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Properties

Label 16.2.96046475217...1875.3
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![-149, 367, -399, 156, -163, 258, -186, 191, -133, 22, 5, 14, -13, -3, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 3*x^14 - 3*x^13 - 13*x^12 + 14*x^11 + 5*x^10 + 22*x^9 - 133*x^8 + 191*x^7 - 186*x^6 + 258*x^5 - 163*x^4 + 156*x^3 - 399*x^2 + 367*x - 149)
 
gp: K = bnfinit(x^16 + 3*x^14 - 3*x^13 - 13*x^12 + 14*x^11 + 5*x^10 + 22*x^9 - 133*x^8 + 191*x^7 - 186*x^6 + 258*x^5 - 163*x^4 + 156*x^3 - 399*x^2 + 367*x - 149, 1)
 

Normalized defining polynomial

\( x^{16} + 3 x^{14} - 3 x^{13} - 13 x^{12} + 14 x^{11} + 5 x^{10} + 22 x^{9} - 133 x^{8} + 191 x^{7} - 186 x^{6} + 258 x^{5} - 163 x^{4} + 156 x^{3} - 399 x^{2} + 367 x - 149 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1146126139267722770243} a^{15} + \frac{158452650777715521472}{1146126139267722770243} a^{14} - \frac{156046343560297367575}{1146126139267722770243} a^{13} - \frac{128738625215337421541}{1146126139267722770243} a^{12} + \frac{11116207026993460459}{1146126139267722770243} a^{11} + \frac{485349126784413658705}{1146126139267722770243} a^{10} - \frac{187599055566786188990}{1146126139267722770243} a^{9} + \frac{3094843453844297902}{1146126139267722770243} a^{8} + \frac{86632586046979312883}{1146126139267722770243} a^{7} + \frac{2970570792816696316}{7590239332898826293} a^{6} - \frac{557141106404095045251}{1146126139267722770243} a^{5} - \frac{39365085002783896339}{1146126139267722770243} a^{4} - \frac{222110995158324884525}{1146126139267722770243} a^{3} + \frac{99012903219021357955}{1146126139267722770243} a^{2} - \frac{383088761748954529378}{1146126139267722770243} a + \frac{547726993357369773714}{1146126139267722770243}$

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