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Properties

Label 16.0.68967557758...5184.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{12}\cdot 17^{14}$
Root discriminant $20.06$
Ramified primes $2, 17$
Class number $2$
Class group $[2]$
Galois group $C_2^5.C_2.C_2$ (as 16T257)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -64, 192, -358, 563, -631, 615, -377, 172, 47, -57, 45, 8, -11, 11, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 11*x^14 - 11*x^13 + 8*x^12 + 45*x^11 - 57*x^10 + 47*x^9 + 172*x^8 - 377*x^7 + 615*x^6 - 631*x^5 + 563*x^4 - 358*x^3 + 192*x^2 - 64*x + 16)
 
gp: K = bnfinit(x^16 - 3*x^15 + 11*x^14 - 11*x^13 + 8*x^12 + 45*x^11 - 57*x^10 + 47*x^9 + 172*x^8 - 377*x^7 + 615*x^6 - 631*x^5 + 563*x^4 - 358*x^3 + 192*x^2 - 64*x + 16, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 11 x^{14} - 11 x^{13} + 8 x^{12} + 45 x^{11} - 57 x^{10} + 47 x^{9} + 172 x^{8} - 377 x^{7} + 615 x^{6} - 631 x^{5} + 563 x^{4} - 358 x^{3} + 192 x^{2} - 64 x + 16 \)

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:  \(689675577587306205184=2^{12}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.06$
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:  $2, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} + \frac{5}{13} a^{11} + \frac{6}{13} a^{10} - \frac{2}{13} a^{9} - \frac{3}{13} a^{8} + \frac{6}{13} a^{7} - \frac{3}{13} a^{6} - \frac{2}{13} a^{5} + \frac{5}{13} a^{4} - \frac{3}{13} a^{3} - \frac{4}{13} a^{2} - \frac{4}{13} a - \frac{2}{13}$, $\frac{1}{26} a^{13} - \frac{1}{26} a^{12} - \frac{11}{26} a^{11} + \frac{1}{26} a^{10} - \frac{2}{13} a^{9} + \frac{11}{26} a^{8} - \frac{1}{2} a^{7} + \frac{3}{26} a^{6} + \frac{2}{13} a^{5} - \frac{7}{26} a^{4} + \frac{1}{26} a^{3} + \frac{7}{26} a^{2} + \frac{9}{26} a + \frac{6}{13}$, $\frac{1}{676} a^{14} - \frac{7}{676} a^{13} - \frac{17}{676} a^{12} + \frac{293}{676} a^{11} + \frac{12}{169} a^{10} - \frac{123}{676} a^{9} + \frac{243}{676} a^{8} + \frac{295}{676} a^{7} + \frac{64}{169} a^{6} + \frac{279}{676} a^{5} - \frac{69}{676} a^{4} + \frac{89}{676} a^{3} + \frac{275}{676} a^{2} - \frac{75}{338} a + \frac{27}{169}$, $\frac{1}{1177592} a^{15} - \frac{213}{1177592} a^{14} - \frac{551}{1177592} a^{13} - \frac{20177}{1177592} a^{12} - \frac{259241}{588796} a^{11} + \frac{44589}{1177592} a^{10} + \frac{419325}{1177592} a^{9} - \frac{418183}{1177592} a^{8} + \frac{134583}{588796} a^{7} + \frac{225639}{1177592} a^{6} - \frac{8143}{1177592} a^{5} + \frac{395931}{1177592} a^{4} + \frac{418793}{1177592} a^{3} - \frac{76717}{294398} a^{2} - \frac{69949}{294398} a - \frac{71928}{147199}$

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:  \( 5147.71826227 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T257):

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 $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.4.6565418768.1, 8.2.386201104.1, 8.2.6565418768.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.12.15$x^{8} + 2 x^{7} + 2 x^{4} + 12$$4$$2$$12$$C_2^2:C_4$$[2, 2]^{4}$
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$