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Properties

Label 16.0.42657346521...8125.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{11}\cdot 29^{3}\cdot 101^{8}\cdot 149^{3}$
Root discriminant $146.01$
Ramified primes $5, 29, 101, 149$
Class number $192$ (GRH)
Class group $[2, 2, 4, 12]$ (GRH)
Galois group 16T1642

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![222521795, -127293585, 343797955, -139729010, 174591890, -45893065, 36338415, -5085440, 3807771, -262273, 225107, -6354, 7596, -58, 136, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 136*x^14 - 58*x^13 + 7596*x^12 - 6354*x^11 + 225107*x^10 - 262273*x^9 + 3807771*x^8 - 5085440*x^7 + 36338415*x^6 - 45893065*x^5 + 174591890*x^4 - 139729010*x^3 + 343797955*x^2 - 127293585*x + 222521795)
 
gp: K = bnfinit(x^16 + 136*x^14 - 58*x^13 + 7596*x^12 - 6354*x^11 + 225107*x^10 - 262273*x^9 + 3807771*x^8 - 5085440*x^7 + 36338415*x^6 - 45893065*x^5 + 174591890*x^4 - 139729010*x^3 + 343797955*x^2 - 127293585*x + 222521795, 1)
 

Normalized defining polynomial

\( x^{16} + 136 x^{14} - 58 x^{13} + 7596 x^{12} - 6354 x^{11} + 225107 x^{10} - 262273 x^{9} + 3807771 x^{8} - 5085440 x^{7} + 36338415 x^{6} - 45893065 x^{5} + 174591890 x^{4} - 139729010 x^{3} + 343797955 x^{2} - 127293585 x + 222521795 \)

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:  \(42657346521926437784823093798828125=5^{11}\cdot 29^{3}\cdot 101^{8}\cdot 149^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $146.01$
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:  $5, 29, 101, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{24} a^{14} + \frac{1}{24} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{1}{4} a^{9} + \frac{1}{3} a^{8} + \frac{1}{24} a^{7} - \frac{1}{2} a^{6} + \frac{11}{24} a^{5} - \frac{5}{12} a^{4} + \frac{1}{4} a^{3} - \frac{1}{12} a^{2} + \frac{1}{12} a + \frac{7}{24}$, $\frac{1}{37619102941759905930969876219389682408750385146834936} a^{15} - \frac{27928487645331650650425648403826860551938328509443}{1567462622573329413790411509141236767031266047784789} a^{14} - \frac{4394072411833581321384183069391525614611690425941025}{37619102941759905930969876219389682408750385146834936} a^{13} + \frac{507235666191683463226775516422614246948052515462891}{12539700980586635310323292073129894136250128382278312} a^{12} + \frac{735469592134801955500079555694040181555965597471201}{3134925245146658827580823018282473534062532095569578} a^{11} + \frac{1888682386967646844137943182160555622816681573780923}{12539700980586635310323292073129894136250128382278312} a^{10} + \frac{4688499970180709925267358400394762386118349780280371}{18809551470879952965484938109694841204375192573417468} a^{9} + \frac{3658375196753179256735360271518403956225626792540985}{37619102941759905930969876219389682408750385146834936} a^{8} - \frac{12759101500979196434781437195931828461712654291597457}{37619102941759905930969876219389682408750385146834936} a^{7} + \frac{1204980814709693772650607994947760344733640965062899}{37619102941759905930969876219389682408750385146834936} a^{6} - \frac{6065104120381171382391719703406461667879556125922499}{12539700980586635310323292073129894136250128382278312} a^{5} + \frac{4122207178473683123102021656347105628062170282037601}{9404775735439976482742469054847420602187596286708734} a^{4} + \frac{816836720901995520257366131817180812010237440290380}{4702387867719988241371234527423710301093798143354367} a^{3} - \frac{4438678574181652550328902259477902538626995909289657}{9404775735439976482742469054847420602187596286708734} a^{2} + \frac{18254927132112630959676759939182892635577725593629173}{37619102941759905930969876219389682408750385146834936} a - \frac{7020773638194523923680855668137061215957277882801363}{37619102941759905930969876219389682408750385146834936}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{12}$, 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:  \( 250214598.048 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1642:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 73 conjugacy class representatives for t16n1642 are not computed
Character table for t16n1642 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.0.13912283190625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
$101$101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$149$149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.4.3.3$x^{4} + 298$$4$$1$$3$$C_4$$[\ ]_{4}$
149.8.0.1$x^{8} - x + 11$$1$$8$$0$$C_8$$[\ ]^{8}$