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Properties

Label 16.0.10271172687...5625.5
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{10}$
Root discriminant $27.43$
Ramified primes $5, 29$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $(C_2\times C_4).D_4$ (as 16T121)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![631, -2841, 6953, -10079, 9045, -4075, 104, 1052, -1034, 737, -191, -127, 93, -21, 14, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 14*x^14 - 21*x^13 + 93*x^12 - 127*x^11 - 191*x^10 + 737*x^9 - 1034*x^8 + 1052*x^7 + 104*x^6 - 4075*x^5 + 9045*x^4 - 10079*x^3 + 6953*x^2 - 2841*x + 631)
 
gp: K = bnfinit(x^16 - 6*x^15 + 14*x^14 - 21*x^13 + 93*x^12 - 127*x^11 - 191*x^10 + 737*x^9 - 1034*x^8 + 1052*x^7 + 104*x^6 - 4075*x^5 + 9045*x^4 - 10079*x^3 + 6953*x^2 - 2841*x + 631, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 14 x^{14} - 21 x^{13} + 93 x^{12} - 127 x^{11} - 191 x^{10} + 737 x^{9} - 1034 x^{8} + 1052 x^{7} + 104 x^{6} - 4075 x^{5} + 9045 x^{4} - 10079 x^{3} + 6953 x^{2} - 2841 x + 631 \)

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:  \(102711726879931884765625=5^{12}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.43$
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:  $5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{3}{8} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a + \frac{1}{8}$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{10} + \frac{1}{16} a^{9} - \frac{1}{4} a^{8} + \frac{5}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{16} a^{3} + \frac{1}{16} a^{2} - \frac{1}{16} a + \frac{1}{16}$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{12} - \frac{1}{16} a^{11} + \frac{3}{32} a^{10} - \frac{5}{32} a^{9} - \frac{7}{32} a^{8} - \frac{3}{16} a^{7} + \frac{13}{32} a^{6} - \frac{1}{4} a^{5} + \frac{11}{32} a^{4} - \frac{7}{16} a^{3} - \frac{1}{16} a^{2} + \frac{1}{16} a + \frac{15}{32}$, $\frac{1}{64} a^{14} + \frac{1}{64} a^{12} + \frac{1}{64} a^{11} + \frac{3}{32} a^{10} - \frac{1}{8} a^{9} - \frac{13}{64} a^{8} - \frac{21}{64} a^{7} - \frac{15}{64} a^{6} - \frac{13}{64} a^{5} - \frac{19}{64} a^{4} + \frac{3}{16} a^{3} + \frac{1}{16} a^{2} - \frac{3}{64} a + \frac{3}{64}$, $\frac{1}{13561838415752825469568} a^{15} - \frac{40928785877108342693}{13561838415752825469568} a^{14} - \frac{172440448027129901719}{13561838415752825469568} a^{13} - \frac{54170451606003401157}{3390459603938206367392} a^{12} + \frac{316282725509902563745}{13561838415752825469568} a^{11} + \frac{614337541510282197433}{6780919207876412734784} a^{10} + \frac{864336309326140208123}{13561838415752825469568} a^{9} - \frac{857510680596452893883}{3390459603938206367392} a^{8} + \frac{148759735312548819313}{6780919207876412734784} a^{7} - \frac{2439369284270115295361}{6780919207876412734784} a^{6} - \frac{2308045082465437394985}{6780919207876412734784} a^{5} - \frac{2743112614631812006493}{13561838415752825469568} a^{4} - \frac{16881609612856354661}{38527950044752345084} a^{3} - \frac{5376292636698596518271}{13561838415752825469568} a^{2} - \frac{1370720336014105616219}{6780919207876412734784} a - \frac{4494215452501685030127}{13561838415752825469568}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $7$
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:  \( 14143.2557982 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 4.0.105125.2, 4.0.3625.1, 8.4.381078125.1, 8.4.12819468125.1, 8.0.11051265625.5

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$29$29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$