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Properties

Label 16.0.10511416756...681.12
Degree $16$
Signature $[0, 8]$
Discriminant $23^{8}\cdot 41^{10}$
Root discriminant $48.85$
Ramified primes $23, 41$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group $C_4\wr C_2$ (as 16T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![431649, 0, -50607, 0, 49627, 0, -17469, 0, 3274, 0, 177, 0, -55, 0, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 2*x^14 - 55*x^12 + 177*x^10 + 3274*x^8 - 17469*x^6 + 49627*x^4 - 50607*x^2 + 431649)
 
gp: K = bnfinit(x^16 + 2*x^14 - 55*x^12 + 177*x^10 + 3274*x^8 - 17469*x^6 + 49627*x^4 - 50607*x^2 + 431649, 1)
 

Normalized defining polynomial

\( x^{16} + 2 x^{14} - 55 x^{12} + 177 x^{10} + 3274 x^{8} - 17469 x^{6} + 49627 x^{4} - 50607 x^{2} + 431649 \)

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:  \(1051141675669222288577809681=23^{8}\cdot 41^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.85$
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:  $23, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{9} + \frac{1}{6} a^{7} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a$, $\frac{1}{276} a^{12} + \frac{43}{276} a^{10} - \frac{29}{276} a^{8} + \frac{59}{276} a^{6} - \frac{25}{69} a^{4} - \frac{1}{2} a^{3} - \frac{32}{69} a^{2} - \frac{33}{92}$, $\frac{1}{276} a^{13} - \frac{1}{92} a^{11} + \frac{21}{92} a^{9} + \frac{13}{276} a^{7} - \frac{9}{46} a^{5} + \frac{14}{69} a^{3} - \frac{7}{276} a$, $\frac{1}{3517027322972616} a^{14} - \frac{1}{552} a^{13} - \frac{22277136355}{76457115716796} a^{12} + \frac{1}{184} a^{11} + \frac{42815138158991}{879256830743154} a^{10} - \frac{21}{184} a^{9} - \frac{27402593861371}{293085610247718} a^{8} - \frac{13}{552} a^{7} + \frac{92040379001971}{3517027322972616} a^{6} + \frac{9}{92} a^{5} + \frac{16597709699393}{195390406831812} a^{4} + \frac{55}{138} a^{3} - \frac{12069287646001}{152914231433592} a^{2} + \frac{7}{552} a - \frac{135704008976807}{390780813663624}$, $\frac{1}{770228983731002904} a^{15} - \frac{1383624257484391}{770228983731002904} a^{13} - \frac{1}{552} a^{12} + \frac{27765907900089587}{770228983731002904} a^{11} + \frac{95}{552} a^{10} + \frac{19117230418558799}{256742994577000968} a^{9} + \frac{29}{552} a^{8} + \frac{94225258186819325}{385114491865501452} a^{7} + \frac{79}{552} a^{6} - \frac{215255365949638}{465114120610509} a^{5} + \frac{119}{276} a^{4} + \frac{18428914029518065}{770228983731002904} a^{3} + \frac{16}{69} a^{2} - \frac{3586820067246253}{10697624774041707} a + \frac{33}{184}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  \( 1179753.33769 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{-23}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{-943}) \), 4.0.21689.1 x2, 4.2.38663.1 x2, \(\Q(\sqrt{-23}, \sqrt{41})\), 8.0.32421315144041.2, 8.0.19286921561.1, 8.0.790763784001.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
$41$41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$