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Properties

Label 16.0.25699221913...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 5^{12}\cdot 89^{4}$
Root discriminant $29.05$
Ramified primes $2, 5, 89$
Class number $14$ (GRH)
Class group $[14]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T210)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 0, 128, 112, 1016, -448, 2818, -2088, 2635, -1784, 1482, -798, 398, -128, 37, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 37*x^14 - 128*x^13 + 398*x^12 - 798*x^11 + 1482*x^10 - 1784*x^9 + 2635*x^8 - 2088*x^7 + 2818*x^6 - 448*x^5 + 1016*x^4 + 112*x^3 + 128*x^2 + 16)
 
gp: K = bnfinit(x^16 - 6*x^15 + 37*x^14 - 128*x^13 + 398*x^12 - 798*x^11 + 1482*x^10 - 1784*x^9 + 2635*x^8 - 2088*x^7 + 2818*x^6 - 448*x^5 + 1016*x^4 + 112*x^3 + 128*x^2 + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 37 x^{14} - 128 x^{13} + 398 x^{12} - 798 x^{11} + 1482 x^{10} - 1784 x^{9} + 2635 x^{8} - 2088 x^{7} + 2818 x^{6} - 448 x^{5} + 1016 x^{4} + 112 x^{3} + 128 x^{2} + 16 \)

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:  \(256992219136000000000000=2^{24}\cdot 5^{12}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.05$
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:  $2, 5, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{13} - \frac{1}{4} a^{11} + \frac{1}{6} a^{10} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{2} a^{7} - \frac{1}{3} a^{6} - \frac{1}{12} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{24} a^{14} - \frac{1}{8} a^{12} + \frac{1}{12} a^{11} + \frac{1}{12} a^{10} + \frac{5}{12} a^{9} + \frac{1}{4} a^{8} - \frac{1}{6} a^{7} + \frac{11}{24} a^{6} + \frac{5}{12} a^{5} - \frac{1}{12} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{64104649310902470312} a^{15} + \frac{911184843642413}{763150587034553218} a^{14} - \frac{2453686439736681833}{64104649310902470312} a^{13} + \frac{1803166405954472957}{16026162327725617578} a^{12} + \frac{1451351196825798733}{8013081163862808789} a^{11} - \frac{10723692030432466}{2671027054620936263} a^{10} - \frac{9668471246850858767}{32052324655451235156} a^{9} + \frac{160796790432852327}{381575293517276609} a^{8} + \frac{31183671028084408067}{64104649310902470312} a^{7} + \frac{4431346358987478299}{10684108218483745052} a^{6} + \frac{821008623647141089}{5342054109241872526} a^{5} + \frac{4269523280007899591}{32052324655451235156} a^{4} + \frac{1857046567583337153}{5342054109241872526} a^{3} + \frac{2328783551108383093}{8013081163862808789} a^{2} + \frac{3396024438292837069}{8013081163862808789} a - \frac{1936522562191186730}{8013081163862808789}$

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

Class group and class number

$C_{14}$, which has order $14$ (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:  \( 12304.4930513 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T210):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.0.11125.1, 4.0.712000.3, \(\Q(\sqrt{2}, \sqrt{5})\), 8.4.5696000000.1, 8.4.227840000.1, 8.0.506944000000.11

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
5Data not computed
89Data not computed