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Properties

Label 16.0.16164273696...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 101^{6}$
Root discriminant $89.23$
Ramified primes $2, 5, 29, 101$
Class number $192$ (GRH)
Class group $[2, 96]$ (GRH)
Galois group $C_2^2.C_2^5.C_2$ (as 16T493)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![132381829, 11851192, 72393140, 937078, 17713171, -1108314, 2607307, -325092, 273604, -41224, 21768, -3092, 1235, -154, 44, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 44*x^14 - 154*x^13 + 1235*x^12 - 3092*x^11 + 21768*x^10 - 41224*x^9 + 273604*x^8 - 325092*x^7 + 2607307*x^6 - 1108314*x^5 + 17713171*x^4 + 937078*x^3 + 72393140*x^2 + 11851192*x + 132381829)
 
gp: K = bnfinit(x^16 - 4*x^15 + 44*x^14 - 154*x^13 + 1235*x^12 - 3092*x^11 + 21768*x^10 - 41224*x^9 + 273604*x^8 - 325092*x^7 + 2607307*x^6 - 1108314*x^5 + 17713171*x^4 + 937078*x^3 + 72393140*x^2 + 11851192*x + 132381829, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 44 x^{14} - 154 x^{13} + 1235 x^{12} - 3092 x^{11} + 21768 x^{10} - 41224 x^{9} + 273604 x^{8} - 325092 x^{7} + 2607307 x^{6} - 1108314 x^{5} + 17713171 x^{4} + 937078 x^{3} + 72393140 x^{2} + 11851192 x + 132381829 \)

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:  \(16164273696996018327577600000000=2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 101^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $89.23$
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:  $2, 5, 29, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{29} a^{12} - \frac{3}{29} a^{11} + \frac{8}{29} a^{10} + \frac{2}{29} a^{9} - \frac{9}{29} a^{8} - \frac{9}{29} a^{7} - \frac{12}{29} a^{6} - \frac{14}{29} a^{5} - \frac{8}{29} a^{4} + \frac{13}{29} a^{3} - \frac{13}{29} a^{2} - \frac{11}{29} a - \frac{8}{29}$, $\frac{1}{29} a^{13} - \frac{1}{29} a^{11} - \frac{3}{29} a^{10} - \frac{3}{29} a^{9} - \frac{7}{29} a^{8} - \frac{10}{29} a^{7} + \frac{8}{29} a^{6} + \frac{8}{29} a^{5} - \frac{11}{29} a^{4} - \frac{3}{29} a^{3} + \frac{8}{29} a^{2} - \frac{12}{29} a + \frac{5}{29}$, $\frac{1}{29} a^{14} - \frac{6}{29} a^{11} + \frac{5}{29} a^{10} - \frac{5}{29} a^{9} + \frac{10}{29} a^{8} - \frac{1}{29} a^{7} - \frac{4}{29} a^{6} + \frac{4}{29} a^{5} - \frac{11}{29} a^{4} - \frac{8}{29} a^{3} + \frac{4}{29} a^{2} - \frac{6}{29} a - \frac{8}{29}$, $\frac{1}{3173977541533152535450152212276393414316073} a^{15} + \frac{31416876359723410724821509815336535022051}{3173977541533152535450152212276393414316073} a^{14} - \frac{26291462738575869759609534572392544225266}{3173977541533152535450152212276393414316073} a^{13} - \frac{26918698842319314072690778436651089256565}{3173977541533152535450152212276393414316073} a^{12} + \frac{1325432615210950462867844099597423645213838}{3173977541533152535450152212276393414316073} a^{11} - \frac{1238859535449226826635872020503675380363305}{3173977541533152535450152212276393414316073} a^{10} - \frac{104617751104460711840618499717721253156484}{288543412866650230495468382934217583119643} a^{9} + \frac{432895157266766358700644435569668450949082}{3173977541533152535450152212276393414316073} a^{8} - \frac{1431481051456731561681747111567894701249655}{3173977541533152535450152212276393414316073} a^{7} - \frac{584161325510240812714068265251782228017438}{3173977541533152535450152212276393414316073} a^{6} + \frac{1217368167215737101082898574601252640889040}{3173977541533152535450152212276393414316073} a^{5} + \frac{1320412844398024488088195313421450339289618}{3173977541533152535450152212276393414316073} a^{4} + \frac{833358858126040505655794723465360450596109}{3173977541533152535450152212276393414316073} a^{3} - \frac{1506317946701162135444009530473776724596359}{3173977541533152535450152212276393414316073} a^{2} + \frac{3362599190077230690555767400671355284616}{9949772857470697603292013204628192521367} a + \frac{406018985231134181752991028187309512642477}{3173977541533152535450152212276393414316073}$

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

Class group and class number

$C_{2}\times C_{96}$, which has order $192$ (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:  \( 3199036.50724 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T493):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.1171600.6, 4.0.40400.1, 4.4.725.1, 8.0.1372646560000.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: 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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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

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
2Data not computed
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
101Data not computed