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Properties

Label 16.4.11805916207...3424.3
Degree $16$
Signature $[4, 6]$
Discriminant $2^{70}$
Root discriminant $20.75$
Ramified prime $2$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T257)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -80, 544, -1472, 2140, -1904, 1152, -528, 206, -64, 16, -16, 4, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 4*x^12 - 16*x^11 + 16*x^10 - 64*x^9 + 206*x^8 - 528*x^7 + 1152*x^6 - 1904*x^5 + 2140*x^4 - 1472*x^3 + 544*x^2 - 80*x - 1)
 
gp: K = bnfinit(x^16 + 4*x^12 - 16*x^11 + 16*x^10 - 64*x^9 + 206*x^8 - 528*x^7 + 1152*x^6 - 1904*x^5 + 2140*x^4 - 1472*x^3 + 544*x^2 - 80*x - 1, 1)
 

Normalized defining polynomial

\( x^{16} + 4 x^{12} - 16 x^{11} + 16 x^{10} - 64 x^{9} + 206 x^{8} - 528 x^{7} + 1152 x^{6} - 1904 x^{5} + 2140 x^{4} - 1472 x^{3} + 544 x^{2} - 80 x - 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1180591620717411303424=2^{70}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.75$
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:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{17} a^{13} - \frac{4}{17} a^{12} + \frac{1}{17} a^{10} - \frac{3}{17} a^{9} - \frac{3}{17} a^{8} - \frac{4}{17} a^{7} + \frac{2}{17} a^{6} - \frac{8}{17} a^{5} - \frac{4}{17} a^{4} - \frac{3}{17} a^{3} + \frac{2}{17} a^{2} + \frac{1}{17} a + \frac{7}{17}$, $\frac{1}{17} a^{14} + \frac{1}{17} a^{12} + \frac{1}{17} a^{11} + \frac{1}{17} a^{10} + \frac{2}{17} a^{9} + \frac{1}{17} a^{8} + \frac{3}{17} a^{7} - \frac{2}{17} a^{5} - \frac{2}{17} a^{4} + \frac{7}{17} a^{3} - \frac{8}{17} a^{2} - \frac{6}{17} a - \frac{6}{17}$, $\frac{1}{6305466158561} a^{15} - \frac{51129145661}{6305466158561} a^{14} - \frac{69328707498}{6305466158561} a^{13} - \frac{1995649833746}{6305466158561} a^{12} - \frac{95816799466}{6305466158561} a^{11} + \frac{62783189326}{6305466158561} a^{10} + \frac{2028954742504}{6305466158561} a^{9} + \frac{2935992461989}{6305466158561} a^{8} + \frac{76478643764}{6305466158561} a^{7} + \frac{1259535463181}{6305466158561} a^{6} + \frac{978754848085}{6305466158561} a^{5} - \frac{1205947127012}{6305466158561} a^{4} - \frac{633248264226}{6305466158561} a^{3} + \frac{1871437813268}{6305466158561} a^{2} + \frac{1691175543194}{6305466158561} a + \frac{42335385523}{370909774033}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $9$
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:  \( 33106.6035108 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T257):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.6.1073741824.1, 8.4.2147483648.1, 8.2.2147483648.3

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed