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Properties

Label 16.0.22819400690...2368.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 73^{5}$
Root discriminant $18.72$
Ramified primes $2, 3, 73$
Class number $2$
Class group $[2]$
Galois group $C_2.D_4^2.C_2$ (as 16T659)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1369, 6590, 11656, 7048, -4051, -7460, -2176, 1402, 829, 58, -2, -36, -7, 14, -4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 4*x^14 + 14*x^13 - 7*x^12 - 36*x^11 - 2*x^10 + 58*x^9 + 829*x^8 + 1402*x^7 - 2176*x^6 - 7460*x^5 - 4051*x^4 + 7048*x^3 + 11656*x^2 + 6590*x + 1369)
 
gp: K = bnfinit(x^16 - 2*x^15 - 4*x^14 + 14*x^13 - 7*x^12 - 36*x^11 - 2*x^10 + 58*x^9 + 829*x^8 + 1402*x^7 - 2176*x^6 - 7460*x^5 - 4051*x^4 + 7048*x^3 + 11656*x^2 + 6590*x + 1369, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 4 x^{14} + 14 x^{13} - 7 x^{12} - 36 x^{11} - 2 x^{10} + 58 x^{9} + 829 x^{8} + 1402 x^{7} - 2176 x^{6} - 7460 x^{5} - 4051 x^{4} + 7048 x^{3} + 11656 x^{2} + 6590 x + 1369 \)

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:  \(228194006908815802368=2^{24}\cdot 3^{8}\cdot 73^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.72$
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:  $2, 3, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{105943373349547299513809} a^{15} + \frac{52575237858946595281399}{105943373349547299513809} a^{14} + \frac{49082535909958241058000}{105943373349547299513809} a^{13} + \frac{1618937331989616126552}{3653219770674044810821} a^{12} - \frac{32511614408397332213062}{105943373349547299513809} a^{11} + \frac{39481902384064698987445}{105943373349547299513809} a^{10} - \frac{44861141755824370478430}{105943373349547299513809} a^{9} + \frac{1220365880827375989476}{105943373349547299513809} a^{8} + \frac{42606489437446495349314}{105943373349547299513809} a^{7} + \frac{51663548518204236754247}{105943373349547299513809} a^{6} - \frac{8435237162231402199437}{105943373349547299513809} a^{5} + \frac{10504733901132778225507}{105943373349547299513809} a^{4} + \frac{3467095716654454443587}{105943373349547299513809} a^{3} + \frac{1574672127419090772568}{105943373349547299513809} a^{2} - \frac{34674693989531209514440}{105943373349547299513809} a - \frac{45010967895888748292220}{105943373349547299513809}$

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:  \( \frac{11611744361195356479}{149426478631237375901} a^{15} - \frac{35125618183247845393}{149426478631237375901} a^{14} - \frac{11040549237982846617}{149426478631237375901} a^{13} + \frac{6088807634781977763}{5152637194180599169} a^{12} - \frac{264819333671593145253}{149426478631237375901} a^{11} - \frac{153991835926244838419}{149426478631237375901} a^{10} + \frac{158537427084981885519}{149426478631237375901} a^{9} + \frac{493354058280643943056}{149426478631237375901} a^{8} + \frac{9121849748576957909159}{149426478631237375901} a^{7} + \frac{6897789312086004885155}{149426478631237375901} a^{6} - \frac{32771912436577086066674}{149426478631237375901} a^{5} - \frac{52735803845174221965564}{149426478631237375901} a^{4} + \frac{8940073732562417238050}{149426478631237375901} a^{3} + \frac{73499326690394231031197}{149426478631237375901} a^{2} + \frac{57615086531635872599796}{149426478631237375901} a + \frac{14691914943656109576805}{149426478631237375901} \) (order $12$)
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:  \( 8789.29006692 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2.D_4^2.C_2$ (as 16T659):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 25 conjugacy class representatives for $C_2.D_4^2.C_2$
Character table for $C_2.D_4^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{12})\), 8.0.1513728.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
3Data not computed
73Data not computed