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Properties

Label 16.0.85661805113...7696.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 19^{4}\cdot 59^{2}\cdot 103^{4}$
Root discriminant $31.32$
Ramified primes $2, 19, 59, 103$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1665

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3481, 0, 1888, 0, 1902, 0, 468, 0, 207, 0, -20, 0, -2, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 - 2*x^12 - 20*x^10 + 207*x^8 + 468*x^6 + 1902*x^4 + 1888*x^2 + 3481)
 
gp: K = bnfinit(x^16 - 4*x^14 - 2*x^12 - 20*x^10 + 207*x^8 + 468*x^6 + 1902*x^4 + 1888*x^2 + 3481, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} - 2 x^{12} - 20 x^{10} + 207 x^{8} + 468 x^{6} + 1902 x^{4} + 1888 x^{2} + 3481 \)

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:  \(856618051132000300957696=2^{24}\cdot 19^{4}\cdot 59^{2}\cdot 103^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.32$
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:  $2, 19, 59, 103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{3}{8} a^{2} + \frac{1}{8} a - \frac{1}{8}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a + \frac{1}{8}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{555214544} a^{14} - \frac{1}{16} a^{13} + \frac{814257}{277607272} a^{12} - \frac{1}{16} a^{11} - \frac{22009893}{555214544} a^{10} - \frac{1}{8} a^{9} + \frac{68303867}{555214544} a^{8} - \frac{1}{16} a^{7} + \frac{104874359}{555214544} a^{6} + \frac{1}{8} a^{5} - \frac{7415773}{555214544} a^{4} + \frac{3}{16} a^{3} - \frac{89432075}{277607272} a^{2} - \frac{5}{16} a + \frac{39399}{9410416}$, $\frac{1}{555214544} a^{15} - \frac{33072395}{555214544} a^{13} - \frac{1}{16} a^{12} + \frac{1586377}{69401818} a^{11} - \frac{1}{16} a^{10} - \frac{1097951}{555214544} a^{9} - \frac{34508001}{138803636} a^{7} + \frac{3}{16} a^{6} - \frac{76817591}{555214544} a^{5} - \frac{1}{4} a^{4} + \frac{133444031}{555214544} a^{3} + \frac{3}{16} a^{2} + \frac{519557}{1176302} a + \frac{1}{16}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( \frac{21255}{69401818} a^{14} - \frac{183315}{138803636} a^{12} + \frac{57937}{138803636} a^{10} - \frac{1174405}{138803636} a^{8} + \frac{9717315}{138803636} a^{6} + \frac{13248035}{138803636} a^{4} + \frac{113462671}{138803636} a^{2} + \frac{243436}{588151} \) (order $4$)
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:  \( 128758.44622 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1665:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6144
The 78 conjugacy class representatives for t16n1665 are not computed
Character table for t16n1665 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.4.1957.1, 8.0.980441344.2

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ 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
2Data not computed
$19$19.4.0.1$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}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$59$59.2.0.1$x^{2} - x + 2$$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$$[\ ]^{2}$
59.4.2.2$x^{4} - 59 x^{2} + 6962$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
59.4.0.1$x^{4} - x + 14$$1$$4$$0$$C_4$$[\ ]^{4}$
$103$103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
103.4.2.1$x^{4} + 927 x^{2} + 265225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
103.4.2.1$x^{4} + 927 x^{2} + 265225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$