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Properties

Label 16.0.135...608.2
Degree $16$
Signature $[0, 8]$
Discriminant $1.352\times 10^{30}$
Root discriminant \(76.42\)
Ramified primes $2,3,13$
Class number $12$ (GRH)
Class group [2, 6] (GRH)
Galois group $C_2^6.\SD_{16}$ (as 16T1250)

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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 + 24*x^14 + 318*x^12 + 3240*x^10 + 24084*x^8 + 129708*x^6 + 505638*x^4 + 1268748*x^2 + 2007837)
 
Copy content gp:K = bnfinit(y^16 + 24*y^14 + 318*y^12 + 3240*y^10 + 24084*y^8 + 129708*y^6 + 505638*y^4 + 1268748*y^2 + 2007837, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 24*x^14 + 318*x^12 + 3240*x^10 + 24084*x^8 + 129708*x^6 + 505638*x^4 + 1268748*x^2 + 2007837);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 + 24*x^14 + 318*x^12 + 3240*x^10 + 24084*x^8 + 129708*x^6 + 505638*x^4 + 1268748*x^2 + 2007837)
 

\( x^{16} + 24x^{14} + 318x^{12} + 3240x^{10} + 24084x^{8} + 129708x^{6} + 505638x^{4} + 1268748x^{2} + 2007837 \) 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:   \(1351639287558027493302608068608\) \(\medspace = 2^{52}\cdot 3^{14}\cdot 13^{7}\) 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:  \(76.42\)
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:  $2^{4}3^{7/8}13^{3/4}\approx 286.45644292474645$
Ramified primes:   \(2\), \(3\), \(13\) 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:  \(\Q(\sqrt{13}) \)
$\Aut(K/\Q)$:   $C_2$
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:  8.0.419853238272.1

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}$, $\frac{1}{3}a^{9}$, $\frac{1}{3}a^{10}$, $\frac{1}{3}a^{11}$, $\frac{1}{3}a^{12}$, $\frac{1}{3}a^{13}$, $\frac{1}{11\cdots 11}a^{14}-\frac{70138341611402}{857645186519847}a^{12}+\frac{18\cdots 18}{11\cdots 11}a^{10}+\frac{15\cdots 88}{11\cdots 11}a^{8}+\frac{293535673992708}{37\cdots 37}a^{6}-\frac{61682930987620}{37\cdots 37}a^{4}+\frac{139311998438200}{285881728839949}a^{2}+\frac{55667698301461}{285881728839949}$, $\frac{1}{14\cdots 41}a^{15}+\frac{41\cdots 18}{37\cdots 19}a^{13}-\frac{31\cdots 15}{14\cdots 41}a^{11}+\frac{36\cdots 87}{48\cdots 47}a^{9}+\frac{48\cdots 89}{48\cdots 47}a^{7}-\frac{96\cdots 82}{48\cdots 47}a^{5}+\frac{15\cdots 48}{37\cdots 19}a^{3}-\frac{11\cdots 99}{37\cdots 19}a$ 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:  $C_{2}\times C_{6}$, which has order $12$ (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:  $C_{2}\times C_{6}$, which has order $12$ (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:   $\frac{5569792741}{857645186519847}a^{14}+\frac{55305152806}{285881728839949}a^{12}+\frac{2239546808977}{857645186519847}a^{10}+\frac{21857987952413}{857645186519847}a^{8}+\frac{55918170038935}{285881728839949}a^{6}+\frac{285373807908727}{285881728839949}a^{4}+\frac{918946477766079}{285881728839949}a^{2}+\frac{18\cdots 23}{285881728839949}$, $\frac{17986629028}{857645186519847}a^{14}+\frac{522484090643}{857645186519847}a^{12}+\frac{7250220404176}{857645186519847}a^{10}+\frac{71730825662651}{857645186519847}a^{8}+\frac{178475098593086}{285881728839949}a^{6}+\frac{898917258882581}{285881728839949}a^{4}+\frac{28\cdots 14}{285881728839949}a^{2}+\frac{45\cdots 68}{285881728839949}$, $\frac{347208}{58666474213}a^{14}+\frac{19539929}{175999422639}a^{12}+\frac{80538248}{58666474213}a^{10}+\frac{787118928}{58666474213}a^{8}+\frac{4793859194}{58666474213}a^{6}+\frac{22641452897}{58666474213}a^{4}+\frac{61128806934}{58666474213}a^{2}-\frac{95684415101}{58666474213}$, $\frac{13\cdots 39}{14\cdots 41}a^{15}+\frac{12377850657212}{11\cdots 11}a^{14}+\frac{21\cdots 74}{11\cdots 57}a^{13}+\frac{21668045485265}{857645186519847}a^{12}+\frac{10\cdots 20}{48\cdots 47}a^{11}+\frac{35\cdots 35}{11\cdots 11}a^{10}+\frac{90\cdots 21}{48\cdots 47}a^{9}+\frac{34\cdots 33}{11\cdots 11}a^{8}+\frac{54\cdots 18}{48\cdots 47}a^{7}+\frac{78\cdots 10}{37\cdots 37}a^{6}+\frac{19\cdots 57}{48\cdots 47}a^{5}+\frac{37\cdots 69}{37\cdots 37}a^{4}+\frac{33\cdots 55}{37\cdots 19}a^{3}+\frac{91\cdots 28}{285881728839949}a^{2}+\frac{13\cdots 02}{37\cdots 19}a+\frac{12\cdots 59}{285881728839949}$, $\frac{36775631494063}{44\cdots 77}a^{15}-\frac{4011161613949}{11\cdots 11}a^{14}+\frac{44678351649698}{34\cdots 29}a^{13}-\frac{5289424142729}{857645186519847}a^{12}+\frac{67\cdots 05}{44\cdots 77}a^{11}-\frac{722119376775058}{11\cdots 11}a^{10}+\frac{60\cdots 32}{44\cdots 77}a^{9}-\frac{74\cdots 33}{11\cdots 11}a^{8}+\frac{36\cdots 02}{44\cdots 77}a^{7}-\frac{15\cdots 86}{37\cdots 37}a^{6}+\frac{16\cdots 97}{44\cdots 77}a^{5}-\frac{79\cdots 49}{37\cdots 37}a^{4}+\frac{34\cdots 08}{34\cdots 29}a^{3}-\frac{17\cdots 64}{285881728839949}a^{2}+\frac{68\cdots 57}{34\cdots 29}a-\frac{38\cdots 09}{285881728839949}$, $\frac{35\cdots 87}{48\cdots 47}a^{15}+\frac{184579987383455}{11\cdots 11}a^{14}+\frac{68\cdots 25}{37\cdots 19}a^{13}+\frac{293719434808192}{857645186519847}a^{12}+\frac{11\cdots 01}{48\cdots 47}a^{11}+\frac{42\cdots 47}{11\cdots 11}a^{10}+\frac{32\cdots 11}{14\cdots 41}a^{9}+\frac{37\cdots 69}{11\cdots 11}a^{8}+\frac{77\cdots 37}{48\cdots 47}a^{7}+\frac{75\cdots 37}{37\cdots 37}a^{6}+\frac{36\cdots 85}{48\cdots 47}a^{5}+\frac{26\cdots 52}{37\cdots 37}a^{4}+\frac{80\cdots 49}{37\cdots 19}a^{3}+\frac{38\cdots 75}{285881728839949}a^{2}+\frac{10\cdots 57}{37\cdots 19}a-\frac{18\cdots 00}{285881728839949}$, $\frac{16\cdots 43}{14\cdots 41}a^{15}-\frac{84350582511230}{11\cdots 11}a^{14}-\frac{414524688922350}{37\cdots 19}a^{13}-\frac{242527210261780}{857645186519847}a^{12}-\frac{32\cdots 88}{48\cdots 47}a^{11}-\frac{49\cdots 50}{11\cdots 11}a^{10}-\frac{12\cdots 69}{14\cdots 41}a^{9}-\frac{44\cdots 88}{11\cdots 11}a^{8}-\frac{29\cdots 46}{48\cdots 47}a^{7}-\frac{85\cdots 00}{37\cdots 37}a^{6}-\frac{13\cdots 19}{48\cdots 47}a^{5}-\frac{33\cdots 14}{37\cdots 37}a^{4}-\frac{28\cdots 97}{37\cdots 19}a^{3}-\frac{64\cdots 39}{285881728839949}a^{2}-\frac{47\cdots 86}{37\cdots 19}a-\frac{93\cdots 04}{285881728839949}$ 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:  \( 51747484.1082 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 51747484.1082 \cdot 12}{2\cdot\sqrt{1351639287558027493302608068608}}\cr\approx \mathstrut & 0.648707671297 \end{aligned}\] (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 + 24*x^14 + 318*x^12 + 3240*x^10 + 24084*x^8 + 129708*x^6 + 505638*x^4 + 1268748*x^2 + 2007837) 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 + 24*x^14 + 318*x^12 + 3240*x^10 + 24084*x^8 + 129708*x^6 + 505638*x^4 + 1268748*x^2 + 2007837, 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 + 24*x^14 + 318*x^12 + 3240*x^10 + 24084*x^8 + 129708*x^6 + 505638*x^4 + 1268748*x^2 + 2007837); 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 + 24*x^14 + 318*x^12 + 3240*x^10 + 24084*x^8 + 129708*x^6 + 505638*x^4 + 1268748*x^2 + 2007837); 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

$C_2^6.\SD_{16}$ (as 16T1250):

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 1024
The 34 conjugacy class representatives for $C_2^6.\SD_{16}$
Character table for $C_2^6.\SD_{16}$

Intermediate fields

\(\Q(\sqrt{3}) \), 4.4.7488.1, 8.0.419853238272.1

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(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: 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 R R $16$ $16$ ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ R ${\href{/padicField/17.4.0.1}{4} }^{4}$ $16$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ $16$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ $16$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{6}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.1.16.52e1.997$x^{16} + 8 x^{15} + 4 x^{14} + 2 x^{12} + 4 x^{10} + 8 x^{9} + 4 x^{6} + 8 x^{5} + 6$$16$$1$$52$16T1250$$[2, 2, \frac{5}{2}, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}, 4, \frac{17}{4}]^{2}$$
\(3\) Copy content Toggle raw display 3.2.8.14a1.3$x^{16} + 16 x^{15} + 128 x^{14} + 672 x^{13} + 2576 x^{12} + 7616 x^{11} + 17920 x^{10} + 34176 x^{9} + 53344 x^{8} + 68352 x^{7} + 71680 x^{6} + 60928 x^{5} + 41216 x^{4} + 21504 x^{3} + 8192 x^{2} + 2051 x + 259$$8$$2$$14$16T49$$[\ ]_{8}^{4}$$
\(13\) Copy content Toggle raw display 13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.2.1.0a1.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
13.1.2.1a1.1$x^{2} + 13$$2$$1$$1$$C_2$$$[\ ]_{2}$$
13.1.2.1a1.1$x^{2} + 13$$2$$1$$1$$C_2$$$[\ ]_{2}$$
13.1.4.3a1.1$x^{4} + 13$$4$$1$$3$$C_4$$$[\ ]_{4}$$
13.2.2.2a1.2$x^{4} + 24 x^{3} + 148 x^{2} + 48 x + 17$$2$$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)