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Properties

Label 16.0.11207235820...3616.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 631^{2}$
Root discriminant $17.91$
Ramified primes $2, 631$
Class number $1$
Class group Trivial
Galois group 16T1497

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -8, 24, -48, 95, -140, 246, -276, 329, -304, 252, -168, 93, -44, 14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 14*x^14 - 44*x^13 + 93*x^12 - 168*x^11 + 252*x^10 - 304*x^9 + 329*x^8 - 276*x^7 + 246*x^6 - 140*x^5 + 95*x^4 - 48*x^3 + 24*x^2 - 8*x + 2)
 
gp: K = bnfinit(x^16 - 4*x^15 + 14*x^14 - 44*x^13 + 93*x^12 - 168*x^11 + 252*x^10 - 304*x^9 + 329*x^8 - 276*x^7 + 246*x^6 - 140*x^5 + 95*x^4 - 48*x^3 + 24*x^2 - 8*x + 2, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 14 x^{14} - 44 x^{13} + 93 x^{12} - 168 x^{11} + 252 x^{10} - 304 x^{9} + 329 x^{8} - 276 x^{7} + 246 x^{6} - 140 x^{5} + 95 x^{4} - 48 x^{3} + 24 x^{2} - 8 x + 2 \)

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:  \(112072358202091503616=2^{48}\cdot 631^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.91$
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:  $2, 631$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{12} + \frac{1}{4} a^{4} - \frac{1}{2}$, $\frac{1}{16} a^{13} + \frac{1}{16} a^{12} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} + \frac{1}{16} a^{5} + \frac{1}{16} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{3}{8} a - \frac{3}{8}$, $\frac{1}{64} a^{14} - \frac{1}{32} a^{13} - \frac{3}{64} a^{12} - \frac{1}{8} a^{11} + \frac{1}{16} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{31}{64} a^{6} - \frac{13}{32} a^{5} + \frac{5}{64} a^{4} - \frac{1}{4} a^{3} - \frac{1}{32} a^{2} - \frac{1}{16} a + \frac{5}{32}$, $\frac{1}{1671424} a^{15} + \frac{3347}{1671424} a^{14} + \frac{38163}{1671424} a^{13} - \frac{188695}{1671424} a^{12} + \frac{935}{417856} a^{11} - \frac{777}{417856} a^{10} + \frac{124}{6529} a^{9} + \frac{14901}{104464} a^{8} - \frac{8327}{1671424} a^{7} - \frac{325557}{1671424} a^{6} - \frac{650493}{1671424} a^{5} - \frac{474215}{1671424} a^{4} - \frac{359625}{835712} a^{3} - \frac{6695}{835712} a^{2} - \frac{393029}{835712} a - \frac{166999}{835712}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( 14485.3357952 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1497:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 40 conjugacy class representatives for t16n1497
Character table for t16n1497 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.0.1323302912.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.1$x^{4} + 12 x^{2} + 2$$4$$1$$11$$C_4$$[3, 4]$
2.8.26.4$x^{8} + 8 x^{7} + 12 x^{6} + 8 x^{5} + 8 x^{4} + 8 x^{3} + 2$$8$$1$$26$$C_2^2:C_4$$[2, 3, 7/2, 4]$
631Data not computed