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Properties

Label 16.0.13853804594...4464.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 17^{10}$
Root discriminant $27.95$
Ramified primes $2, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4.C_2^2:D_4$ (as 16T305)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4624, 0, -4624, 0, 2788, 0, -1768, 0, 1260, 0, -460, 0, 105, 0, -14, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 14*x^14 + 105*x^12 - 460*x^10 + 1260*x^8 - 1768*x^6 + 2788*x^4 - 4624*x^2 + 4624)
 
gp: K = bnfinit(x^16 - 14*x^14 + 105*x^12 - 460*x^10 + 1260*x^8 - 1768*x^6 + 2788*x^4 - 4624*x^2 + 4624, 1)
 

Normalized defining polynomial

\( x^{16} - 14 x^{14} + 105 x^{12} - 460 x^{10} + 1260 x^{8} - 1768 x^{6} + 2788 x^{4} - 4624 x^{2} + 4624 \)

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:  \(138538045941822955454464=2^{36}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.95$
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:  $2, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{10} - \frac{1}{4} a^{7} + \frac{3}{16} a^{6} + \frac{1}{4} a^{5} + \frac{3}{8} a^{4} - \frac{1}{2} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{16} a^{11} - \frac{1}{4} a^{8} + \frac{3}{16} a^{7} - \frac{1}{4} a^{6} + \frac{3}{8} a^{5} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{272} a^{12} + \frac{3}{272} a^{10} - \frac{65}{272} a^{8} - \frac{1}{4} a^{7} + \frac{67}{272} a^{6} + \frac{1}{4} a^{5} + \frac{9}{68} a^{4} - \frac{1}{2} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{272} a^{13} + \frac{3}{272} a^{11} + \frac{3}{272} a^{9} - \frac{1}{272} a^{7} - \frac{25}{68} a^{5} + \frac{1}{8} a^{3} - \frac{1}{4} a$, $\frac{1}{2913959392} a^{14} - \frac{2542047}{1456979696} a^{12} - \frac{68335961}{2913959392} a^{10} - \frac{20614977}{364244924} a^{8} + \frac{11854335}{1456979696} a^{6} - \frac{6310629}{91061231} a^{4} + \frac{9901373}{21426172} a^{2} - \frac{1542078}{5356543}$, $\frac{1}{5827918784} a^{15} + \frac{87953}{91061231} a^{13} - \frac{36196703}{5827918784} a^{11} - \frac{66390279}{2913959392} a^{9} + \frac{203056}{91061231} a^{7} - \frac{1}{4} a^{6} - \frac{9362123}{42852344} a^{5} + \frac{1}{4} a^{4} + \frac{25159289}{85704688} a^{3} - \frac{1}{2} a^{2} - \frac{11524855}{42852344} a - \frac{1}{2}$

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:  $7$
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:  \( 475501.195576 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.C_2^2:D_4$ (as 16T305):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 29 conjugacy class representatives for $C_4.C_2^2:D_4$
Character table for $C_4.C_2^2:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{34}) \), 4.0.1088.2 x2, 4.0.2312.1 x2, \(\Q(\sqrt{2}, \sqrt{17})\), 8.0.342102016.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.4.11.3$x^{4} + 4 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.11.4$x^{4} + 12 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
$17$17.8.6.2$x^{8} + 85 x^{4} + 2601$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$