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Properties

Label 16.0.12913449667...0169.1
Degree $16$
Signature $[0, 8]$
Discriminant $17^{12}\cdot 53^{6}$
Root discriminant $37.11$
Ramified primes $17, 53$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_4\wr C_2$ (as 16T28)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2209, -6345, 11003, -12492, 9867, -2768, -391, -890, 866, -332, 255, -111, 11, 1, 2, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 2*x^14 + x^13 + 11*x^12 - 111*x^11 + 255*x^10 - 332*x^9 + 866*x^8 - 890*x^7 - 391*x^6 - 2768*x^5 + 9867*x^4 - 12492*x^3 + 11003*x^2 - 6345*x + 2209)
 
gp: K = bnfinit(x^16 - 3*x^15 + 2*x^14 + x^13 + 11*x^12 - 111*x^11 + 255*x^10 - 332*x^9 + 866*x^8 - 890*x^7 - 391*x^6 - 2768*x^5 + 9867*x^4 - 12492*x^3 + 11003*x^2 - 6345*x + 2209, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 2 x^{14} + x^{13} + 11 x^{12} - 111 x^{11} + 255 x^{10} - 332 x^{9} + 866 x^{8} - 890 x^{7} - 391 x^{6} - 2768 x^{5} + 9867 x^{4} - 12492 x^{3} + 11003 x^{2} - 6345 x + 2209 \)

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:  \(12913449667746331350360169=17^{12}\cdot 53^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.11$
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:  $17, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{188} a^{12} - \frac{41}{188} a^{11} + \frac{17}{94} a^{10} - \frac{19}{94} a^{9} - \frac{7}{188} a^{8} - \frac{51}{188} a^{6} - \frac{5}{188} a^{5} + \frac{63}{188} a^{4} - \frac{19}{188} a^{3} - \frac{37}{188} a^{2} - \frac{7}{47} a + \frac{1}{4}$, $\frac{1}{188} a^{13} + \frac{45}{188} a^{11} + \frac{10}{47} a^{10} + \frac{33}{188} a^{9} - \frac{5}{188} a^{8} - \frac{51}{188} a^{7} + \frac{33}{94} a^{6} - \frac{12}{47} a^{5} + \frac{13}{94} a^{4} - \frac{16}{47} a^{3} + \frac{53}{188} a^{2} - \frac{67}{188} a + \frac{1}{4}$, $\frac{1}{188} a^{14} + \frac{5}{188} a^{11} + \frac{7}{188} a^{10} + \frac{13}{188} a^{9} - \frac{9}{94} a^{8} - \frac{7}{47} a^{7} - \frac{9}{188} a^{6} - \frac{31}{188} a^{5} + \frac{15}{188} a^{4} + \frac{31}{94} a^{3} - \frac{1}{2} a^{2} + \frac{85}{188} a + \frac{1}{4}$, $\frac{1}{203175372548375705204} a^{15} + \frac{304441568764351989}{203175372548375705204} a^{14} - \frac{50473960440226027}{101587686274187852602} a^{13} - \frac{55985475272200321}{50793843137093926301} a^{12} - \frac{22299618665054336857}{203175372548375705204} a^{11} - \frac{4548504720821676457}{50793843137093926301} a^{10} - \frac{1063418412914603295}{203175372548375705204} a^{9} + \frac{2416678666009298731}{203175372548375705204} a^{8} + \frac{51591313787675496195}{203175372548375705204} a^{7} + \frac{57413975629114574645}{203175372548375705204} a^{6} - \frac{37176046992501820625}{203175372548375705204} a^{5} + \frac{33978865462389660541}{101587686274187852602} a^{4} + \frac{70975180023990531813}{203175372548375705204} a^{3} - \frac{8185553380248368931}{50793843137093926301} a^{2} - \frac{34951980500347042485}{101587686274187852602} a - \frac{311617069201829045}{2161440133493358566}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( 128061.369114 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T28):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 4.4.15317.1, 4.4.260389.1, 8.8.67802431321.1

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
17Data not computed
$53$53.4.2.2$x^{4} - 53 x^{2} + 14045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
53.4.0.1$x^{4} - x + 18$$1$$4$$0$$C_4$$[\ ]^{4}$
53.8.4.1$x^{8} + 101124 x^{4} - 148877 x^{2} + 2556515844$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$