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Properties

Label 16.2.96046475217...1875.2
Degree $16$
Signature $[2, 7]$
Discriminant $-\,5^{8}\cdot 71^{3}\cdot 1901^{3}$
Root discriminant $20.48$
Ramified primes $5, 71, 1901$
Class number $1$
Class group Trivial
Galois group 16T1651

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19, 42, -70, -22, 72, 69, -28, 54, 20, -5, 34, 0, -1, 8, -3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 3*x^14 + 8*x^13 - x^12 + 34*x^10 - 5*x^9 + 20*x^8 + 54*x^7 - 28*x^6 + 69*x^5 + 72*x^4 - 22*x^3 - 70*x^2 + 42*x - 19)
 
gp: K = bnfinit(x^16 - x^15 - 3*x^14 + 8*x^13 - x^12 + 34*x^10 - 5*x^9 + 20*x^8 + 54*x^7 - 28*x^6 + 69*x^5 + 72*x^4 - 22*x^3 - 70*x^2 + 42*x - 19, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 3 x^{14} + 8 x^{13} - x^{12} + 34 x^{10} - 5 x^{9} + 20 x^{8} + 54 x^{7} - 28 x^{6} + 69 x^{5} + 72 x^{4} - 22 x^{3} - 70 x^{2} + 42 x - 19 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-960464752179926171875=-\,5^{8}\cdot 71^{3}\cdot 1901^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.48$
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:  $5, 71, 1901$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17} a^{14} - \frac{5}{17} a^{12} + \frac{3}{17} a^{11} - \frac{5}{17} a^{10} + \frac{6}{17} a^{9} - \frac{1}{17} a^{8} - \frac{1}{17} a^{7} + \frac{4}{17} a^{6} - \frac{8}{17} a^{5} + \frac{7}{17} a^{4} + \frac{7}{17} a^{3} - \frac{3}{17} a^{2} - \frac{5}{17} a - \frac{1}{17}$, $\frac{1}{4442373665767933} a^{15} - \frac{54060724818875}{4442373665767933} a^{14} + \frac{1600608939764145}{4442373665767933} a^{13} - \frac{1504765553611114}{4442373665767933} a^{12} + \frac{1392379324971132}{4442373665767933} a^{11} + \frac{642187085375062}{4442373665767933} a^{10} + \frac{803508702617433}{4442373665767933} a^{9} - \frac{1310715580927748}{4442373665767933} a^{8} - \frac{237571196114455}{634624809395419} a^{7} - \frac{1537462386814881}{4442373665767933} a^{6} + \frac{229663277747863}{4442373665767933} a^{5} - \frac{1438541685912567}{4442373665767933} a^{4} - \frac{1722807248470622}{4442373665767933} a^{3} - \frac{1286752715357093}{4442373665767933} a^{2} - \frac{22084376673049}{261316097986349} a - \frac{1947367702432309}{4442373665767933}$

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:  $8$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 7380.47762635 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1651:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4608
The 44 conjugacy class representatives for t16n1651
Character table for t16n1651 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.6.84356875.1

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
Arithmetically equvalently 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 $16$ $16$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ $16$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
71Data not computed
1901Data not computed