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Properties

Label 16.0.39228809359...4849.7
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 17^{14}$
Root discriminant $81.68$
Ramified primes $13, 17$
Class number $9792$ (GRH)
Class group $[2, 12, 408]$ (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![66232832, 138718144, 79867792, -22705032, -33955384, -4700058, 4327649, 335966, -163742, 75224, 5879, 68, 865, -22, 12, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 12*x^14 - 22*x^13 + 865*x^12 + 68*x^11 + 5879*x^10 + 75224*x^9 - 163742*x^8 + 335966*x^7 + 4327649*x^6 - 4700058*x^5 - 33955384*x^4 - 22705032*x^3 + 79867792*x^2 + 138718144*x + 66232832)
 
gp: K = bnfinit(x^16 - 2*x^15 + 12*x^14 - 22*x^13 + 865*x^12 + 68*x^11 + 5879*x^10 + 75224*x^9 - 163742*x^8 + 335966*x^7 + 4327649*x^6 - 4700058*x^5 - 33955384*x^4 - 22705032*x^3 + 79867792*x^2 + 138718144*x + 66232832, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 12 x^{14} - 22 x^{13} + 865 x^{12} + 68 x^{11} + 5879 x^{10} + 75224 x^{9} - 163742 x^{8} + 335966 x^{7} + 4327649 x^{6} - 4700058 x^{5} - 33955384 x^{4} - 22705032 x^{3} + 79867792 x^{2} + 138718144 x + 66232832 \)

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:  \(3922880935919264967742950184849=13^{12}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.68$
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:  $13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{32} a^{11} + \frac{1}{32} a^{10} - \frac{1}{16} a^{9} + \frac{1}{16} a^{7} + \frac{1}{16} a^{6} + \frac{7}{32} a^{5} + \frac{5}{32} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{64} a^{12} - \frac{1}{64} a^{11} - \frac{1}{16} a^{9} - \frac{3}{32} a^{8} - \frac{5}{32} a^{7} - \frac{13}{64} a^{6} + \frac{15}{64} a^{5} + \frac{1}{32} a^{4} - \frac{1}{8} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{64} a^{13} - \frac{1}{64} a^{11} - \frac{1}{16} a^{10} - \frac{1}{32} a^{9} - \frac{7}{64} a^{7} - \frac{7}{32} a^{6} - \frac{15}{64} a^{5} + \frac{5}{32} a^{4} + \frac{3}{8} a^{3} - \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{512} a^{14} + \frac{3}{512} a^{12} - \frac{1}{64} a^{11} + \frac{3}{256} a^{10} + \frac{1}{32} a^{9} - \frac{47}{512} a^{8} - \frac{35}{256} a^{7} - \frac{99}{512} a^{6} + \frac{43}{256} a^{5} + \frac{5}{64} a^{4} + \frac{3}{64} a^{3} - \frac{11}{32} a^{2} + \frac{3}{8} a$, $\frac{1}{25632862144280391208451093275599852215528746620157952} a^{15} - \frac{8471880761155863616605055849337148672219870817109}{25632862144280391208451093275599852215528746620157952} a^{14} + \frac{140779027141661281799820078111919998655627040998299}{25632862144280391208451093275599852215528746620157952} a^{13} + \frac{171297334901168820540217975423063862616729250642857}{25632862144280391208451093275599852215528746620157952} a^{12} + \frac{196832006253586293507341918791849849170453104110667}{12816431072140195604225546637799926107764373310078976} a^{11} - \frac{242186115621171764694161696862451124087509159546159}{12816431072140195604225546637799926107764373310078976} a^{10} - \frac{1434519117916197026975361758422850880208220119790223}{25632862144280391208451093275599852215528746620157952} a^{9} - \frac{2805184666569710615734403757817082311995502371066059}{25632862144280391208451093275599852215528746620157952} a^{8} + \frac{2371446760557247435253483407669805827068707920018227}{25632862144280391208451093275599852215528746620157952} a^{7} - \frac{4456581130595957363842934116677462017307498062697003}{25632862144280391208451093275599852215528746620157952} a^{6} - \frac{764345723766658971427673475791629330265637460813079}{12816431072140195604225546637799926107764373310078976} a^{5} + \frac{266355412951841967251354829942220689569370740610549}{1602053884017524450528193329724990763470546663759872} a^{4} + \frac{1068386053589475581822031469647228576308788258366811}{3204107768035048901056386659449981526941093327519744} a^{3} - \frac{315772709264637828766934219035008129941258642170361}{1602053884017524450528193329724990763470546663759872} a^{2} - \frac{49213530658934412079102571782453339998509066723731}{400513471004381112632048332431247690867636665939968} a - \frac{10284364914633299172535668289446502915342309633717}{50064183875547639079006041553905961358454583242496}$

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

Class group and class number

$C_{2}\times C_{12}\times C_{408}$, which has order $9792$ (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:  \( 32676067.5694 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.0.3757.1, 4.4.830297.1, 4.0.63869.1, 8.0.1980626399884457.1, 8.8.1980626399884457.1, 8.0.689393108209.2

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

Sibling fields

Galois closure: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$13$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$17$17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$