↑

Properties

Label 16.0.68102711625...9361.1
Degree $16$
Signature $[0, 8]$
Discriminant $31^{8}\cdot 41^{8}$
Root discriminant $35.65$
Ramified primes $31, 41$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2^2.SD_{16}$ (as 16T163)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![67, 474, 433, -1830, 851, 144, 628, -2516, 3849, -3364, 1852, -612, 102, -28, 24, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 24*x^14 - 28*x^13 + 102*x^12 - 612*x^11 + 1852*x^10 - 3364*x^9 + 3849*x^8 - 2516*x^7 + 628*x^6 + 144*x^5 + 851*x^4 - 1830*x^3 + 433*x^2 + 474*x + 67)
 
gp: K = bnfinit(x^16 - 8*x^15 + 24*x^14 - 28*x^13 + 102*x^12 - 612*x^11 + 1852*x^10 - 3364*x^9 + 3849*x^8 - 2516*x^7 + 628*x^6 + 144*x^5 + 851*x^4 - 1830*x^3 + 433*x^2 + 474*x + 67, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 24 x^{14} - 28 x^{13} + 102 x^{12} - 612 x^{11} + 1852 x^{10} - 3364 x^{9} + 3849 x^{8} - 2516 x^{7} + 628 x^{6} + 144 x^{5} + 851 x^{4} - 1830 x^{3} + 433 x^{2} + 474 x + 67 \)

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:  \(6810271162553824314519361=31^{8}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.65$
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:  $31, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{10} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{3}{10} a^{4} - \frac{1}{5} a^{3} - \frac{1}{10} a^{2} - \frac{3}{10} a + \frac{1}{10}$, $\frac{1}{10} a^{9} + \frac{1}{10} a^{5} - \frac{2}{5} a^{4} + \frac{1}{10} a^{3} + \frac{3}{10} a^{2} - \frac{1}{10} a + \frac{2}{5}$, $\frac{1}{10} a^{10} + \frac{1}{10} a^{6} - \frac{2}{5} a^{5} + \frac{1}{10} a^{4} + \frac{3}{10} a^{3} - \frac{1}{10} a^{2} + \frac{2}{5} a$, $\frac{1}{10} a^{11} + \frac{1}{10} a^{7} - \frac{2}{5} a^{6} + \frac{1}{10} a^{5} + \frac{3}{10} a^{4} - \frac{1}{10} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{10} a^{12} - \frac{1}{2} a^{6} - \frac{3}{10} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{10} a^{2} + \frac{3}{10} a - \frac{1}{10}$, $\frac{1}{10} a^{13} - \frac{1}{2} a^{7} - \frac{3}{10} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{10} a^{3} + \frac{3}{10} a^{2} - \frac{1}{10} a$, $\frac{1}{781982750} a^{14} - \frac{7}{781982750} a^{13} - \frac{2648872}{78198275} a^{12} + \frac{2535861}{781982750} a^{11} + \frac{14128689}{390991375} a^{10} - \frac{34201991}{781982750} a^{9} + \frac{1064143}{31279310} a^{8} + \frac{126692403}{781982750} a^{7} - \frac{115420973}{781982750} a^{6} - \frac{6938038}{15639655} a^{5} - \frac{138130998}{390991375} a^{4} - \frac{101556251}{390991375} a^{3} - \frac{175661849}{781982750} a^{2} + \frac{21197797}{78198275} a - \frac{158005092}{390991375}$, $\frac{1}{75852326750} a^{15} + \frac{1}{1850056750} a^{14} - \frac{2919825231}{75852326750} a^{13} - \frac{1268922699}{75852326750} a^{12} + \frac{856972103}{37926163375} a^{11} - \frac{1880367861}{37926163375} a^{10} - \frac{138219891}{3297927250} a^{9} - \frac{1214025736}{37926163375} a^{8} + \frac{15533730323}{37926163375} a^{7} - \frac{4038330152}{37926163375} a^{6} - \frac{10731526573}{37926163375} a^{5} - \frac{1502765381}{7585232675} a^{4} + \frac{7336221991}{15170465350} a^{3} - \frac{5350491}{19072750} a^{2} + \frac{18770560863}{37926163375} a - \frac{2578566557}{75852326750}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 324217.745672 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.SD_{16}$ (as 16T163):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 19 conjugacy class representatives for $C_2^2.SD_{16}$
Character table for $C_2^2.SD_{16}$

Intermediate fields

\(\Q(\sqrt{-31}) \), 4.0.39401.1, 8.0.63649990841.1, 8.0.63649990841.2, 8.0.2609649624481.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$31$31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.8.6.1$x^{8} - 9881 x^{4} + 34857216$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$