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Properties

Label 16.4.14111142256...0561.3
Degree $16$
Signature $[4, 6]$
Discriminant $11^{8}\cdot 37^{12}$
Root discriminant $49.76$
Ramified primes $11, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_4:C_4$ (as 16T26)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21217, -46032, 60104, -88249, 53816, -50087, 30835, -11312, 4119, -1031, -867, 705, -224, 83, -23, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 23*x^14 + 83*x^13 - 224*x^12 + 705*x^11 - 867*x^10 - 1031*x^9 + 4119*x^8 - 11312*x^7 + 30835*x^6 - 50087*x^5 + 53816*x^4 - 88249*x^3 + 60104*x^2 - 46032*x + 21217)
 
gp: K = bnfinit(x^16 - 2*x^15 - 23*x^14 + 83*x^13 - 224*x^12 + 705*x^11 - 867*x^10 - 1031*x^9 + 4119*x^8 - 11312*x^7 + 30835*x^6 - 50087*x^5 + 53816*x^4 - 88249*x^3 + 60104*x^2 - 46032*x + 21217, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 23 x^{14} + 83 x^{13} - 224 x^{12} + 705 x^{11} - 867 x^{10} - 1031 x^{9} + 4119 x^{8} - 11312 x^{7} + 30835 x^{6} - 50087 x^{5} + 53816 x^{4} - 88249 x^{3} + 60104 x^{2} - 46032 x + 21217 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1411114225648575427835680561=11^{8}\cdot 37^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.76$
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:  $11, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{3}{7} a^{11} + \frac{1}{7} a^{10} - \frac{1}{7} a^{9} - \frac{1}{7} a^{8} + \frac{1}{7} a^{7} + \frac{1}{7} a^{6} + \frac{2}{7} a^{5} - \frac{3}{7} a^{4} - \frac{1}{7} a^{3} + \frac{2}{7} a^{2} - \frac{3}{7} a$, $\frac{1}{7} a^{13} - \frac{1}{7} a^{11} + \frac{3}{7} a^{10} + \frac{2}{7} a^{9} - \frac{3}{7} a^{8} - \frac{2}{7} a^{7} - \frac{1}{7} a^{6} - \frac{2}{7} a^{5} + \frac{1}{7} a^{4} - \frac{2}{7} a^{3} - \frac{2}{7} a^{2} + \frac{2}{7} a$, $\frac{1}{7} a^{14} - \frac{1}{7} a^{11} + \frac{3}{7} a^{10} + \frac{3}{7} a^{9} - \frac{3}{7} a^{8} - \frac{1}{7} a^{6} + \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{3}{7} a^{3} - \frac{3}{7} a^{2} - \frac{3}{7} a$, $\frac{1}{207022302892968681265663346572032529} a^{15} + \frac{3771018949034947697219760715571982}{207022302892968681265663346572032529} a^{14} - \frac{8370033844853877827394063815910068}{207022302892968681265663346572032529} a^{13} + \frac{967899312036420326858274367889517}{29574614698995525895094763796004647} a^{12} + \frac{13682824470312332859551439213767839}{29574614698995525895094763796004647} a^{11} + \frac{73246894818387448582300548961262338}{207022302892968681265663346572032529} a^{10} + \frac{31968887967934725064916958440486592}{207022302892968681265663346572032529} a^{9} + \frac{86227410350427693093268531806521573}{207022302892968681265663346572032529} a^{8} - \frac{6472330286431587092673431709739649}{207022302892968681265663346572032529} a^{7} - \frac{49843179413763870757320458285553282}{207022302892968681265663346572032529} a^{6} - \frac{101103799125790751787960478563439517}{207022302892968681265663346572032529} a^{5} - \frac{85626167341987603748543540248666019}{207022302892968681265663346572032529} a^{4} + \frac{67665072694605107252540712565052635}{207022302892968681265663346572032529} a^{3} - \frac{9501615419420958906954517561889555}{29574614698995525895094763796004647} a^{2} + \frac{55783701385449415816054755639182224}{207022302892968681265663346572032529} a - \frac{7162739496874916054836189073240176}{29574614698995525895094763796004647}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $9$
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:  \( 16768421.8387 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:C_4$ (as 16T26):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $D_4:C_4$
Character table for $D_4:C_4$

Intermediate fields

\(\Q(\sqrt{37}) \), 4.4.6129013.1, 4.2.15059.1, 4.2.557183.1, 8.2.3414981850379.1, 8.2.2494508291.1, 8.4.37564800354169.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$37$37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.2$x^{4} - 148$$4$$1$$3$$C_4$$[\ ]_{4}$