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Properties

Label 16.0.21988037298...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{12}\cdot 5^{10}\cdot 19^{4}\cdot 59^{6}$
Root discriminant $44.30$
Ramified primes $2, 5, 19, 59$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1174

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1936, 0, 5256, 0, 5049, 0, 4759, 0, 3734, 0, 951, 0, 74, 0, 19, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 19*x^14 + 74*x^12 + 951*x^10 + 3734*x^8 + 4759*x^6 + 5049*x^4 + 5256*x^2 + 1936)
 
gp: K = bnfinit(x^16 + 19*x^14 + 74*x^12 + 951*x^10 + 3734*x^8 + 4759*x^6 + 5049*x^4 + 5256*x^2 + 1936, 1)
 

Normalized defining polynomial

\( x^{16} + 19 x^{14} + 74 x^{12} + 951 x^{10} + 3734 x^{8} + 4759 x^{6} + 5049 x^{4} + 5256 x^{2} + 1936 \)

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:  \(219880372985150440000000000=2^{12}\cdot 5^{10}\cdot 19^{4}\cdot 59^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.30$
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:  $2, 5, 19, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{8} a^{9} - \frac{1}{16} a^{8} - \frac{1}{16} a^{7} - \frac{3}{16} a^{6} + \frac{3}{16} a^{4} - \frac{7}{16} a^{3} + \frac{3}{16} a^{2} - \frac{1}{4}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{10} + \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} + \frac{3}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{2} a^{4} + \frac{3}{8} a^{3} - \frac{7}{16} a^{2} - \frac{1}{4}$, $\frac{1}{1087597677184} a^{14} + \frac{1262893023}{1087597677184} a^{12} - \frac{1}{16} a^{11} - \frac{6714119257}{543798838592} a^{10} - \frac{1}{8} a^{9} - \frac{40689109737}{1087597677184} a^{8} - \frac{1}{16} a^{7} + \frac{129050633045}{543798838592} a^{6} - \frac{1}{4} a^{5} + \frac{454272443815}{1087597677184} a^{4} + \frac{5}{16} a^{3} - \frac{138625940323}{1087597677184} a^{2} + \frac{1}{4} a - \frac{54350168085}{271899419296}$, $\frac{1}{47854297796096} a^{15} - \frac{1}{2175195354368} a^{14} - \frac{1018359929337}{47854297796096} a^{13} + \frac{66711961801}{2175195354368} a^{12} - \frac{1196274078677}{23927148898048} a^{11} + \frac{40701546669}{1087597677184} a^{10} - \frac{3915255834705}{47854297796096} a^{9} - \frac{27285745087}{2175195354368} a^{8} + \frac{910761463521}{23927148898048} a^{7} - \frac{27088350809}{1087597677184} a^{6} + \frac{3105291781951}{47854297796096} a^{5} + \frac{565350378545}{2175195354368} a^{4} + \frac{413785367503}{4350390708736} a^{3} + \frac{750399633739}{2175195354368} a^{2} + \frac{1237172073571}{11963574449024} a - \frac{13624686739}{543798838592}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $7$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2576844.17557 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1174:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 70 conjugacy class representatives for t16n1174 are not computed
Character table for t16n1174 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.28025.1, 8.0.62832050000.1, 8.4.741418190000.1, 8.4.3707090950000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ R

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.8.12.15$x^{8} + 2 x^{7} + 2 x^{4} + 12$$4$$2$$12$$C_2^2:C_4$$[2, 2]^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$59$59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.4.3.1$x^{4} + 177$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
59.4.3.1$x^{4} + 177$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$