↑

Properties

Label 16.0.145220537353515625.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.452\times 10^{17}$
Root discriminant \(11.82\)
Ramified primes $5,29$
Class number $1$
Class group trivial
Galois group $C_4\wr C_2$ (as 16T28)

Related objects

Downloads

Learn more

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 - 6*x^15 + 18*x^14 - 45*x^13 + 105*x^12 - 208*x^11 + 345*x^10 - 496*x^9 + 621*x^8 - 645*x^7 + 535*x^6 - 359*x^5 + 208*x^4 - 106*x^3 + 42*x^2 - 10*x + 1)
 
Copy content gp:K = bnfinit(y^16 - 6*y^15 + 18*y^14 - 45*y^13 + 105*y^12 - 208*y^11 + 345*y^10 - 496*y^9 + 621*y^8 - 645*y^7 + 535*y^6 - 359*y^5 + 208*y^4 - 106*y^3 + 42*y^2 - 10*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 18*x^14 - 45*x^13 + 105*x^12 - 208*x^11 + 345*x^10 - 496*x^9 + 621*x^8 - 645*x^7 + 535*x^6 - 359*x^5 + 208*x^4 - 106*x^3 + 42*x^2 - 10*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 6*x^15 + 18*x^14 - 45*x^13 + 105*x^12 - 208*x^11 + 345*x^10 - 496*x^9 + 621*x^8 - 645*x^7 + 535*x^6 - 359*x^5 + 208*x^4 - 106*x^3 + 42*x^2 - 10*x + 1)
 

\( x^{16} - 6 x^{15} + 18 x^{14} - 45 x^{13} + 105 x^{12} - 208 x^{11} + 345 x^{10} - 496 x^{9} + 621 x^{8} + \cdots + 1 \) 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:  $[0, 8]$
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:   \(145220537353515625\) \(\medspace = 5^{12}\cdot 29^{6}\) 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:  \(11.82\)
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:  \(\Q\)
$\Aut(K/\Q)$:   $C_4$
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.
Maximal CM subfield:  8.0.13140625.1

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3548501}a^{15}+\frac{1051588}{3548501}a^{14}-\frac{574847}{3548501}a^{13}-\frac{768308}{3548501}a^{12}-\frac{535660}{3548501}a^{11}-\frac{696506}{3548501}a^{10}+\frac{1012690}{3548501}a^{9}+\frac{1640755}{3548501}a^{8}-\frac{818145}{3548501}a^{7}-\frac{1015319}{3548501}a^{6}+\frac{140858}{322591}a^{5}+\frac{305639}{3548501}a^{4}-\frac{887802}{3548501}a^{3}-\frac{17445}{322591}a^{2}+\frac{524280}{3548501}a-\frac{898060}{3548501}$ 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:  $7$
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:   \( -\frac{7221127}{3548501} a^{15} + \frac{42282296}{3548501} a^{14} - \frac{124571768}{3548501} a^{13} + \frac{308985712}{3548501} a^{12} - \frac{716262939}{3548501} a^{11} + \frac{1406883783}{3548501} a^{10} - \frac{2304572476}{3548501} a^{9} + \frac{3265024438}{3548501} a^{8} - \frac{4029850629}{3548501} a^{7} + \frac{4089331012}{3548501} a^{6} - \frac{296995952}{322591} a^{5} + \frac{2086101904}{3548501} a^{4} - \frac{1171280635}{3548501} a^{3} + \frac{53034757}{322591} a^{2} - \frac{209656720}{3548501} a + \frac{34572096}{3548501} \)  (order $10$) 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:   $\frac{45514}{322591}a^{15}-\frac{205256}{322591}a^{14}+\frac{479288}{322591}a^{13}-\frac{1196276}{322591}a^{12}+\frac{2688904}{322591}a^{11}-\frac{4595379}{322591}a^{10}+\frac{6867582}{322591}a^{9}-\frac{8945250}{322591}a^{8}+\frac{9305330}{322591}a^{7}-\frac{7165218}{322591}a^{6}+\frac{3696305}{322591}a^{5}-\frac{560838}{322591}a^{4}-\frac{961932}{322591}a^{3}+\frac{887477}{322591}a^{2}-\frac{298941}{322591}a+\frac{234997}{322591}$, $\frac{4725815}{3548501}a^{15}-\frac{22724268}{3548501}a^{14}+\frac{55014279}{3548501}a^{13}-\frac{133314842}{3548501}a^{12}+\frac{304751566}{3548501}a^{11}-\frac{540283451}{3548501}a^{10}+\frac{805870399}{3548501}a^{9}-\frac{1067562097}{3548501}a^{8}+\frac{1194490749}{3548501}a^{7}-\frac{1009000590}{3548501}a^{6}+\frac{58451604}{322591}a^{5}-\frac{364617361}{3548501}a^{4}+\frac{191510279}{3548501}a^{3}-\frac{5970762}{322591}a^{2}+\frac{8822978}{3548501}a-\frac{1446884}{3548501}$, $\frac{10200933}{3548501}a^{15}-\frac{60440430}{3548501}a^{14}+\frac{177662321}{3548501}a^{13}-\frac{437838820}{3548501}a^{12}+\frac{1016883877}{3548501}a^{11}-\frac{1994330903}{3548501}a^{10}+\frac{3251349490}{3548501}a^{9}-\frac{4592345084}{3548501}a^{8}+\frac{5643486740}{3548501}a^{7}-\frac{5684303475}{3548501}a^{6}+\frac{406911111}{322591}a^{5}-\frac{2812691231}{3548501}a^{4}+\frac{1555257851}{3548501}a^{3}-\frac{69243646}{322591}a^{2}+\frac{263169158}{3548501}a-\frac{37443823}{3548501}$, $\frac{2011678}{3548501}a^{15}-\frac{10814494}{3548501}a^{14}+\frac{31670129}{3548501}a^{13}-\frac{81272286}{3548501}a^{12}+\frac{185931743}{3548501}a^{11}-\frac{364381815}{3548501}a^{10}+\frac{607917888}{3548501}a^{9}-\frac{868787514}{3548501}a^{8}+\frac{1082438309}{3548501}a^{7}-\frac{1120337004}{3548501}a^{6}+\frac{85072667}{322591}a^{5}-\frac{615406701}{3548501}a^{4}+\frac{347200145}{3548501}a^{3}-\frac{16145143}{322591}a^{2}+\frac{71593141}{3548501}a-\frac{12458065}{3548501}$, $\frac{12135118}{3548501}a^{15}-\frac{70638343}{3548501}a^{14}+\frac{200972465}{3548501}a^{13}-\frac{488981032}{3548501}a^{12}+\frac{1138478284}{3548501}a^{11}-\frac{2200939426}{3548501}a^{10}+\frac{3524421729}{3548501}a^{9}-\frac{4929545332}{3548501}a^{8}+\frac{5980182957}{3548501}a^{7}-\frac{5879279124}{3548501}a^{6}+\frac{407790200}{322591}a^{5}-\frac{2791555105}{3548501}a^{4}+\frac{1547710894}{3548501}a^{3}-\frac{66792007}{322591}a^{2}+\frac{230613679}{3548501}a-\frac{31997415}{3548501}$, $\frac{254763}{322591}a^{15}-\frac{1272991}{322591}a^{14}+\frac{3341829}{322591}a^{13}-\frac{8232846}{322591}a^{12}+\frac{18677610}{322591}a^{11}-\frac{34715037}{322591}a^{10}+\frac{54114416}{322591}a^{9}-\frac{73302162}{322591}a^{8}+\frac{85876673}{322591}a^{7}-\frac{79349525}{322591}a^{6}+\frac{56804287}{322591}a^{5}-\frac{32798350}{322591}a^{4}+\frac{18159773}{322591}a^{3}-\frac{8112383}{322591}a^{2}+\frac{2213182}{322591}a+\frac{45514}{322591}$, $\frac{14646001}{3548501}a^{15}-\frac{81606235}{3548501}a^{14}+\frac{229873328}{3548501}a^{13}-\frac{566620871}{3548501}a^{12}+\frac{1310455580}{3548501}a^{11}-\frac{2522146474}{3548501}a^{10}+\frac{4056989080}{3548501}a^{9}-\frac{5680230360}{3548501}a^{8}+\frac{6895586591}{3548501}a^{7}-\frac{6803081132}{3548501}a^{6}+\frac{479236119}{322591}a^{5}-\frac{3302435779}{3548501}a^{4}+\frac{1822324505}{3548501}a^{3}-\frac{78708056}{322591}a^{2}+\frac{284421959}{3548501}a-\frac{39332939}{3548501}$ 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:  \( 367.977318378 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 367.977318378 \cdot 1}{10\cdot\sqrt{145220537353515625}}\cr\approx \mathstrut & 0.234555692445 \end{aligned}\]

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 - 6*x^15 + 18*x^14 - 45*x^13 + 105*x^12 - 208*x^11 + 345*x^10 - 496*x^9 + 621*x^8 - 645*x^7 + 535*x^6 - 359*x^5 + 208*x^4 - 106*x^3 + 42*x^2 - 10*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 - 6*x^15 + 18*x^14 - 45*x^13 + 105*x^12 - 208*x^11 + 345*x^10 - 496*x^9 + 621*x^8 - 645*x^7 + 535*x^6 - 359*x^5 + 208*x^4 - 106*x^3 + 42*x^2 - 10*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 - 6*x^15 + 18*x^14 - 45*x^13 + 105*x^12 - 208*x^11 + 345*x^10 - 496*x^9 + 621*x^8 - 645*x^7 + 535*x^6 - 359*x^5 + 208*x^4 - 106*x^3 + 42*x^2 - 10*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 - 6*x^15 + 18*x^14 - 45*x^13 + 105*x^12 - 208*x^11 + 345*x^10 - 496*x^9 + 621*x^8 - 645*x^7 + 535*x^6 - 359*x^5 + 208*x^4 - 106*x^3 + 42*x^2 - 10*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 16T28):

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

\(\Q(\sqrt{5}) \), 4.4.725.1, \(\Q(\zeta_{5})\), 4.0.3625.1, 8.0.13140625.1

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: 16.8.4885218876572265625.1
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 ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ R ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(5\) 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 29.4.1.0a1.1$x^{4} + 2 x^{2} + 15 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
29.2.2.2a1.1$x^{4} + 48 x^{3} + 580 x^{2} + 125 x + 4$$2$$2$$2$$C_4$$$[\ ]_{2}^{2}$$
29.4.2.4a1.2$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)