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Properties

Label 16.0.13573982477...7041.3
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 17^{12}$
Root discriminant $57.32$
Ramified primes $13, 17$
Class number $32$ (GRH)
Class group $[4, 8]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![26711, -62675, 59905, -50012, 53241, -42862, 25877, -16629, 10309, -4899, 2090, -944, 319, -84, 29, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 29*x^14 - 84*x^13 + 319*x^12 - 944*x^11 + 2090*x^10 - 4899*x^9 + 10309*x^8 - 16629*x^7 + 25877*x^6 - 42862*x^5 + 53241*x^4 - 50012*x^3 + 59905*x^2 - 62675*x + 26711)
 
gp: K = bnfinit(x^16 - 4*x^15 + 29*x^14 - 84*x^13 + 319*x^12 - 944*x^11 + 2090*x^10 - 4899*x^9 + 10309*x^8 - 16629*x^7 + 25877*x^6 - 42862*x^5 + 53241*x^4 - 50012*x^3 + 59905*x^2 - 62675*x + 26711, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 29 x^{14} - 84 x^{13} + 319 x^{12} - 944 x^{11} + 2090 x^{10} - 4899 x^{9} + 10309 x^{8} - 16629 x^{7} + 25877 x^{6} - 42862 x^{5} + 53241 x^{4} - 50012 x^{3} + 59905 x^{2} - 62675 x + 26711 \)

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:  \(13573982477229290545823357041=13^{12}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.32$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{424} a^{12} - \frac{3}{424} a^{11} + \frac{2}{53} a^{10} - \frac{19}{212} a^{9} - \frac{37}{212} a^{8} - \frac{1}{8} a^{7} + \frac{81}{424} a^{6} + \frac{77}{212} a^{5} - \frac{85}{424} a^{4} + \frac{137}{424} a^{3} + \frac{45}{424} a^{2} - \frac{17}{424} a - \frac{187}{424}$, $\frac{1}{424} a^{13} + \frac{7}{424} a^{11} + \frac{5}{212} a^{10} + \frac{3}{53} a^{9} - \frac{63}{424} a^{8} - \frac{39}{212} a^{7} - \frac{27}{424} a^{6} + \frac{165}{424} a^{5} + \frac{47}{212} a^{4} - \frac{45}{106} a^{3} + \frac{59}{212} a^{2} + \frac{93}{212} a + \frac{75}{424}$, $\frac{1}{424} a^{14} + \frac{31}{424} a^{11} - \frac{11}{53} a^{10} - \frac{9}{424} a^{9} + \frac{2}{53} a^{8} - \frac{10}{53} a^{7} + \frac{11}{212} a^{6} + \frac{19}{106} a^{5} - \frac{9}{424} a^{4} - \frac{205}{424} a^{3} + \frac{83}{424} a^{2} - \frac{9}{212} a + \frac{37}{424}$, $\frac{1}{2582374431483340904} a^{15} - \frac{363657674993583}{1291187215741670452} a^{14} - \frac{689835413876075}{2582374431483340904} a^{13} + \frac{983215842004861}{1291187215741670452} a^{12} + \frac{112811203546094891}{645593607870835226} a^{11} - \frac{96962666339977319}{2582374431483340904} a^{10} + \frac{30859427979584765}{322796803935417613} a^{9} + \frac{510435876437688663}{2582374431483340904} a^{8} - \frac{996684759929018515}{2582374431483340904} a^{7} - \frac{302793408732604877}{645593607870835226} a^{6} + \frac{515373396929503723}{1291187215741670452} a^{5} - \frac{209858791956068507}{645593607870835226} a^{4} + \frac{46461942668999185}{322796803935417613} a^{3} + \frac{292468114061011637}{2582374431483340904} a^{2} - \frac{75908622359206683}{645593607870835226} a + \frac{4468468561387484}{322796803935417613}$

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

Class group and class number

$C_{4}\times C_{8}$, which has order $32$ (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:  \( 506567.420659 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

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 $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{221}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{13}, \sqrt{17})\), 4.0.830297.1 x2, 4.0.63869.1 x2, 4.0.634933.1, 4.0.2197.1, 8.0.689393108209.3, 8.0.403139914489.1, 8.8.116507435287321.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$