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Properties

Label 16.16.3922880935...4849.1
Degree $16$
Signature $[16, 0]$
Discriminant $13^{12}\cdot 17^{14}$
Root discriminant $81.68$
Ramified primes $13, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-13312, -33280, 70720, 189696, -71148, -325110, -39947, 210109, 68499, -51610, -23624, 3444, 2368, -32, -85, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 85*x^14 - 32*x^13 + 2368*x^12 + 3444*x^11 - 23624*x^10 - 51610*x^9 + 68499*x^8 + 210109*x^7 - 39947*x^6 - 325110*x^5 - 71148*x^4 + 189696*x^3 + 70720*x^2 - 33280*x - 13312)
 
gp: K = bnfinit(x^16 - x^15 - 85*x^14 - 32*x^13 + 2368*x^12 + 3444*x^11 - 23624*x^10 - 51610*x^9 + 68499*x^8 + 210109*x^7 - 39947*x^6 - 325110*x^5 - 71148*x^4 + 189696*x^3 + 70720*x^2 - 33280*x - 13312, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 85 x^{14} - 32 x^{13} + 2368 x^{12} + 3444 x^{11} - 23624 x^{10} - 51610 x^{9} + 68499 x^{8} + 210109 x^{7} - 39947 x^{6} - 325110 x^{5} - 71148 x^{4} + 189696 x^{3} + 70720 x^{2} - 33280 x - 13312 \)

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:  \(3922880935919264967742950184849=13^{12}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.68$
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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{8} + \frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{4} a^{4} - \frac{7}{16} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{9} + \frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{4} a^{5} + \frac{1}{16} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{9} - \frac{1}{32} a^{8} - \frac{1}{16} a^{7} + \frac{3}{32} a^{6} + \frac{7}{32} a^{5} + \frac{5}{32} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{8} - \frac{5}{64} a^{7} - \frac{3}{32} a^{6} - \frac{3}{32} a^{5} - \frac{5}{64} a^{4} + \frac{13}{32} a^{3} + \frac{3}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{128} a^{12} - \frac{1}{128} a^{11} - \frac{1}{128} a^{9} - \frac{1}{32} a^{8} - \frac{1}{128} a^{7} + \frac{1}{128} a^{5} + \frac{31}{128} a^{4} - \frac{7}{64} a^{3} - \frac{7}{32} a^{2} - \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{256} a^{13} - \frac{1}{256} a^{12} + \frac{3}{256} a^{10} - \frac{1}{32} a^{9} + \frac{3}{256} a^{8} - \frac{1}{32} a^{7} - \frac{11}{256} a^{6} - \frac{45}{256} a^{5} - \frac{13}{128} a^{4} + \frac{23}{64} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{512} a^{14} - \frac{1}{512} a^{12} + \frac{3}{512} a^{11} - \frac{5}{512} a^{10} - \frac{5}{512} a^{9} - \frac{5}{512} a^{8} - \frac{19}{512} a^{7} - \frac{7}{64} a^{6} + \frac{57}{512} a^{5} - \frac{31}{256} a^{4} + \frac{55}{128} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{7948569438620280889856} a^{15} + \frac{4835454400161078209}{7948569438620280889856} a^{14} - \frac{4102594148452693347}{7948569438620280889856} a^{13} + \frac{1917916643420349595}{993571179827535111232} a^{12} - \frac{4301316489594789071}{3974284719310140444928} a^{11} + \frac{7354489722941682533}{496785589913767555616} a^{10} + \frac{29935251391987295209}{3974284719310140444928} a^{9} - \frac{105935060352843911715}{3974284719310140444928} a^{8} + \frac{476553088904594888601}{7948569438620280889856} a^{7} + \frac{797906520685737714567}{7948569438620280889856} a^{6} + \frac{1502569749323400755897}{7948569438620280889856} a^{5} + \frac{880197411387970907507}{3974284719310140444928} a^{4} + \frac{914530996566014002423}{1987142359655070222464} a^{3} + \frac{245702544008896254955}{496785589913767555616} a^{2} - \frac{31400952865671310259}{124196397478441888904} a - \frac{5589091152317377838}{15524549684805236113}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 56949968052.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

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 $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{221}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{13}, \sqrt{17})\), 4.4.830297.1, 4.4.4913.1, 8.8.689393108209.1, 8.8.1980626399884457.1 x2

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$17$17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$