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Properties

Label 16.16.2699666436...0625.1
Degree $16$
Signature $[16, 0]$
Discriminant $5^{14}\cdot 89^{7}$
Root discriminant $29.14$
Ramified primes $5, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 16, 56, -156, -687, 901, 2259, -3558, -507, 2591, -449, -718, 198, 82, -26, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 26*x^14 + 82*x^13 + 198*x^12 - 718*x^11 - 449*x^10 + 2591*x^9 - 507*x^8 - 3558*x^7 + 2259*x^6 + 901*x^5 - 687*x^4 - 156*x^3 + 56*x^2 + 16*x + 1)
 
gp: K = bnfinit(x^16 - 3*x^15 - 26*x^14 + 82*x^13 + 198*x^12 - 718*x^11 - 449*x^10 + 2591*x^9 - 507*x^8 - 3558*x^7 + 2259*x^6 + 901*x^5 - 687*x^4 - 156*x^3 + 56*x^2 + 16*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 26 x^{14} + 82 x^{13} + 198 x^{12} - 718 x^{11} - 449 x^{10} + 2591 x^{9} - 507 x^{8} - 3558 x^{7} + 2259 x^{6} + 901 x^{5} - 687 x^{4} - 156 x^{3} + 56 x^{2} + 16 x + 1 \)

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:  \(269966643649468994140625=5^{14}\cdot 89^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.14$
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:  $5, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{19} a^{14} - \frac{4}{19} a^{13} - \frac{8}{19} a^{12} - \frac{4}{19} a^{11} - \frac{5}{19} a^{10} - \frac{9}{19} a^{9} + \frac{3}{19} a^{8} - \frac{8}{19} a^{7} - \frac{1}{19} a^{6} - \frac{2}{19} a^{5} + \frac{5}{19} a^{4} - \frac{6}{19} a^{3} - \frac{3}{19} a^{2} - \frac{9}{19} a + \frac{4}{19}$, $\frac{1}{95084536909} a^{15} + \frac{375428074}{95084536909} a^{14} - \frac{41384536662}{95084536909} a^{13} - \frac{43474828082}{95084536909} a^{12} + \frac{40560291128}{95084536909} a^{11} - \frac{43892316710}{95084536909} a^{10} - \frac{37001006820}{95084536909} a^{9} - \frac{18783691761}{95084536909} a^{8} + \frac{5817481460}{95084536909} a^{7} - \frac{9457454091}{95084536909} a^{6} + \frac{12468908939}{95084536909} a^{5} - \frac{25050927182}{95084536909} a^{4} + \frac{15455738835}{95084536909} a^{3} + \frac{38430598215}{95084536909} a^{2} + \frac{9886354236}{95084536909} a + \frac{44995844633}{95084536909}$

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:  \( 2982102.00199 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2225.1, 8.8.11015140625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ R $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
5Data not computed
$89$89.8.7.6$x^{8} + 2403$$8$$1$$7$$C_8$$[\ ]_{8}$
89.8.0.1$x^{8} - x + 62$$1$$8$$0$$C_8$$[\ ]^{8}$