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Properties

Label 16.2.10221881505...1875.2
Degree $16$
Signature $[2, 7]$
Discriminant $-\,5^{10}\cdot 11^{4}\cdot 59^{5}$
Root discriminant $17.81$
Ramified primes $5, 11, 59$
Class number $1$
Class group Trivial
Galois group 16T1574

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![61, -521, 1469, -2814, 3831, -3605, 2303, -909, 131, 0, 108, -123, 50, -7, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 4*x^14 - 7*x^13 + 50*x^12 - 123*x^11 + 108*x^10 + 131*x^8 - 909*x^7 + 2303*x^6 - 3605*x^5 + 3831*x^4 - 2814*x^3 + 1469*x^2 - 521*x + 61)
 
gp: K = bnfinit(x^16 - 4*x^15 + 4*x^14 - 7*x^13 + 50*x^12 - 123*x^11 + 108*x^10 + 131*x^8 - 909*x^7 + 2303*x^6 - 3605*x^5 + 3831*x^4 - 2814*x^3 + 1469*x^2 - 521*x + 61, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 4 x^{14} - 7 x^{13} + 50 x^{12} - 123 x^{11} + 108 x^{10} + 131 x^{8} - 909 x^{7} + 2303 x^{6} - 3605 x^{5} + 3831 x^{4} - 2814 x^{3} + 1469 x^{2} - 521 x + 61 \)

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:  \(-102218815055263671875=-\,5^{10}\cdot 11^{4}\cdot 59^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.81$
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:  $5, 11, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{15} a^{12} + \frac{1}{15} a^{10} + \frac{1}{3} a^{9} + \frac{2}{5} a^{8} - \frac{1}{15} a^{7} - \frac{7}{15} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{15} a^{3} - \frac{7}{15} a^{2} - \frac{2}{15}$, $\frac{1}{15} a^{13} + \frac{1}{15} a^{11} - \frac{1}{15} a^{10} - \frac{2}{5} a^{9} - \frac{4}{15} a^{8} - \frac{4}{15} a^{7} - \frac{4}{15} a^{4} - \frac{7}{15} a^{3} - \frac{2}{5} a^{2} + \frac{4}{15} a + \frac{2}{5}$, $\frac{1}{15} a^{14} - \frac{1}{15} a^{11} - \frac{1}{15} a^{10} + \frac{1}{5} a^{9} - \frac{7}{15} a^{8} - \frac{2}{15} a^{7} + \frac{1}{15} a^{6} + \frac{1}{3} a^{5} - \frac{4}{15} a^{4} + \frac{7}{15} a^{3} + \frac{2}{15} a^{2} - \frac{4}{15}$, $\frac{1}{2329301324009865} a^{15} - \frac{5224471364962}{465860264801973} a^{14} + \frac{424935481664}{75138752387415} a^{13} - \frac{11925801468076}{776433774669955} a^{12} - \frac{19845446178537}{776433774669955} a^{11} + \frac{6333349658414}{80320735310685} a^{10} + \frac{1046651325241591}{2329301324009865} a^{9} - \frac{106679822769062}{2329301324009865} a^{8} - \frac{2107809631271}{15027750477483} a^{7} + \frac{123537461792737}{465860264801973} a^{6} - \frac{98543096883214}{465860264801973} a^{5} + \frac{36082658319037}{75138752387415} a^{4} + \frac{6691700644903}{26773578436895} a^{3} + \frac{626815914808201}{2329301324009865} a^{2} + \frac{645503232634054}{2329301324009865} a + \frac{220643241463327}{776433774669955}$

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:  \( 3232.81111967 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1574:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 88 conjugacy class representatives for t16n1574 are not computed
Character table for t16n1574 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 8.2.4461875.1

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 $16$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ R

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.8.0.1$x^{8} + x^{2} - 2 x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
59Data not computed