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Properties

Label 16.0.30232191979...5625.3
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 29^{4}\cdot 149^{6}$
Root discriminant $33.89$
Ramified primes $5, 29, 149$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T392)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2224, -160, -1640, 1494, -373, 1563, -888, -115, 1639, -1475, 880, -310, 124, -36, 13, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 13*x^14 - 36*x^13 + 124*x^12 - 310*x^11 + 880*x^10 - 1475*x^9 + 1639*x^8 - 115*x^7 - 888*x^6 + 1563*x^5 - 373*x^4 + 1494*x^3 - 1640*x^2 - 160*x + 2224)
 
gp: K = bnfinit(x^16 - 2*x^15 + 13*x^14 - 36*x^13 + 124*x^12 - 310*x^11 + 880*x^10 - 1475*x^9 + 1639*x^8 - 115*x^7 - 888*x^6 + 1563*x^5 - 373*x^4 + 1494*x^3 - 1640*x^2 - 160*x + 2224, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 13 x^{14} - 36 x^{13} + 124 x^{12} - 310 x^{11} + 880 x^{10} - 1475 x^{9} + 1639 x^{8} - 115 x^{7} - 888 x^{6} + 1563 x^{5} - 373 x^{4} + 1494 x^{3} - 1640 x^{2} - 160 x + 2224 \)

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:  \(3023219197928805422265625=5^{8}\cdot 29^{4}\cdot 149^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.89$
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:  $5, 29, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{10} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{3}{10} a^{9} + \frac{1}{10} a^{8} - \frac{2}{5} a^{7} - \frac{3}{10} a^{6} - \frac{1}{10} a^{4} - \frac{1}{10} a^{3} - \frac{3}{10} a^{2} - \frac{1}{2} a + \frac{2}{5}$, $\frac{1}{10} a^{13} + \frac{1}{5} a^{11} - \frac{1}{2} a^{10} - \frac{3}{10} a^{9} - \frac{1}{5} a^{8} - \frac{1}{10} a^{7} + \frac{2}{5} a^{6} - \frac{1}{10} a^{5} - \frac{3}{10} a^{4} - \frac{1}{2} a^{3} - \frac{1}{10} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{77140} a^{14} + \frac{199}{5510} a^{13} + \frac{85}{2204} a^{12} - \frac{9587}{38570} a^{11} + \frac{1877}{7714} a^{10} + \frac{13557}{38570} a^{9} - \frac{345}{1102} a^{8} + \frac{31211}{77140} a^{7} + \frac{24259}{77140} a^{6} - \frac{7739}{77140} a^{5} - \frac{7803}{38570} a^{4} + \frac{17191}{77140} a^{3} - \frac{5233}{11020} a^{2} + \frac{7421}{38570} a + \frac{341}{3857}$, $\frac{1}{11346132749172815399960} a^{15} + \frac{182990614523823}{39124595686802811724} a^{14} + \frac{27483233602665850607}{1620876107024687914280} a^{13} + \frac{653599263195809747}{2836533187293203849990} a^{12} + \frac{48224748082578544}{193884701797211473} a^{11} + \frac{2156119452785886726753}{5673066374586407699980} a^{10} + \frac{702577553577551086444}{1418266593646601924995} a^{9} - \frac{5010918563737181905147}{11346132749172815399960} a^{8} + \frac{33276898031435511337}{119432976307082267368} a^{7} - \frac{30798237080579608589}{1031466613561165036360} a^{6} + \frac{930930865158114207847}{2836533187293203849990} a^{5} + \frac{414348398397492189251}{2269226549834563079992} a^{4} + \frac{721364863675622743371}{11346132749172815399960} a^{3} - \frac{247730019907300053569}{515733306780582518180} a^{2} - \frac{961080788647284490463}{2836533187293203849990} a + \frac{120644945700005243936}{1418266593646601924995}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  $7$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 75079.232878 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T392):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.3725.1, 4.4.725.1, 4.0.108025.1, 8.0.11669400625.4, 8.4.11669400625.2, 8.4.11669400625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$5.4.2.1$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$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$149$149.2.1.2$x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.1.2$x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$
149.4.2.1$x^{4} + 745 x^{2} + 199809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
149.4.2.1$x^{4} + 745 x^{2} + 199809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$