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Properties

Label 16.0.28500297912...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{4}\cdot 5^{14}\cdot 7^{8}$
Root discriminant $14.24$
Ramified primes $3, 5, 7$
Class number $1$
Class group Trivial
Galois group $(C_8:C_2):C_2$ (as 16T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 39, -80, 157, -190, 247, -195, 165, -90, 48, -35, 27, -25, 16, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 16*x^14 - 25*x^13 + 27*x^12 - 35*x^11 + 48*x^10 - 90*x^9 + 165*x^8 - 195*x^7 + 247*x^6 - 190*x^5 + 157*x^4 - 80*x^3 + 39*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^16 - 5*x^15 + 16*x^14 - 25*x^13 + 27*x^12 - 35*x^11 + 48*x^10 - 90*x^9 + 165*x^8 - 195*x^7 + 247*x^6 - 190*x^5 + 157*x^4 - 80*x^3 + 39*x^2 - 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 16 x^{14} - 25 x^{13} + 27 x^{12} - 35 x^{11} + 48 x^{10} - 90 x^{9} + 165 x^{8} - 195 x^{7} + 247 x^{6} - 190 x^{5} + 157 x^{4} - 80 x^{3} + 39 x^{2} - 10 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2850029791259765625=3^{4}\cdot 5^{14}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.24$
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:  $3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11} a^{12} - \frac{2}{11} a^{11} - \frac{1}{11} a^{10} + \frac{3}{11} a^{9} - \frac{2}{11} a^{8} + \frac{1}{11} a^{7} - \frac{1}{11} a^{6} + \frac{5}{11} a^{5} - \frac{2}{11} a^{4} + \frac{1}{11} a^{3} - \frac{4}{11} a^{2} - \frac{1}{11} a + \frac{5}{11}$, $\frac{1}{11} a^{13} - \frac{5}{11} a^{11} + \frac{1}{11} a^{10} + \frac{4}{11} a^{9} - \frac{3}{11} a^{8} + \frac{1}{11} a^{7} + \frac{3}{11} a^{6} - \frac{3}{11} a^{5} - \frac{3}{11} a^{4} - \frac{2}{11} a^{3} + \frac{2}{11} a^{2} + \frac{3}{11} a - \frac{1}{11}$, $\frac{1}{121} a^{14} + \frac{2}{121} a^{13} + \frac{5}{121} a^{12} - \frac{40}{121} a^{11} - \frac{15}{121} a^{10} + \frac{13}{121} a^{9} + \frac{30}{121} a^{8} + \frac{37}{121} a^{7} + \frac{37}{121} a^{6} - \frac{14}{121} a^{5} - \frac{50}{121} a^{4} + \frac{52}{121} a^{3} - \frac{4}{11} a^{2} + \frac{6}{121} a - \frac{29}{121}$, $\frac{1}{145388881} a^{15} - \frac{65214}{145388881} a^{14} - \frac{308120}{13217171} a^{13} - \frac{3365607}{145388881} a^{12} - \frac{29691907}{145388881} a^{11} + \frac{9400909}{145388881} a^{10} + \frac{54601902}{145388881} a^{9} + \frac{11833740}{145388881} a^{8} - \frac{21474173}{145388881} a^{7} + \frac{55816880}{145388881} a^{6} - \frac{44812035}{145388881} a^{5} - \frac{36346265}{145388881} a^{4} + \frac{13707993}{145388881} a^{3} + \frac{72062139}{145388881} a^{2} + \frac{3675921}{13217171} a + \frac{11854085}{145388881}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{2783516}{13217171} a^{15} + \frac{151647940}{145388881} a^{14} - \frac{497846033}{145388881} a^{13} + \frac{814852256}{145388881} a^{12} - \frac{1010411364}{145388881} a^{11} + \frac{1322767881}{145388881} a^{10} - \frac{1626060021}{145388881} a^{9} + \frac{2927974070}{145388881} a^{8} - \frac{5363434863}{145388881} a^{7} + \frac{6846673476}{145388881} a^{6} - \frac{9265228842}{145388881} a^{5} + \frac{7205668930}{145388881} a^{4} - \frac{6482271137}{145388881} a^{3} + \frac{276381085}{13217171} a^{2} - \frac{1657006668}{145388881} a + \frac{374978565}{145388881} \) (order $10$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2117.22006174 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}:C_2$ (as 16T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $(C_8:C_2):C_2$
Character table for $(C_8:C_2):C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-7}) \), 4.4.6125.1, \(\Q(\zeta_{5})\), \(\Q(\sqrt{5}, \sqrt{-7})\), 8.0.37515625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$3$3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
5Data not computed
7Data not computed