}a^{3}-\frac{2843995}{5356543}a^{2}-\frac{45898049}{42852344}a+\frac{670835}{10713086}$, $\frac{5103773}{5827918784}a^{15}-\frac{5024169}{2913959392}a^{14}-\frac{3548897}{364244924}a^{13}+\frac{15381621}{728489848}a^{12}+\frac{341972821}{5827918784}a^{11}-\frac{395560217}{2913959392}a^{10}-\frac{530187283}{2913959392}a^{9}+\frac{695847233}{1456979696}a^{8}+\frac{197631339}{728489848}a^{7}-\frac{320577799}{364244924}a^{6}+\frac{109931413}{728489848}a^{5}+\frac{2046101}{21426172}a^{4}+\frac{143970537}{85704688}a^{3}-\frac{111078165}{42852344}a^{2}-\frac{2242571}{42852344}a+\frac{35657105}{21426172}$, $\frac{1537}{1937473}a^{14}-\frac{49955}{3874946}a^{12}+\frac{204684}{1937473}a^{10}-\frac{1935855}{3874946}a^{8}+\frac{2740924}{1937473}a^{6}-\frac{224516}{113969}a^{4}+\frac{126096}{113969}a^{2}-\frac{904387}{113969}$ 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:  \( 475501.195576 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 475501.195576 \cdot 1}{2\cdot\sqrt{138538045941822955454464}}\cr\approx \mathstrut & 1.55158665489 \end{aligned}\] (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 - 14*x^14 + 105*x^12 - 460*x^10 + 1260*x^8 - 1768*x^6 + 2788*x^4 - 4624*x^2 + 4624) 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 - 14*x^14 + 105*x^12 - 460*x^10 + 1260*x^8 - 1768*x^6 + 2788*x^4 - 4624*x^2 + 4624, 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 - 14*x^14 + 105*x^12 - 460*x^10 + 1260*x^8 - 1768*x^6 + 2788*x^4 - 4624*x^2 + 4624); 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 - 14*x^14 + 105*x^12 - 460*x^10 + 1260*x^8 - 1768*x^6 + 2788*x^4 - 4624*x^2 + 4624); 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

$C_4^2:D_4$ (as 16T305):

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 128
The 29 conjugacy class representatives for $C_4^2:D_4$
Character table for $C_4^2:D_4$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{34}) \), 4.0.1088.2 x2, 4.0.2312.1 x2, \(\Q(\sqrt{2}, \sqrt{17})\), 8.0.342102016.2

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: 16.0.490875290811165073997824.4

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ R ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

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 2.1.4.11a1.10$x^{4} + 4 x^{2} + 18$$4$$1$$11$$C_4$$$[3, 4]$$
2.1.4.11a1.12$x^{4} + 8 x^{3} + 4 x^{2} + 18$$4$$1$$11$$C_4$$$[3, 4]$$
2.2.2.6a1.5$x^{4} + 2 x^{3} + 7 x^{2} + 6 x + 7$$2$$2$$6$$C_2^2$$$[3]^{2}$$
2.1.4.8b1.6$x^{4} + 4 x^{3} + 2 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$$[2, 3]$$
\(17\) Copy content Toggle raw display 17.2.4.6a1.3$x^{8} + 64 x^{7} + 1548 x^{6} + 16960 x^{5} + 74806 x^{4} + 50880 x^{3} + 13932 x^{2} + 1745 x + 319$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
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)