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Properties

Label 16.16.2944392860...6304.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{20}\cdot 3^{12}\cdot 11^{8}\cdot 157^{2}$
Root discriminant $33.83$
Ramified primes $2, 3, 11, 157$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4.C_2^3$ (as 16T364)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -18, -60, 498, 622, -1908, -2058, 2322, 2427, -1188, -1254, 258, 298, -18, -30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 30*x^14 - 18*x^13 + 298*x^12 + 258*x^11 - 1254*x^10 - 1188*x^9 + 2427*x^8 + 2322*x^7 - 2058*x^6 - 1908*x^5 + 622*x^4 + 498*x^3 - 60*x^2 - 18*x + 1)
 
gp: K = bnfinit(x^16 - 30*x^14 - 18*x^13 + 298*x^12 + 258*x^11 - 1254*x^10 - 1188*x^9 + 2427*x^8 + 2322*x^7 - 2058*x^6 - 1908*x^5 + 622*x^4 + 498*x^3 - 60*x^2 - 18*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 30 x^{14} - 18 x^{13} + 298 x^{12} + 258 x^{11} - 1254 x^{10} - 1188 x^{9} + 2427 x^{8} + 2322 x^{7} - 2058 x^{6} - 1908 x^{5} + 622 x^{4} + 498 x^{3} - 60 x^{2} - 18 x + 1 \)

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:  \(2944392860546173636706304=2^{20}\cdot 3^{12}\cdot 11^{8}\cdot 157^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.83$
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:  $2, 3, 11, 157$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{2} - \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{3}{8} a$, $\frac{1}{8} a^{14} - \frac{1}{4} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{3}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{90294788216} a^{15} + \frac{2305111915}{45147394108} a^{14} + \frac{692181691}{22573697054} a^{13} + \frac{867017347}{90294788216} a^{12} + \frac{4423593179}{90294788216} a^{11} + \frac{21316825435}{90294788216} a^{10} - \frac{2502915403}{90294788216} a^{9} - \frac{3342107827}{22573697054} a^{8} + \frac{11488974849}{90294788216} a^{7} + \frac{33286182311}{90294788216} a^{6} - \frac{25261011531}{90294788216} a^{5} - \frac{17258922543}{45147394108} a^{4} + \frac{7090417433}{45147394108} a^{3} - \frac{18717171441}{90294788216} a^{2} - \frac{17977763617}{90294788216} a - \frac{42245958645}{90294788216}$

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:  $15$
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:  \( 19814614.8251 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T364):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{33}) \), 4.4.13068.1 x2, \(\Q(\sqrt{3}, \sqrt{11})\), 4.4.4752.1 x2, 8.8.2732361984.1, 8.8.1715923325952.1, 8.8.190658147328.1

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$157$$\Q_{157}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{157}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{157}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{157}$$x + 5$$1$$1$$0$Trivial$[\ ]$
157.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
157.2.1.1$x^{2} - 157$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
157.2.1.1$x^{2} - 157$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
157.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$