Properties

Label 16.0.474...625.3
Degree $16$
Signature $(0, 8)$
Discriminant $4.749\times 10^{19}$
Root discriminant \(16.97\)
Ramified primes $3,5,11$
Class number $2$
Class group [2]
Galois group $C_2^2:\OD_{16}$ (as 16T95)

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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 - 5*x^15 + 12*x^14 - 15*x^13 + 8*x^12 + 29*x^10 - 80*x^9 + 45*x^8 + 80*x^7 + 29*x^6 + 8*x^4 + 15*x^3 + 12*x^2 + 5*x + 1)
 
Copy content gp:K = bnfinit(y^16 - 5*y^15 + 12*y^14 - 15*y^13 + 8*y^12 + 29*y^10 - 80*y^9 + 45*y^8 + 80*y^7 + 29*y^6 + 8*y^4 + 15*y^3 + 12*y^2 + 5*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 5*x^15 + 12*x^14 - 15*x^13 + 8*x^12 + 29*x^10 - 80*x^9 + 45*x^8 + 80*x^7 + 29*x^6 + 8*x^4 + 15*x^3 + 12*x^2 + 5*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 5*x^15 + 12*x^14 - 15*x^13 + 8*x^12 + 29*x^10 - 80*x^9 + 45*x^8 + 80*x^7 + 29*x^6 + 8*x^4 + 15*x^3 + 12*x^2 + 5*x + 1)
 

\( x^{16} - 5 x^{15} + 12 x^{14} - 15 x^{13} + 8 x^{12} + 29 x^{10} - 80 x^{9} + 45 x^{8} + 80 x^{7} + \cdots + 1 \) 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:  $(0, 8)$
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:   \(47490403326416015625\) \(\medspace = 3^{12}\cdot 5^{14}\cdot 11^{4}\) 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:  \(16.97\)
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:  $3^{3/4}5^{7/8}11^{1/2}\approx 30.912635812282684$
Ramified primes:   \(3\), \(5\), \(11\) 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:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:  8.0.6891328125.1

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{112561}a^{14}-\frac{14749}{112561}a^{13}-\frac{8583}{112561}a^{12}+\frac{14424}{112561}a^{11}-\frac{48302}{112561}a^{10}+\frac{5665}{112561}a^{9}-\frac{52771}{112561}a^{8}+\frac{39577}{112561}a^{7}+\frac{52771}{112561}a^{6}+\frac{5665}{112561}a^{5}+\frac{48302}{112561}a^{4}+\frac{14424}{112561}a^{3}+\frac{8583}{112561}a^{2}-\frac{14749}{112561}a-\frac{1}{112561}$, $\frac{1}{20373541}a^{15}+\frac{14}{20373541}a^{14}-\frac{8722293}{20373541}a^{13}+\frac{7926551}{20373541}a^{12}+\frac{1053408}{20373541}a^{11}-\frac{8107218}{20373541}a^{10}+\frac{1747777}{20373541}a^{9}+\frac{5306352}{20373541}a^{8}-\frac{1664544}{20373541}a^{7}+\frac{5207063}{20373541}a^{6}-\frac{2315907}{20373541}a^{5}+\frac{9590600}{20373541}a^{4}+\frac{6062977}{20373541}a^{3}+\frac{5580444}{20373541}a^{2}-\frac{9276516}{20373541}a-\frac{2603666}{20373541}$ 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:  $C_{2}$, which has order $2$
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:  $C_{2}$, which has order $2$
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:   $\frac{4371}{3631}a^{15}-\frac{24547}{3631}a^{14}+\frac{67039}{3631}a^{13}-\frac{103597}{3631}a^{12}+\frac{89138}{3631}a^{11}-\frac{38447}{3631}a^{10}+\frac{134125}{3631}a^{9}-\frac{423717}{3631}a^{8}+\frac{440355}{3631}a^{7}+\frac{138128}{3631}a^{6}-\frac{37481}{3631}a^{5}+\frac{28100}{3631}a^{4}+\frac{39021}{3631}a^{3}+\frac{37385}{3631}a^{2}+\frac{19897}{3631}a+\frac{3287}{3631}$, $\frac{24130745}{20373541}a^{15}-\frac{135120760}{20373541}a^{14}+\frac{367119262}{20373541}a^{13}-\frac{562958581}{20373541}a^{12}+\frac{479530649}{20373541}a^{11}-\frac{211278505}{20373541}a^{10}+\frac{764706965}{20373541}a^{9}-\frac{2364444599}{20373541}a^{8}+\frac{2395648260}{20373541}a^{7}+\frac{824714703}{20373541}a^{6}-\frac{106521106}{20373541}a^{5}-\frac{91043101}{20373541}a^{4}+\frac{254495170}{20373541}a^{3}+\frac{237852841}{20373541}a^{2}+\frac{112269265}{20373541}a+\frac{18749361}{20373541}$, $\frac{10014809}{20373541}a^{15}-\frac{56717054}{20373541}a^{14}+\frac{157350058}{20373541}a^{13}-\frac{252263346}{20373541}a^{12}+\frac{241097749}{20373541}a^{11}-\frac{148714761}{20373541}a^{10}+\frac{374031051}{20373541}a^{9}-\frac{1031999539}{20373541}a^{8}+\frac{1107590100}{20373541}a^{7}+\frac{109671159}{20373541}a^{6}+\frac{175409048}{20373541}a^{5}-\frac{83254663}{20373541}a^{4}+\frac{48983619}{20373541}a^{3}+\frac{101882801}{20373541}a^{2}+\frac{83636281}{20373541}a+\frac{14936354}{20373541}$, $\frac{34995101}{20373541}a^{15}-\frac{189214976}{20373541}a^{14}+\frac{493837983}{20373541}a^{13}-\frac{707973936}{20373541}a^{12}+\frac{517952642}{20373541}a^{11}-\frac{130627641}{20373541}a^{10}+\frac{994955566}{20373541}a^{9}-\frac{3168606903}{20373541}a^{8}+\frac{2764393314}{20373541}a^{7}+\frac{1992121561}{20373541}a^{6}-\frac{150417370}{20373541}a^{5}+\frac{1156830}{20373541}a^{4}+\frac{319500655}{20373541}a^{3}+\frac{396048142}{20373541}a^{2}+\frac{261187443}{20373541}a+\frac{61663603}{20373541}$, $\frac{14115936}{20373541}a^{15}-\frac{78403706}{20373541}a^{14}+\frac{209769204}{20373541}a^{13}-\frac{310695235}{20373541}a^{12}+\frac{238432900}{20373541}a^{11}-\frac{62563744}{20373541}a^{10}+\frac{390675914}{20373541}a^{9}-\frac{1332445060}{20373541}a^{8}+\frac{1288058160}{20373541}a^{7}+\frac{715043544}{20373541}a^{6}-\frac{281930154}{20373541}a^{5}-\frac{7788438}{20373541}a^{4}+\frac{205511551}{20373541}a^{3}+\frac{135970040}{20373541}a^{2}+\frac{28632984}{20373541}a+\frac{3813007}{20373541}$, $\frac{3568601}{20373541}a^{15}-\frac{416035}{657211}a^{14}+\frac{14457679}{20373541}a^{13}+\frac{26928365}{20373541}a^{12}-\frac{105813565}{20373541}a^{11}+\frac{139218992}{20373541}a^{10}+\frac{4673082}{20373541}a^{9}-\frac{81121248}{20373541}a^{8}-\frac{370880381}{20373541}a^{7}+\frac{897990688}{20373541}a^{6}+\frac{41008929}{20373541}a^{5}+\frac{148461757}{20373541}a^{4}+\frac{18722921}{20373541}a^{3}+\frac{103225370}{20373541}a^{2}+\frac{36318854}{20373541}a+\frac{11886365}{20373541}$, $\frac{10327304}{20373541}a^{15}-\frac{45608743}{20373541}a^{14}+\frac{87463550}{20373541}a^{13}-\frac{48961277}{20373541}a^{12}-\frac{94203828}{20373541}a^{11}+\frac{169435797}{20373541}a^{10}+\frac{214063823}{20373541}a^{9}-\frac{631256180}{20373541}a^{8}-\frac{6452096}{657211}a^{7}+\frac{1657933583}{20373541}a^{6}+\frac{322923885}{20373541}a^{5}-\frac{207390094}{20373541}a^{4}+\frac{63153536}{20373541}a^{3}+\frac{204603238}{20373541}a^{2}+\frac{119173134}{20373541}a+\frac{33703220}{20373541}$ 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:  \( 881.214818634 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 881.214818634 \cdot 2}{2\cdot\sqrt{47490403326416015625}}\cr\approx \mathstrut & 0.31061169931 \end{aligned}\]

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 - 5*x^15 + 12*x^14 - 15*x^13 + 8*x^12 + 29*x^10 - 80*x^9 + 45*x^8 + 80*x^7 + 29*x^6 + 8*x^4 + 15*x^3 + 12*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 - 5*x^15 + 12*x^14 - 15*x^13 + 8*x^12 + 29*x^10 - 80*x^9 + 45*x^8 + 80*x^7 + 29*x^6 + 8*x^4 + 15*x^3 + 12*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 - 5*x^15 + 12*x^14 - 15*x^13 + 8*x^12 + 29*x^10 - 80*x^9 + 45*x^8 + 80*x^7 + 29*x^6 + 8*x^4 + 15*x^3 + 12*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 = polynomial_ring(QQ); K, a = number_field(x^16 - 5*x^15 + 12*x^14 - 15*x^13 + 8*x^12 + 29*x^10 - 80*x^9 + 45*x^8 + 80*x^7 + 29*x^6 + 8*x^4 + 15*x^3 + 12*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

$C_2^2:\OD_{16}$ (as 16T95):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma: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 64
The 28 conjugacy class representatives for $C_2^2:\OD_{16}$
Character table for $C_2^2:\OD_{16}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-10 -14 \sqrt{5}})\), \(\Q(\zeta_{15})^+\), \(\Q(\sqrt{6 +18 \sqrt{5}})\), 8.0.6891328125.1, 8.4.56953125.1, 8.4.153140625.1

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

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.8.47490403326416015625.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ R R ${\href{/padicField/7.8.0.1}{8} }^{2}$ R ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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
\(3\) Copy content Toggle raw display 3.4.4.12a1.1$x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 24 x^{12} + 48 x^{11} + 96 x^{10} + 64 x^{9} + 24 x^{8} + 96 x^{7} + 96 x^{6} + 32 x^{4} + 64 x^{3} + 3 x^{2} + 16$$4$$4$$12$$C_8: C_2$$$[\ ]_{4}^{4}$$
\(5\) Copy content Toggle raw display 5.1.8.7a1.4$x^{8} + 20$$8$$1$$7$$C_8:C_2$$$[\ ]_{8}^{2}$$
5.1.8.7a1.4$x^{8} + 20$$8$$1$$7$$C_8:C_2$$$[\ ]_{8}^{2}$$
\(11\) Copy content Toggle raw display 11.4.1.0a1.1$x^{4} + 8 x^{2} + 10 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
11.4.1.0a1.1$x^{4} + 8 x^{2} + 10 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
11.4.2.4a1.2$x^{8} + 16 x^{6} + 20 x^{5} + 68 x^{4} + 160 x^{3} + 132 x^{2} + 40 x + 15$$2$$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)