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Properties

Label 16.16.8012980262...0625.1
Degree $16$
Signature $[16, 0]$
Discriminant $3^{8}\cdot 5^{12}\cdot 29^{8}$
Root discriminant $31.19$
Ramified primes $3, 5, 29$
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![631, -838, -5542, 6174, 14471, -11402, -17127, 8602, 10407, -2950, -3318, 436, 532, -22, -39, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 39*x^14 - 22*x^13 + 532*x^12 + 436*x^11 - 3318*x^10 - 2950*x^9 + 10407*x^8 + 8602*x^7 - 17127*x^6 - 11402*x^5 + 14471*x^4 + 6174*x^3 - 5542*x^2 - 838*x + 631)
 
gp: K = bnfinit(x^16 - 39*x^14 - 22*x^13 + 532*x^12 + 436*x^11 - 3318*x^10 - 2950*x^9 + 10407*x^8 + 8602*x^7 - 17127*x^6 - 11402*x^5 + 14471*x^4 + 6174*x^3 - 5542*x^2 - 838*x + 631, 1)
 

Normalized defining polynomial

\( x^{16} - 39 x^{14} - 22 x^{13} + 532 x^{12} + 436 x^{11} - 3318 x^{10} - 2950 x^{9} + 10407 x^{8} + 8602 x^{7} - 17127 x^{6} - 11402 x^{5} + 14471 x^{4} + 6174 x^{3} - 5542 x^{2} - 838 x + 631 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(801298026229765869140625=3^{8}\cdot 5^{12}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.19$
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:  $3, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{12} - \frac{1}{5} a^{11} + \frac{1}{10} a^{10} + \frac{1}{10} a^{9} + \frac{1}{10} a^{8} - \frac{1}{10} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{10} a^{4} - \frac{1}{2} a^{3} - \frac{3}{10} a^{2} - \frac{1}{10} a + \frac{1}{10}$, $\frac{1}{10} a^{13} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{10} a^{8} - \frac{1}{10} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{3}{10} a^{4} - \frac{3}{10} a^{3} - \frac{1}{5} a^{2} - \frac{1}{10} a + \frac{1}{5}$, $\frac{1}{40} a^{14} - \frac{1}{20} a^{12} + \frac{3}{20} a^{11} - \frac{3}{20} a^{10} + \frac{1}{20} a^{9} + \frac{1}{10} a^{7} + \frac{11}{40} a^{6} - \frac{7}{20} a^{5} + \frac{2}{5} a^{4} - \frac{3}{10} a^{3} - \frac{9}{40} a^{2} - \frac{7}{20} a - \frac{19}{40}$, $\frac{1}{6455717151597731600} a^{15} - \frac{1218532197917413}{6455717151597731600} a^{14} - \frac{11698181820126337}{645571715159773160} a^{13} - \frac{40230159777186323}{1613929287899432900} a^{12} + \frac{142420243932215291}{806964643949716450} a^{11} - \frac{1513113156529747}{1613929287899432900} a^{10} + \frac{406888793370194343}{3227858575798865800} a^{9} + \frac{327445328253558473}{1613929287899432900} a^{8} + \frac{804246622147608931}{6455717151597731600} a^{7} - \frac{88123115910082421}{6455717151597731600} a^{6} + \frac{1182526399792634723}{3227858575798865800} a^{5} + \frac{710142832732750}{16139292878994329} a^{4} + \frac{2559685667905982631}{6455717151597731600} a^{3} + \frac{2213437000750984211}{6455717151597731600} a^{2} - \frac{52048888114434369}{258228686063909264} a + \frac{2314253546063457}{10230930509663600}$

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:  $15$
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:  \( 4926687.56772 \) (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

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{5}, \sqrt{29})\), 4.4.32625.1 x2, 4.4.946125.2 x2, 4.4.4205.1 x2, 4.4.725.1 x2, \(\Q(\zeta_{15})^+\), 4.4.946125.1, 8.8.895152515625.2, 8.8.442050625.1, 8.8.895152515625.1, 8.8.1064390625.1 x2, 8.8.895152515625.3 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{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}$