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Properties

Label 16.0.68158711342...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 29^{6}\cdot 41^{2}\cdot 1321^{2}$
Root discriminant $30.87$
Ramified primes $5, 29, 41, 1321$
Class number $2$
Class group $[2]$
Galois group 16T1429

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![121531, 21948, 122354, -4835, 52207, -4790, 12153, 293, 1134, 337, 168, -90, 77, -40, 14, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 14*x^14 - 40*x^13 + 77*x^12 - 90*x^11 + 168*x^10 + 337*x^9 + 1134*x^8 + 293*x^7 + 12153*x^6 - 4790*x^5 + 52207*x^4 - 4835*x^3 + 122354*x^2 + 21948*x + 121531)
 
gp: K = bnfinit(x^16 - 3*x^15 + 14*x^14 - 40*x^13 + 77*x^12 - 90*x^11 + 168*x^10 + 337*x^9 + 1134*x^8 + 293*x^7 + 12153*x^6 - 4790*x^5 + 52207*x^4 - 4835*x^3 + 122354*x^2 + 21948*x + 121531, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 14 x^{14} - 40 x^{13} + 77 x^{12} - 90 x^{11} + 168 x^{10} + 337 x^{9} + 1134 x^{8} + 293 x^{7} + 12153 x^{6} - 4790 x^{5} + 52207 x^{4} - 4835 x^{3} + 122354 x^{2} + 21948 x + 121531 \)

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:  \(681587113420645172265625=5^{8}\cdot 29^{6}\cdot 41^{2}\cdot 1321^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.87$
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:  $5, 29, 41, 1321$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{22015798211359885930676869844911344187} a^{15} + \frac{6189977566531771856948523606817040761}{22015798211359885930676869844911344187} a^{14} - \frac{6912536229679456965988077260270612581}{22015798211359885930676869844911344187} a^{13} - \frac{7166190570583382758510938820198897729}{22015798211359885930676869844911344187} a^{12} + \frac{1188289324222775287176219887318529765}{22015798211359885930676869844911344187} a^{11} + \frac{9240182263706516554416591325457706187}{22015798211359885930676869844911344187} a^{10} - \frac{8494237129109846917873466849100363666}{22015798211359885930676869844911344187} a^{9} + \frac{4533782854826731303221814959521615882}{22015798211359885930676869844911344187} a^{8} + \frac{9171935850813529919605817865572780136}{22015798211359885930676869844911344187} a^{7} + \frac{5063016967173014219298457844733468279}{22015798211359885930676869844911344187} a^{6} - \frac{1307984960614722482724073764050930420}{22015798211359885930676869844911344187} a^{5} + \frac{5045828636409213280801933998237020246}{22015798211359885930676869844911344187} a^{4} - \frac{10619813232774143990701684093395887747}{22015798211359885930676869844911344187} a^{3} + \frac{373515503419308689559271222111139612}{22015798211359885930676869844911344187} a^{2} + \frac{1767332972634515274445060378970244904}{22015798211359885930676869844911344187} a - \frac{3820408695628036805463783567160731410}{22015798211359885930676869844911344187}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 57513.6723658 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1429:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 77 conjugacy class representatives for t16n1429 are not computed
Character table for t16n1429 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.4.15243125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$5.8.4.1$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$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
41Data not computed
1321Data not computed