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Properties

Label 16.0.20253399449...6336.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{34}\cdot 13^{2}\cdot 17^{8}$
Root discriminant $24.78$
Ramified primes $2, 13, 17$
Class number $2$
Class group $[2]$
Galois group 16T1177

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -8, 176, 904, 1958, 2100, 1176, 598, 465, 136, 40, -52, 24, -4, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 4*x^13 + 24*x^12 - 52*x^11 + 40*x^10 + 136*x^9 + 465*x^8 + 598*x^7 + 1176*x^6 + 2100*x^5 + 1958*x^4 + 904*x^3 + 176*x^2 - 8*x + 4)
 
gp: K = bnfinit(x^16 - 2*x^15 - 4*x^13 + 24*x^12 - 52*x^11 + 40*x^10 + 136*x^9 + 465*x^8 + 598*x^7 + 1176*x^6 + 2100*x^5 + 1958*x^4 + 904*x^3 + 176*x^2 - 8*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 4 x^{13} + 24 x^{12} - 52 x^{11} + 40 x^{10} + 136 x^{9} + 465 x^{8} + 598 x^{7} + 1176 x^{6} + 2100 x^{5} + 1958 x^{4} + 904 x^{3} + 176 x^{2} - 8 x + 4 \)

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:  \(20253399449972387086336=2^{34}\cdot 13^{2}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.78$
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:  $2, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{158067202926354693028} a^{15} - \frac{9906418605152495675}{158067202926354693028} a^{14} + \frac{12376993606852382167}{158067202926354693028} a^{13} - \frac{4545708770713162939}{79033601463177346514} a^{12} - \frac{25271817247062431749}{158067202926354693028} a^{11} - \frac{1343045546594707709}{158067202926354693028} a^{10} + \frac{29389634238751147631}{158067202926354693028} a^{9} + \frac{15577394683286462053}{79033601463177346514} a^{8} + \frac{9308930919330533127}{79033601463177346514} a^{7} + \frac{1966290828152723102}{39516800731588673257} a^{6} + \frac{9611231401776722015}{39516800731588673257} a^{5} - \frac{29042231813265536845}{79033601463177346514} a^{4} - \frac{10703376574275471170}{39516800731588673257} a^{3} - \frac{13727482948217571035}{39516800731588673257} a^{2} + \frac{9306258233414996191}{39516800731588673257} a - \frac{14905089579351011592}{39516800731588673257}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( 47827.5778933 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1177:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 76 conjugacy class representatives for t16n1177 are not computed
Character table for t16n1177 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.9248.1, 8.4.2736816128.4, 8.0.142314438656.6, 8.4.4447326208.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.4.11.6$x^{4} + 18$$4$$1$$11$$D_{4}$$[2, 3, 4]$
2.4.11.6$x^{4} + 18$$4$$1$$11$$D_{4}$$[2, 3, 4]$
2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$