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Properties

Label 16.0.16252179065...1121.2
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 17^{8}$
Root discriminant $28.23$
Ramified primes $13, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![937, 1179, -1080, 1659, 2412, -1230, 1299, -588, 2032, -1881, 1026, -570, 261, -84, 24, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 24*x^14 - 84*x^13 + 261*x^12 - 570*x^11 + 1026*x^10 - 1881*x^9 + 2032*x^8 - 588*x^7 + 1299*x^6 - 1230*x^5 + 2412*x^4 + 1659*x^3 - 1080*x^2 + 1179*x + 937)
 
gp: K = bnfinit(x^16 - 6*x^15 + 24*x^14 - 84*x^13 + 261*x^12 - 570*x^11 + 1026*x^10 - 1881*x^9 + 2032*x^8 - 588*x^7 + 1299*x^6 - 1230*x^5 + 2412*x^4 + 1659*x^3 - 1080*x^2 + 1179*x + 937, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 24 x^{14} - 84 x^{13} + 261 x^{12} - 570 x^{11} + 1026 x^{10} - 1881 x^{9} + 2032 x^{8} - 588 x^{7} + 1299 x^{6} - 1230 x^{5} + 2412 x^{4} + 1659 x^{3} - 1080 x^{2} + 1179 x + 937 \)

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:  \(162521790654198232131121=13^{12}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.23$
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:  $13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{2} a^{6} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{10} - \frac{1}{2} a^{7} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{42} a^{14} - \frac{1}{21} a^{13} - \frac{1}{14} a^{12} + \frac{5}{42} a^{11} + \frac{1}{21} a^{10} + \frac{1}{42} a^{9} + \frac{5}{42} a^{8} - \frac{3}{7} a^{7} + \frac{17}{42} a^{6} + \frac{11}{42} a^{5} + \frac{2}{7} a^{4} - \frac{5}{42} a^{3} + \frac{1}{42} a^{2} + \frac{4}{21} a - \frac{17}{42}$, $\frac{1}{113256044154493110892518} a^{15} - \frac{1168656088176401499331}{113256044154493110892518} a^{14} - \frac{4371212538677141848819}{113256044154493110892518} a^{13} - \frac{1057270585315673622753}{37752014718164370297506} a^{12} + \frac{4087851542241551761111}{37752014718164370297506} a^{11} - \frac{2182507562369391651811}{16179434879213301556074} a^{10} + \frac{1200693651320228169305}{37752014718164370297506} a^{9} - \frac{8138621948969468151863}{113256044154493110892518} a^{8} - \frac{20357349677599992586909}{113256044154493110892518} a^{7} + \frac{1530923072365666219225}{113256044154493110892518} a^{6} - \frac{6932097730676859469145}{113256044154493110892518} a^{5} + \frac{14641608622103854235517}{37752014718164370297506} a^{4} - \frac{2375082848544324824075}{5393144959737767185358} a^{3} + \frac{43861185636161074271605}{113256044154493110892518} a^{2} - \frac{1855105532264530255535}{5393144959737767185358} a + \frac{9423134364258980928289}{56628022077246555446259}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 113461.432048 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_4$ (as 16T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{221}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{13}, \sqrt{17})\), 4.4.634933.1 x2, 4.4.37349.1 x2, 4.0.3757.1 x2, 4.0.2873.1 x2, 4.0.2197.1, 4.0.634933.1, 8.8.403139914489.1, 8.0.2385443281.2, 8.0.403139914489.1, 8.0.1394947801.1 x2, 8.0.403139914489.3 x2

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$