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Properties

Label 16.0.13670807573...7641.1
Degree $16$
Signature $[0, 8]$
Discriminant $7^{12}\cdot 61^{14}$
Root discriminant $157.03$
Ramified primes $7, 61$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![87593557, 738777220, 1566061688, 190246216, 696973786, 24673950, 107694699, 1047639, 8227470, -599430, 287286, -49434, 6157, -995, 92, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 92*x^14 - 995*x^13 + 6157*x^12 - 49434*x^11 + 287286*x^10 - 599430*x^9 + 8227470*x^8 + 1047639*x^7 + 107694699*x^6 + 24673950*x^5 + 696973786*x^4 + 190246216*x^3 + 1566061688*x^2 + 738777220*x + 87593557)
 
gp: K = bnfinit(x^16 - 5*x^15 + 92*x^14 - 995*x^13 + 6157*x^12 - 49434*x^11 + 287286*x^10 - 599430*x^9 + 8227470*x^8 + 1047639*x^7 + 107694699*x^6 + 24673950*x^5 + 696973786*x^4 + 190246216*x^3 + 1566061688*x^2 + 738777220*x + 87593557, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 92 x^{14} - 995 x^{13} + 6157 x^{12} - 49434 x^{11} + 287286 x^{10} - 599430 x^{9} + 8227470 x^{8} + 1047639 x^{7} + 107694699 x^{6} + 24673950 x^{5} + 696973786 x^{4} + 190246216 x^{3} + 1566061688 x^{2} + 738777220 x + 87593557 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(136708075730434417658369530269167641=7^{12}\cdot 61^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $157.03$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $7, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{15} a^{11} - \frac{2}{15} a^{10} + \frac{1}{15} a^{9} - \frac{1}{15} a^{8} - \frac{1}{15} a^{7} + \frac{7}{15} a^{6} - \frac{1}{15} a^{5} + \frac{1}{15} a^{4} + \frac{1}{15} a^{3} + \frac{1}{5} a^{2} + \frac{1}{15} a - \frac{1}{15}$, $\frac{1}{45} a^{12} + \frac{1}{45} a^{11} + \frac{1}{9} a^{10} + \frac{7}{45} a^{9} + \frac{2}{15} a^{8} - \frac{1}{45} a^{7} - \frac{1}{9} a^{6} - \frac{7}{45} a^{5} - \frac{2}{15} a^{4} + \frac{16}{45} a^{3} - \frac{2}{9} a^{2} + \frac{22}{45} a + \frac{7}{45}$, $\frac{1}{45} a^{13} + \frac{1}{45} a^{11} - \frac{7}{45} a^{10} - \frac{4}{45} a^{9} - \frac{4}{45} a^{8} - \frac{1}{45} a^{7} + \frac{22}{45} a^{6} + \frac{4}{45} a^{5} + \frac{19}{45} a^{4} + \frac{16}{45} a^{3} - \frac{7}{45} a^{2} - \frac{2}{5} a - \frac{4}{45}$, $\frac{1}{1845} a^{14} + \frac{1}{1845} a^{13} - \frac{1}{123} a^{12} + \frac{32}{1845} a^{11} + \frac{131}{1845} a^{10} - \frac{92}{615} a^{9} + \frac{29}{369} a^{8} - \frac{137}{1845} a^{7} + \frac{874}{1845} a^{6} - \frac{73}{615} a^{5} - \frac{155}{369} a^{4} + \frac{362}{1845} a^{3} + \frac{254}{615} a^{2} - \frac{136}{369} a - \frac{124}{369}$, $\frac{1}{670022391118640654913533635687189006506946499509057239311957085} a^{15} + \frac{41435058095829419252450412499416795017163620257422887071694}{223340797039546884971177878562396335502315499836352413103985695} a^{14} - \frac{753264435837026149414545387735662407904523777370231930031288}{670022391118640654913533635687189006506946499509057239311957085} a^{13} + \frac{231624884328726932889484161171019889685545503991477966566007}{223340797039546884971177878562396335502315499836352413103985695} a^{12} + \frac{2066918752897018838362193380528398936086156109687175495690477}{74446932346515628323725959520798778500771833278784137701328565} a^{11} - \frac{51517526367938517854213254229225352079151226639129125823817232}{670022391118640654913533635687189006506946499509057239311957085} a^{10} - \frac{21190600708986450086137220826799975645569717826707647134654}{5447336513159680121248240940546252085422329264301278368389895} a^{9} + \frac{3972956773394458828622264737803408170269519486727410009144003}{670022391118640654913533635687189006506946499509057239311957085} a^{8} + \frac{60410011466755864741534887921734922748741408662977582967869}{5447336513159680121248240940546252085422329264301278368389895} a^{7} - \frac{162694218674891204887674620255400496728944564391524548335600793}{670022391118640654913533635687189006506946499509057239311957085} a^{6} + \frac{88949935474193588825582543113626714665020432642362716916967669}{223340797039546884971177878562396335502315499836352413103985695} a^{5} + \frac{264203289167800516286369957599549880856749886599664171991652237}{670022391118640654913533635687189006506946499509057239311957085} a^{4} + \frac{244420424246446773479387652764119994991108776575285147739682589}{670022391118640654913533635687189006506946499509057239311957085} a^{3} - \frac{12749159752447481047797751396124467636674109012395405959642476}{134004478223728130982706727137437801301389299901811447862391417} a^{2} - \frac{1701959731527826719660946141802383372199922615502920601185433}{7052867274933059525405617217759884279020489468516391992757443} a - \frac{58805164101952973827093286721247499000120470843582382873185091}{670022391118640654913533635687189006506946499509057239311957085}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{5}$, which has order $5$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 26057831853.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{-427}) \), \(\Q(\sqrt{-7}, \sqrt{61})\), 4.4.11122069.1, 4.0.226981.1, 8.0.123700418840761.1, 8.4.369740551915034629.1 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
7Data not computed
61Data not computed