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Properties

Label 16.0.23216514189...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 29^{8}\cdot 109^{2}$
Root discriminant $21.65$
Ramified primes $5, 29, 109$
Class number $2$
Class group $[2]$
Galois group $D_4^2.C_2$ (as 16T396)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![121, -110, 391, -202, 273, 52, -131, 177, -64, -9, 37, -39, 36, -16, 7, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 7*x^14 - 16*x^13 + 36*x^12 - 39*x^11 + 37*x^10 - 9*x^9 - 64*x^8 + 177*x^7 - 131*x^6 + 52*x^5 + 273*x^4 - 202*x^3 + 391*x^2 - 110*x + 121)
 
gp: K = bnfinit(x^16 - 3*x^15 + 7*x^14 - 16*x^13 + 36*x^12 - 39*x^11 + 37*x^10 - 9*x^9 - 64*x^8 + 177*x^7 - 131*x^6 + 52*x^5 + 273*x^4 - 202*x^3 + 391*x^2 - 110*x + 121, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 7 x^{14} - 16 x^{13} + 36 x^{12} - 39 x^{11} + 37 x^{10} - 9 x^{9} - 64 x^{8} + 177 x^{7} - 131 x^{6} + 52 x^{5} + 273 x^{4} - 202 x^{3} + 391 x^{2} - 110 x + 121 \)

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:  \(2321651418902203515625=5^{8}\cdot 29^{8}\cdot 109^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.65$
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:  $5, 29, 109$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{13} a^{13} + \frac{5}{13} a^{12} - \frac{3}{13} a^{11} + \frac{6}{13} a^{10} - \frac{2}{13} a^{9} + \frac{5}{13} a^{8} + \frac{3}{13} a^{7} + \frac{5}{13} a^{6} + \frac{6}{13} a^{5} + \frac{5}{13} a^{4} - \frac{3}{13} a^{3} - \frac{6}{13} a^{2} - \frac{4}{13} a - \frac{3}{13}$, $\frac{1}{65} a^{14} - \frac{2}{65} a^{13} + \frac{1}{65} a^{12} - \frac{12}{65} a^{11} - \frac{1}{13} a^{10} + \frac{19}{65} a^{9} - \frac{19}{65} a^{8} - \frac{29}{65} a^{7} + \frac{23}{65} a^{6} - \frac{24}{65} a^{5} - \frac{12}{65} a^{4} - \frac{24}{65} a^{3} + \frac{12}{65} a^{2} - \frac{14}{65} a - \frac{31}{65}$, $\frac{1}{16772654101228025} a^{15} - \frac{7235061162554}{3354530820245605} a^{14} - \frac{489924898097408}{16772654101228025} a^{13} + \frac{44207043307877}{145849166097635} a^{12} - \frac{1317637789095779}{16772654101228025} a^{11} - \frac{6589879823186061}{16772654101228025} a^{10} - \frac{400362238862727}{1290204161632925} a^{9} + \frac{36678186391563}{16772654101228025} a^{8} - \frac{226305956219859}{479218688606515} a^{7} + \frac{6490579723797577}{16772654101228025} a^{6} - \frac{125731069427657}{258040832326585} a^{5} - \frac{21658152257238}{104177975784025} a^{4} + \frac{482139501461327}{2396093443032575} a^{3} + \frac{378005996850611}{3354530820245605} a^{2} + \frac{5694251628135571}{16772654101228025} a - \frac{327416836545377}{1524786736475275}$

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:  \( 3793.72993285 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4^2.C_2$ (as 16T396):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 20 conjugacy class representatives for $D_4^2.C_2$
Character table for $D_4^2.C_2$

Intermediate fields

\(\Q(\sqrt{145}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{29}) \), 4.4.4205.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 4.4.725.1 x2, 8.0.1927340725.1, 8.0.48183518125.1, 8.8.442050625.1

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$109$$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$