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Properties

Label 16.0.27404580206...7024.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{30}\cdot 761^{5}$
Root discriminant $29.17$
Ramified primes $2, 761$
Class number $3$
Class group $[3]$
Galois group 16T1870

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![252, -780, 1573, -2118, 2067, -1736, 1304, -668, 124, 192, -239, 114, -13, -20, 17, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 17*x^14 - 20*x^13 - 13*x^12 + 114*x^11 - 239*x^10 + 192*x^9 + 124*x^8 - 668*x^7 + 1304*x^6 - 1736*x^5 + 2067*x^4 - 2118*x^3 + 1573*x^2 - 780*x + 252)
 
gp: K = bnfinit(x^16 - 6*x^15 + 17*x^14 - 20*x^13 - 13*x^12 + 114*x^11 - 239*x^10 + 192*x^9 + 124*x^8 - 668*x^7 + 1304*x^6 - 1736*x^5 + 2067*x^4 - 2118*x^3 + 1573*x^2 - 780*x + 252, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 17 x^{14} - 20 x^{13} - 13 x^{12} + 114 x^{11} - 239 x^{10} + 192 x^{9} + 124 x^{8} - 668 x^{7} + 1304 x^{6} - 1736 x^{5} + 2067 x^{4} - 2118 x^{3} + 1573 x^{2} - 780 x + 252 \)

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:  \(274045802062918423937024=2^{30}\cdot 761^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.17$
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:  $2, 761$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{38} a^{14} + \frac{9}{38} a^{13} + \frac{8}{19} a^{12} - \frac{8}{19} a^{11} + \frac{3}{38} a^{10} + \frac{17}{38} a^{9} - \frac{6}{19} a^{8} + \frac{9}{19} a^{7} + \frac{6}{19} a^{6} - \frac{5}{19} a^{5} + \frac{8}{19} a^{4} + \frac{8}{19} a^{3} + \frac{17}{38} a^{2} - \frac{11}{38} a + \frac{4}{19}$, $\frac{1}{36193423922106467688} a^{15} - \frac{13968657597655183}{12064474640702155896} a^{14} - \frac{1886370218070022051}{9048355980526616922} a^{13} - \frac{207167364104766007}{4524177990263308461} a^{12} - \frac{10590910143365937565}{36193423922106467688} a^{11} + \frac{4389952218649316383}{12064474640702155896} a^{10} - \frac{1254903693569785285}{18096711961053233844} a^{9} + \frac{314756604087245245}{6032237320351077948} a^{8} - \frac{1465007258816745607}{18096711961053233844} a^{7} - \frac{6879609287857188229}{18096711961053233844} a^{6} - \frac{2455978178734836029}{18096711961053233844} a^{5} + \frac{2135445248864918483}{18096711961053233844} a^{4} - \frac{584095496029057997}{1340497182300239544} a^{3} + \frac{1548025221505089281}{12064474640702155896} a^{2} + \frac{2078972757490770056}{4524177990263308461} a + \frac{64290147401580809}{430874094310791282}$

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

Class group and class number

$C_{3}$, which has order $3$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 233892.233015 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1870:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 83 conjugacy class representatives for t16n1870 are not computed
Character table for t16n1870 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 8.4.2372079616.2

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 R ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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
$2$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.12.24.244$x^{12} - 8 x^{11} + 4 x^{10} - 8 x^{9} - 14 x^{8} + 8 x^{7} + 8 x^{5} + 16 x^{3} - 8 x^{2} + 16 x + 8$$4$$3$$24$$C_2^2 \times A_4$$[2, 2, 3]^{6}$
761Data not computed