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Properties

Label 16.0.102...625.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.030\times 10^{19}$
Root discriminant \(15.43\)
Ramified primes $5,59$
Class number $1$
Class group trivial
Galois group $D_4:C_4$ (as 16T26)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 8*x^14 - 8*x^13 + 8*x^12 - 10*x^11 - 6*x^10 + 71*x^9 - 137*x^8 + 103*x^7 + 51*x^6 - 194*x^5 + 213*x^4 - 124*x^3 + 37*x^2 + x + 1)
 
Copy content gp:K = bnfinit(y^16 - 4*y^15 + 8*y^14 - 8*y^13 + 8*y^12 - 10*y^11 - 6*y^10 + 71*y^9 - 137*y^8 + 103*y^7 + 51*y^6 - 194*y^5 + 213*y^4 - 124*y^3 + 37*y^2 + y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 8*x^14 - 8*x^13 + 8*x^12 - 10*x^11 - 6*x^10 + 71*x^9 - 137*x^8 + 103*x^7 + 51*x^6 - 194*x^5 + 213*x^4 - 124*x^3 + 37*x^2 + x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 + 8*x^14 - 8*x^13 + 8*x^12 - 10*x^11 - 6*x^10 + 71*x^9 - 137*x^8 + 103*x^7 + 51*x^6 - 194*x^5 + 213*x^4 - 124*x^3 + 37*x^2 + x + 1)
 

\( x^{16} - 4 x^{15} + 8 x^{14} - 8 x^{13} + 8 x^{12} - 10 x^{11} - 6 x^{10} + 71 x^{9} - 137 x^{8} + \cdots + 1 \) 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:   \(10297981845947265625\) \(\medspace = 5^{12}\cdot 59^{6}\) 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:  \(15.43\)
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:  $5^{3/4}59^{1/2}\approx 25.683458749990002$
Ramified primes:   \(5\), \(59\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2^2$
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:  \(\Q(\zeta_{5})\)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{22592480419}a^{15}+\frac{397076423}{22592480419}a^{14}-\frac{2828720454}{22592480419}a^{13}+\frac{5973300131}{22592480419}a^{12}+\frac{5071993066}{22592480419}a^{11}-\frac{7908439260}{22592480419}a^{10}+\frac{2007821899}{22592480419}a^{9}+\frac{1572359281}{22592480419}a^{8}-\frac{5916163190}{22592480419}a^{7}-\frac{3006449652}{22592480419}a^{6}-\frac{5634726313}{22592480419}a^{5}+\frac{4359637712}{22592480419}a^{4}+\frac{60351811}{22592480419}a^{3}+\frac{8237297912}{22592480419}a^{2}+\frac{10052361464}{22592480419}a-\frac{7436319314}{22592480419}$ 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:   \( \frac{833636690}{22592480419} a^{15} - \frac{2263968184}{22592480419} a^{14} + \frac{5869917190}{22592480419} a^{13} - \frac{9428642146}{22592480419} a^{12} + \frac{14614390223}{22592480419} a^{11} - \frac{8080348884}{22592480419} a^{10} - \frac{3033658798}{22592480419} a^{9} + \frac{28694129788}{22592480419} a^{8} - \frac{85926926912}{22592480419} a^{7} + \frac{160637929179}{22592480419} a^{6} - \frac{110747899082}{22592480419} a^{5} - \frac{95883854070}{22592480419} a^{4} + \frac{289312474490}{22592480419} a^{3} - \frac{259282473116}{22592480419} a^{2} + \frac{131699487676}{22592480419} a - \frac{6354477578}{22592480419} \)  (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:   $\frac{692491013}{22592480419}a^{15}-\frac{4784019875}{22592480419}a^{14}+\frac{12872790373}{22592480419}a^{13}-\frac{16433476822}{22592480419}a^{12}+\frac{12157163994}{22592480419}a^{11}-\frac{15070007365}{22592480419}a^{10}+\frac{13469049846}{22592480419}a^{9}+\frac{77505936653}{22592480419}a^{8}-\frac{234885762567}{22592480419}a^{7}+\frac{247204501501}{22592480419}a^{6}+\frac{3042493016}{22592480419}a^{5}-\frac{314119942878}{22592480419}a^{4}+\frac{409425167812}{22592480419}a^{3}-\frac{277978933550}{22592480419}a^{2}+\frac{74568429798}{22592480419}a-\frac{6942687213}{22592480419}$, $\frac{2787887269}{22592480419}a^{15}-\frac{12090336318}{22592480419}a^{14}+\frac{21085098036}{22592480419}a^{13}-\frac{15269893005}{22592480419}a^{12}+\frac{10479598467}{22592480419}a^{11}-\frac{30746572517}{22592480419}a^{10}-\frac{22075840711}{22592480419}a^{9}+\frac{231729373827}{22592480419}a^{8}-\frac{374170266473}{22592480419}a^{7}+\frac{115468814822}{22592480419}a^{6}+\frac{337548114010}{22592480419}a^{5}-\frac{492431198285}{22592480419}a^{4}+\frac{320571721632}{22592480419}a^{3}-\frac{139799402971}{22592480419}a^{2}+\frac{57237820557}{22592480419}a-\frac{13166120401}{22592480419}$, $\frac{421741987}{22592480419}a^{15}-\frac{1864030110}{22592480419}a^{14}+\frac{5097766476}{22592480419}a^{13}-\frac{7285608953}{22592480419}a^{12}+\frac{8085838367}{22592480419}a^{11}-\frac{6993169510}{22592480419}a^{10}+\frac{3056664929}{22592480419}a^{9}+\frac{28461868404}{22592480419}a^{8}-\frac{85300854680}{22592480419}a^{7}+\frac{115970663110}{22592480419}a^{6}-\frac{45292452839}{22592480419}a^{5}-\frac{94169236021}{22592480419}a^{4}+\frac{204250143057}{22592480419}a^{3}-\frac{170081301055}{22592480419}a^{2}+\frac{66992436085}{22592480419}a+\frac{3154618565}{22592480419}$, $\frac{943477247}{22592480419}a^{15}-\frac{6533685939}{22592480419}a^{14}+\frac{13653097344}{22592480419}a^{13}-\frac{14562532593}{22592480419}a^{12}+\frac{7889079219}{22592480419}a^{11}-\frac{19592157266}{22592480419}a^{10}+\frac{2155000472}{22592480419}a^{9}+\frac{111733194168}{22592480419}a^{8}-\frac{250651277271}{22592480419}a^{7}+\frac{185537918919}{22592480419}a^{6}+\frac{113935022566}{22592480419}a^{5}-\frac{373672441563}{22592480419}a^{4}+\frac{348021571494}{22592480419}a^{3}-\frac{149801192463}{22592480419}a^{2}+\frac{14823292494}{22592480419}a+\frac{1526127703}{22592480419}$, $\frac{39374705}{22592480419}a^{15}+\frac{2423797969}{22592480419}a^{14}-\frac{8637262830}{22592480419}a^{13}+\frac{14233265822}{22592480419}a^{12}-\frac{10658712261}{22592480419}a^{11}+\frac{11036884566}{22592480419}a^{10}-\frac{19914933525}{22592480419}a^{9}-\frac{30502897869}{22592480419}a^{8}+\frac{157502930915}{22592480419}a^{7}-\frac{248493418360}{22592480419}a^{6}+\frac{85050368538}{22592480419}a^{5}+\frac{201119683630}{22592480419}a^{4}-\frac{371593257626}{22592480419}a^{3}+\frac{273069771624}{22592480419}a^{2}-\frac{103132380704}{22592480419}a+\frac{4711928078}{22592480419}$, $\frac{3663092553}{22592480419}a^{15}-\frac{13786403707}{22592480419}a^{14}+\frac{25483461481}{22592480419}a^{13}-\frac{21281210820}{22592480419}a^{12}+\frac{20220001709}{22592480419}a^{11}-\frac{31464741640}{22592480419}a^{10}-\frac{30330991510}{22592480419}a^{9}+\frac{256185984121}{22592480419}a^{8}-\frac{440900694975}{22592480419}a^{7}+\frac{221005958577}{22592480419}a^{6}+\frac{297877826858}{22592480419}a^{5}-\frac{623670175435}{22592480419}a^{4}+\frac{545232211582}{22592480419}a^{3}-\frac{246782804890}{22592480419}a^{2}+\frac{43136451242}{22592480419}a-\frac{14584454186}{22592480419}$, $\frac{2031921719}{22592480419}a^{15}-\frac{10476979937}{22592480419}a^{14}+\frac{24525622671}{22592480419}a^{13}-\frac{30426721970}{22592480419}a^{12}+\frac{26096976489}{22592480419}a^{11}-\frac{31748284594}{22592480419}a^{10}+\frac{4407469077}{22592480419}a^{9}+\frac{166880568286}{22592480419}a^{8}-\frac{439167794070}{22592480419}a^{7}+\frac{443956607508}{22592480419}a^{6}+\frac{14167176533}{22592480419}a^{5}-\frac{608274924340}{22592480419}a^{4}+\frac{769972799294}{22592480419}a^{3}-\frac{510061534556}{22592480419}a^{2}+\frac{183065535784}{22592480419}a-\frac{24956508853}{22592480419}$ 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:  \( 3370.63382124 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3370.63382124 \cdot 1}{10\cdot\sqrt{10297981845947265625}}\cr\approx \mathstrut & 0.255137596596 \end{aligned}\]

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 - 4*x^15 + 8*x^14 - 8*x^13 + 8*x^12 - 10*x^11 - 6*x^10 + 71*x^9 - 137*x^8 + 103*x^7 + 51*x^6 - 194*x^5 + 213*x^4 - 124*x^3 + 37*x^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 - 4*x^15 + 8*x^14 - 8*x^13 + 8*x^12 - 10*x^11 - 6*x^10 + 71*x^9 - 137*x^8 + 103*x^7 + 51*x^6 - 194*x^5 + 213*x^4 - 124*x^3 + 37*x^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 - 4*x^15 + 8*x^14 - 8*x^13 + 8*x^12 - 10*x^11 - 6*x^10 + 71*x^9 - 137*x^8 + 103*x^7 + 51*x^6 - 194*x^5 + 213*x^4 - 124*x^3 + 37*x^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 - 4*x^15 + 8*x^14 - 8*x^13 + 8*x^12 - 10*x^11 - 6*x^10 + 71*x^9 - 137*x^8 + 103*x^7 + 51*x^6 - 194*x^5 + 213*x^4 - 124*x^3 + 37*x^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

$D_4:C_4$ (as 16T26):

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 $D_4:C_4$
Character table for $D_4:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.2.1475.1, 4.2.7375.1, 8.2.128361875.1, 8.2.3209046875.2, 8.0.54390625.1

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 16 sibling: 16.4.35847274805742431640625.2
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 ${\href{/padicField/2.8.0.1}{8} }^{2}$ ${\href{/padicField/3.4.0.1}{4} }^{4}$ R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ R

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 5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
\(59\) Copy content Toggle raw display 59.2.1.0a1.1$x^{2} + 58 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
59.2.1.0a1.1$x^{2} + 58 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
59.2.2.2a1.2$x^{4} + 116 x^{3} + 3368 x^{2} + 232 x + 63$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
59.2.2.2a1.2$x^{4} + 116 x^{3} + 3368 x^{2} + 232 x + 63$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
59.2.2.2a1.2$x^{4} + 116 x^{3} + 3368 x^{2} + 232 x + 63$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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