↑

Properties

Label 16.0.13364932132...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{22}\cdot 5^{8}\cdot 13^{8}$
Root discriminant $20.91$
Ramified primes $2, 5, 13$
Class number $1$
Class group Trivial
Galois group $(C_4\times C_8):C_2$ (as 16T114)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![191, -591, 1285, -2189, 3037, -3235, 2450, -1198, 204, 193, -172, 54, 17, -32, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 21*x^14 - 32*x^13 + 17*x^12 + 54*x^11 - 172*x^10 + 193*x^9 + 204*x^8 - 1198*x^7 + 2450*x^6 - 3235*x^5 + 3037*x^4 - 2189*x^3 + 1285*x^2 - 591*x + 191)
 
gp: K = bnfinit(x^16 - 7*x^15 + 21*x^14 - 32*x^13 + 17*x^12 + 54*x^11 - 172*x^10 + 193*x^9 + 204*x^8 - 1198*x^7 + 2450*x^6 - 3235*x^5 + 3037*x^4 - 2189*x^3 + 1285*x^2 - 591*x + 191, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 21 x^{14} - 32 x^{13} + 17 x^{12} + 54 x^{11} - 172 x^{10} + 193 x^{9} + 204 x^{8} - 1198 x^{7} + 2450 x^{6} - 3235 x^{5} + 3037 x^{4} - 2189 x^{3} + 1285 x^{2} - 591 x + 191 \)

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:  \(1336493213286400000000=2^{22}\cdot 5^{8}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.91$
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:  $2, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{31} a^{14} - \frac{10}{31} a^{13} + \frac{3}{31} a^{12} + \frac{5}{31} a^{11} + \frac{13}{31} a^{10} - \frac{8}{31} a^{9} + \frac{3}{31} a^{8} + \frac{10}{31} a^{7} - \frac{1}{31} a^{6} - \frac{1}{31} a^{5} - \frac{10}{31} a^{4} + \frac{5}{31} a^{3} - \frac{1}{31} a^{2} - \frac{8}{31} a - \frac{7}{31}$, $\frac{1}{8812032731610029263} a^{15} + \frac{31483269881770887}{8812032731610029263} a^{14} - \frac{141034562821931871}{303863197641725147} a^{13} + \frac{3413408710399901640}{8812032731610029263} a^{12} - \frac{1446389389206577792}{8812032731610029263} a^{11} + \frac{3454261136560223408}{8812032731610029263} a^{10} + \frac{863206808363917013}{8812032731610029263} a^{9} - \frac{3493994607856882703}{8812032731610029263} a^{8} + \frac{3219275186639292016}{8812032731610029263} a^{7} - \frac{321698181551309234}{8812032731610029263} a^{6} + \frac{985158193656430433}{8812032731610029263} a^{5} + \frac{2739186573758294181}{8812032731610029263} a^{4} - \frac{2927247218682714395}{8812032731610029263} a^{3} + \frac{3102203382393018579}{8812032731610029263} a^{2} + \frac{1254778618362192586}{8812032731610029263} a + \frac{594244250535839117}{8812032731610029263}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( 7679.94606709 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_4\times C_8):C_2$ (as 16T114):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_4\times C_8):C_2$
Character table for $(C_4\times C_8):C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{5}, \sqrt{13})\), 8.8.1142440000.1

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.22.31$x^{8} + 12 x^{6} + 6 x^{4} + 8 x^{2} + 52$$4$$2$$22$$C_8$$[3, 4]^{2}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
5Data not computed
13Data not computed