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Properties

Label 16.10.732...008.1
Degree $16$
Signature $[10, 3]$
Discriminant $-7.328\times 10^{20}$
Root discriminant \(20.14\)
Ramified primes $2,17$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2\wr C_8$ (as 16T1351)

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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 - 34*x^12 + 102*x^10 - 51*x^8 - 136*x^6 + 153*x^4 - 17*x^2 - 17)
 
Copy content gp:K = bnfinit(y^16 - 34*y^12 + 102*y^10 - 51*y^8 - 136*y^6 + 153*y^4 - 17*y^2 - 17, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 34*x^12 + 102*x^10 - 51*x^8 - 136*x^6 + 153*x^4 - 17*x^2 - 17);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 34*x^12 + 102*x^10 - 51*x^8 - 136*x^6 + 153*x^4 - 17*x^2 - 17)
 

\( x^{16} - 34x^{12} + 102x^{10} - 51x^{8} - 136x^{6} + 153x^{4} - 17x^{2} - 17 \) 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:  $[10, 3]$
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:   \(-732780301186512843008\) \(\medspace = -\,2^{8}\cdot 17^{15}\) 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:  \(20.14\)
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:  $2^{15/8}17^{15/16}\approx 52.236865100328075$
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:  \(\Q(\sqrt{-17}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{270478}a^{14}+\frac{48429}{270478}a^{12}-\frac{40985}{135239}a^{10}-\frac{1}{2}a^{9}+\frac{40289}{135239}a^{8}-\frac{1}{2}a^{7}+\frac{125805}{270478}a^{6}-\frac{20990}{135239}a^{4}-\frac{690}{135239}a^{2}-\frac{1}{2}a+\frac{55634}{135239}$, $\frac{1}{270478}a^{15}+\frac{48429}{270478}a^{13}-\frac{40985}{135239}a^{11}-\frac{1}{2}a^{10}+\frac{40289}{135239}a^{9}-\frac{1}{2}a^{8}+\frac{125805}{270478}a^{7}-\frac{20990}{135239}a^{5}-\frac{690}{135239}a^{3}-\frac{1}{2}a^{2}+\frac{55634}{135239}a$ 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 $1$ (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:  $C_{2}$, which has order $2$ (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:  $12$
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:   $\frac{29557}{135239}a^{14}+\frac{46377}{135239}a^{12}-\frac{927278}{135239}a^{10}+\frac{1572785}{135239}a^{8}+\frac{833514}{135239}a^{6}-\frac{2554576}{135239}a^{4}+\frac{459235}{135239}a^{2}+\frac{276752}{135239}$, $\frac{1407}{135239}a^{14}-\frac{20853}{135239}a^{12}-\frac{108162}{135239}a^{10}+\frac{719159}{135239}a^{8}-\frac{425933}{135239}a^{6}-\frac{1318807}{135239}a^{4}+\frac{763120}{135239}a^{2}+\frac{353031}{135239}$, $\frac{11383}{135239}a^{14}+\frac{33143}{135239}a^{12}-\frac{321127}{135239}a^{10}+\frac{163715}{135239}a^{8}+\frac{803978}{135239}a^{6}-\frac{464670}{135239}a^{4}-\frac{426533}{135239}a^{2}+\frac{320887}{135239}$, $\frac{14463}{135239}a^{14}+\frac{25846}{135239}a^{12}-\frac{432753}{135239}a^{10}+\frac{721346}{135239}a^{8}+\frac{147448}{135239}a^{6}-\frac{1015542}{135239}a^{4}+\frac{732627}{135239}a^{2}+\frac{60223}{135239}$, $\frac{36407}{135239}a^{14}+\frac{43760}{135239}a^{12}-\frac{1179928}{135239}a^{10}+\frac{2297921}{135239}a^{8}+\frac{719617}{135239}a^{6}-\frac{3681374}{135239}a^{4}+\frac{1149160}{135239}a^{2}+\frac{526026}{135239}$, $\frac{3080}{135239}a^{14}-\frac{7297}{135239}a^{12}-\frac{111626}{135239}a^{10}+\frac{557631}{135239}a^{8}-\frac{656530}{135239}a^{6}-\frac{550872}{135239}a^{4}+\frac{1159160}{135239}a^{2}-\frac{260664}{135239}$, $\frac{49960}{135239}a^{14}+\frac{87130}{135239}a^{12}-\frac{1536670}{135239}a^{10}+\frac{2451869}{135239}a^{8}+\frac{1472904}{135239}a^{6}-\frac{4226797}{135239}a^{4}+\frac{973763}{135239}a^{2}+\frac{626380}{135239}$, $\frac{32371}{270478}a^{15}-\frac{8701}{270478}a^{14}+\frac{4671}{270478}a^{13}+\frac{23995}{270478}a^{12}-\frac{571801}{135239}a^{11}+\frac{376201}{270478}a^{10}+\frac{3011103}{270478}a^{9}-\frac{1517831}{270478}a^{8}-\frac{9176}{135239}a^{7}+\frac{536117}{270478}a^{6}-\frac{2596095}{135239}a^{5}+\frac{1278491}{135239}a^{4}+\frac{1060357}{135239}a^{3}-\frac{1381281}{270478}a^{2}+\frac{1118053}{270478}a-\frac{186292}{135239}$, $\frac{2729}{10403}a^{15}+\frac{37357}{270478}a^{14}+\frac{3029}{10403}a^{13}+\frac{35025}{135239}a^{12}-\frac{177693}{20806}a^{11}-\frac{576882}{135239}a^{10}+\frac{361601}{20806}a^{9}+\frac{812776}{135239}a^{8}+\frac{43651}{10403}a^{7}+\frac{814882}{135239}a^{6}-\frac{296868}{10403}a^{5}-\frac{2855435}{270478}a^{4}+\frac{218195}{20806}a^{3}-\frac{297079}{270478}a^{2}+\frac{28414}{10403}a+\frac{608967}{270478}$, $\frac{37627}{270478}a^{15}-\frac{35497}{270478}a^{14}+\frac{27697}{270478}a^{13}-\frac{30642}{135239}a^{12}-\frac{1241707}{270478}a^{11}+\frac{1103917}{270478}a^{10}+\frac{2960523}{270478}a^{9}-\frac{1730523}{270478}a^{8}-\frac{241221}{270478}a^{7}-\frac{662728}{135239}a^{6}-\frac{2294033}{135239}a^{5}+\frac{3211255}{270478}a^{4}+\frac{2846535}{270478}a^{3}-\frac{120568}{135239}a^{2}+\frac{246515}{135239}a-\frac{836635}{270478}$, $\frac{32371}{270478}a^{15}-\frac{71267}{270478}a^{14}+\frac{4671}{270478}a^{13}-\frac{90263}{270478}a^{12}-\frac{571801}{135239}a^{11}+\frac{2271209}{270478}a^{10}+\frac{3011103}{270478}a^{9}-\frac{4496795}{270478}a^{8}-\frac{9176}{135239}a^{7}-\frac{1022103}{270478}a^{6}-\frac{2596095}{135239}a^{5}+\frac{3802443}{135239}a^{4}+\frac{1060357}{135239}a^{3}-\frac{2945551}{270478}a^{2}+\frac{1118053}{270478}a-\frac{607471}{135239}$, $\frac{37357}{270478}a^{15}+\frac{8413}{135239}a^{14}+\frac{35025}{135239}a^{13}+\frac{51379}{270478}a^{12}-\frac{576882}{135239}a^{11}-\frac{465615}{270478}a^{10}+\frac{812776}{135239}a^{9}+\frac{84846}{135239}a^{8}+\frac{814882}{135239}a^{7}+\frac{558007}{135239}a^{6}-\frac{2855435}{270478}a^{5}-\frac{272661}{270478}a^{4}-\frac{297079}{270478}a^{3}-\frac{249864}{135239}a^{2}+\frac{608967}{270478}a-\frac{26674}{135239}$ 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:  \( 38695.4736663 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{3}\cdot 38695.4736663 \cdot 1}{2\cdot\sqrt{732780301186512843008}}\cr\approx \mathstrut & 0.181544392984 \end{aligned}\] (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 - 34*x^12 + 102*x^10 - 51*x^8 - 136*x^6 + 153*x^4 - 17*x^2 - 17) 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 - 34*x^12 + 102*x^10 - 51*x^8 - 136*x^6 + 153*x^4 - 17*x^2 - 17, 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 - 34*x^12 + 102*x^10 - 51*x^8 - 136*x^6 + 153*x^4 - 17*x^2 - 17); 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 - 34*x^12 + 102*x^10 - 51*x^8 - 136*x^6 + 153*x^4 - 17*x^2 - 17); 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

$C_2\wr C_8$ (as 16T1351):

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 solvable group of order 2048
The 56 conjugacy class representatives for $C_2\wr C_8$
Character table for $C_2\wr C_8$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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: 16.6.732780301186512843008.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }^{2}$ $16$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ R ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}$

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
\(2\) Copy content Toggle raw display 2.4.1.0a1.1$x^{4} + x + 1$$1$$4$$0$$C_4$$$[\ ]^{4}$$
2.4.1.0a1.1$x^{4} + x + 1$$1$$4$$0$$C_4$$$[\ ]^{4}$$
2.4.2.8a4.2$x^{8} + 2 x^{7} + 2 x^{5} + 4 x^{4} + 2 x^{3} + x^{2} + 2 x + 7$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$$[2, 2, 2, 2]^{4}$$
\(17\) Copy content Toggle raw display 17.1.16.15a1.16$x^{16} + 272$$16$$1$$15$$C_{16}$$$[\ ]_{16}$$

Spectrum of ring of integers

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