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Properties

Label 16.0.17303073984...5625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 29^{8}\cdot 941^{2}$
Root discriminant $28.34$
Ramified primes $5, 29, 941$
Class number $20$ (GRH)
Class group $[20]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T364)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![625, -375, 3100, -985, 5166, -4, 2311, 626, 542, 95, 170, -74, 57, -24, 10, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 10*x^14 - 24*x^13 + 57*x^12 - 74*x^11 + 170*x^10 + 95*x^9 + 542*x^8 + 626*x^7 + 2311*x^6 - 4*x^5 + 5166*x^4 - 985*x^3 + 3100*x^2 - 375*x + 625)
 
gp: K = bnfinit(x^16 - 2*x^15 + 10*x^14 - 24*x^13 + 57*x^12 - 74*x^11 + 170*x^10 + 95*x^9 + 542*x^8 + 626*x^7 + 2311*x^6 - 4*x^5 + 5166*x^4 - 985*x^3 + 3100*x^2 - 375*x + 625, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 10 x^{14} - 24 x^{13} + 57 x^{12} - 74 x^{11} + 170 x^{10} + 95 x^{9} + 542 x^{8} + 626 x^{7} + 2311 x^{6} - 4 x^{5} + 5166 x^{4} - 985 x^{3} + 3100 x^{2} - 375 x + 625 \)

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:  \(173030739841843453515625=5^{8}\cdot 29^{8}\cdot 941^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.34$
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:  $5, 29, 941$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{10} a^{12} - \frac{2}{5} a^{10} - \frac{1}{10} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{2} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{3}{10} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{1}{2}$, $\frac{1}{10} a^{13} + \frac{3}{10} a^{10} - \frac{3}{10} a^{7} - \frac{1}{10} a^{4} - \frac{1}{10} a$, $\frac{1}{3250} a^{14} - \frac{11}{1625} a^{13} - \frac{1}{130} a^{12} + \frac{201}{3250} a^{11} - \frac{594}{1625} a^{10} + \frac{1011}{3250} a^{9} + \frac{21}{130} a^{8} - \frac{118}{325} a^{7} + \frac{1617}{3250} a^{6} + \frac{11}{3250} a^{5} - \frac{142}{1625} a^{4} - \frac{1499}{3250} a^{3} - \frac{1229}{3250} a^{2} - \frac{148}{325} a - \frac{41}{130}$, $\frac{1}{186667212413063346250} a^{15} - \frac{15663802258397717}{186667212413063346250} a^{14} + \frac{327914396806074703}{37333442482612669250} a^{13} + \frac{5214162835870724301}{186667212413063346250} a^{12} + \frac{11100326047802223467}{186667212413063346250} a^{11} - \frac{44716008089599149129}{186667212413063346250} a^{10} - \frac{8301869925939247629}{37333442482612669250} a^{9} - \frac{18280415379894222751}{37333442482612669250} a^{8} + \frac{82958791009763677367}{186667212413063346250} a^{7} - \frac{49285348146026602529}{186667212413063346250} a^{6} + \frac{6137710133593571817}{14359016339466411250} a^{5} - \frac{5731775767441410169}{186667212413063346250} a^{4} - \frac{77538831022592489899}{186667212413063346250} a^{3} - \frac{2522218843000496349}{7466688496522533850} a^{2} + \frac{1999871273121149983}{7466688496522533850} a - \frac{290249424415367503}{746668849652253385}$

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

Class group and class number

$C_{20}$, which has order $20$ (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:  \( 3793.72993285 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T364):

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

\(\Q(\sqrt{145}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{29}) \), 4.4.4205.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 4.4.725.1 x2, 8.0.415969638125.2, 8.8.442050625.1, 8.0.415969638125.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$29.4.2.1$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}$
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}$
941Data not computed