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Properties

Label 16.0.26873856000...000.11
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{12}$
Root discriminant $16.38$
Ramified primes $2, 3, 5$
Class number $2$
Class group $[2]$
Galois group $C_4 \times D_4$ (as 16T19)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![361, -798, 1071, -1540, 1666, -1124, 1110, -454, 267, -128, 45, -62, 41, -30, 14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 14*x^14 - 30*x^13 + 41*x^12 - 62*x^11 + 45*x^10 - 128*x^9 + 267*x^8 - 454*x^7 + 1110*x^6 - 1124*x^5 + 1666*x^4 - 1540*x^3 + 1071*x^2 - 798*x + 361)
 
gp: K = bnfinit(x^16 - 4*x^15 + 14*x^14 - 30*x^13 + 41*x^12 - 62*x^11 + 45*x^10 - 128*x^9 + 267*x^8 - 454*x^7 + 1110*x^6 - 1124*x^5 + 1666*x^4 - 1540*x^3 + 1071*x^2 - 798*x + 361, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 14 x^{14} - 30 x^{13} + 41 x^{12} - 62 x^{11} + 45 x^{10} - 128 x^{9} + 267 x^{8} - 454 x^{7} + 1110 x^{6} - 1124 x^{5} + 1666 x^{4} - 1540 x^{3} + 1071 x^{2} - 798 x + 361 \)

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:  \(26873856000000000000=2^{24}\cdot 3^{8}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.38$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{27474} a^{14} - \frac{239}{4579} a^{13} - \frac{2069}{27474} a^{12} + \frac{1595}{13737} a^{11} + \frac{4937}{13737} a^{10} - \frac{2144}{13737} a^{9} - \frac{2845}{9158} a^{8} - \frac{2068}{13737} a^{7} + \frac{3073}{13737} a^{6} + \frac{6182}{13737} a^{5} - \frac{1999}{4579} a^{4} - \frac{1890}{4579} a^{3} - \frac{5131}{13737} a^{2} - \frac{6503}{13737} a - \frac{475}{1446}$, $\frac{1}{12754434692239806} a^{15} - \frac{15755822554}{2125739115373301} a^{14} - \frac{1700841035991073}{12754434692239806} a^{13} - \frac{592277838126173}{6377217346119903} a^{12} - \frac{751200758188061}{2125739115373301} a^{11} - \frac{396058893816431}{6377217346119903} a^{10} - \frac{2159561221216985}{12754434692239806} a^{9} - \frac{483584240210977}{6377217346119903} a^{8} - \frac{1711096416288554}{6377217346119903} a^{7} - \frac{2294511892842011}{6377217346119903} a^{6} + \frac{534470531442427}{2125739115373301} a^{5} - \frac{111388409689270}{6377217346119903} a^{4} + \frac{3170479386359923}{6377217346119903} a^{3} - \frac{554444734176579}{2125739115373301} a^{2} + \frac{32165052804613}{411433377169026} a - \frac{18045844430611}{335643018216837}$

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:  \( \frac{39588464}{6140979987} a^{15} - \frac{425424005}{12281959974} a^{14} + \frac{718735085}{6140979987} a^{13} - \frac{1102067725}{4093986658} a^{12} + \frac{2207046845}{6140979987} a^{11} - \frac{2167136359}{6140979987} a^{10} + \frac{473539990}{2046993329} a^{9} - \frac{2229001775}{4093986658} a^{8} + \frac{14455901725}{6140979987} a^{7} - \frac{26757015230}{6140979987} a^{6} + \frac{48270245006}{6140979987} a^{5} - \frac{21114357805}{2046993329} a^{4} + \frac{48719748340}{6140979987} a^{3} - \frac{46854375710}{6140979987} a^{2} + \frac{20790331760}{6140979987} a + \frac{216648805}{646418946} \) (order $6$)
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:  \( 2522.8409077 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times D_4$ (as 16T19):

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 $C_4 \times D_4$
Character table for $C_4 \times D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), 4.0.8000.2, 4.4.72000.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 4.0.9000.1, 4.0.9000.2, 8.0.81000000.1, 8.0.5184000000.3, 8.0.3240000.1

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 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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed