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Properties

Label 16.6.11878851749...6875.2
Degree $16$
Signature $[6, 5]$
Discriminant $-\,5^{8}\cdot 13^{4}\cdot 29^{6}\cdot 179$
Root discriminant $20.76$
Ramified primes $5, 13, 29, 179$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1775

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11, -37, -20, -61, -69, 22, -53, 76, 55, -31, 24, -30, -3, 8, 1, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + x^14 + 8*x^13 - 3*x^12 - 30*x^11 + 24*x^10 - 31*x^9 + 55*x^8 + 76*x^7 - 53*x^6 + 22*x^5 - 69*x^4 - 61*x^3 - 20*x^2 - 37*x - 11)
 
gp: K = bnfinit(x^16 - 3*x^15 + x^14 + 8*x^13 - 3*x^12 - 30*x^11 + 24*x^10 - 31*x^9 + 55*x^8 + 76*x^7 - 53*x^6 + 22*x^5 - 69*x^4 - 61*x^3 - 20*x^2 - 37*x - 11, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + x^{14} + 8 x^{13} - 3 x^{12} - 30 x^{11} + 24 x^{10} - 31 x^{9} + 55 x^{8} + 76 x^{7} - 53 x^{6} + 22 x^{5} - 69 x^{4} - 61 x^{3} - 20 x^{2} - 37 x - 11 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1187885174970116796875=-\,5^{8}\cdot 13^{4}\cdot 29^{6}\cdot 179\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.76$
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:  $5, 13, 29, 179$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{145} a^{14} + \frac{2}{145} a^{13} - \frac{46}{145} a^{12} - \frac{46}{145} a^{11} + \frac{69}{145} a^{10} + \frac{37}{145} a^{9} + \frac{46}{145} a^{8} - \frac{5}{29} a^{7} + \frac{63}{145} a^{6} - \frac{69}{145} a^{5} + \frac{71}{145} a^{4} - \frac{8}{29} a^{3} + \frac{34}{145} a^{2} + \frac{69}{145} a - \frac{18}{145}$, $\frac{1}{565795808845} a^{15} + \frac{198057508}{565795808845} a^{14} + \frac{63895197336}{565795808845} a^{13} + \frac{123777850503}{565795808845} a^{12} + \frac{227566048638}{565795808845} a^{11} + \frac{217986659271}{565795808845} a^{10} + \frac{138983706718}{565795808845} a^{9} - \frac{170914754494}{565795808845} a^{8} + \frac{16085752708}{565795808845} a^{7} + \frac{149508436294}{565795808845} a^{6} - \frac{21552726208}{565795808845} a^{5} - \frac{43249253244}{565795808845} a^{4} + \frac{23008832559}{565795808845} a^{3} + \frac{93447285488}{565795808845} a^{2} - \frac{259477150049}{565795808845} a + \frac{60905385152}{565795808845}$

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:  $10$
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:  \( 28359.6284306 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1775:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16384
The 136 conjugacy class representatives for t16n1775 are not computed
Character table for t16n1775 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.8.2576088125.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 $16$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
$179$179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.1.1$x^{2} - 179$$2$$1$$1$$C_2$$[\ ]_{2}$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$