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Properties

Label 16.0.19021871292...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{14}\cdot 41^{6}$
Root discriminant $28.51$
Ramified primes $3, 5, 41$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $C_4\wr C_2$ (as 16T28)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![271, -763, 1931, -3097, 4061, -3974, 3084, -1676, 612, 14, -136, 121, -39, 8, 6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 6*x^14 + 8*x^13 - 39*x^12 + 121*x^11 - 136*x^10 + 14*x^9 + 612*x^8 - 1676*x^7 + 3084*x^6 - 3974*x^5 + 4061*x^4 - 3097*x^3 + 1931*x^2 - 763*x + 271)
 
gp: K = bnfinit(x^16 - 3*x^15 + 6*x^14 + 8*x^13 - 39*x^12 + 121*x^11 - 136*x^10 + 14*x^9 + 612*x^8 - 1676*x^7 + 3084*x^6 - 3974*x^5 + 4061*x^4 - 3097*x^3 + 1931*x^2 - 763*x + 271, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 6 x^{14} + 8 x^{13} - 39 x^{12} + 121 x^{11} - 136 x^{10} + 14 x^{9} + 612 x^{8} - 1676 x^{7} + 3084 x^{6} - 3974 x^{5} + 4061 x^{4} - 3097 x^{3} + 1931 x^{2} - 763 x + 271 \)

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:  \(190218712922369384765625=3^{8}\cdot 5^{14}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.51$
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:  $3, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{4}{9} a^{7} + \frac{2}{9} a^{6} + \frac{1}{9} a^{5} - \frac{4}{9} a^{4} + \frac{1}{9} a^{3} - \frac{2}{9}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{9} a^{3} - \frac{1}{3} a^{2} + \frac{4}{9} a + \frac{1}{9}$, $\frac{1}{27} a^{14} + \frac{1}{27} a^{13} - \frac{1}{27} a^{12} + \frac{2}{27} a^{11} - \frac{2}{27} a^{10} + \frac{1}{27} a^{9} - \frac{2}{27} a^{8} - \frac{5}{27} a^{7} + \frac{2}{27} a^{6} - \frac{2}{27} a^{5} + \frac{4}{9} a^{4} + \frac{2}{27} a^{3} + \frac{4}{27} a^{2} + \frac{2}{27} a + \frac{5}{27}$, $\frac{1}{21019280343363} a^{15} - \frac{313436079571}{21019280343363} a^{14} - \frac{5205460988}{2335475593707} a^{13} - \frac{996795239711}{21019280343363} a^{12} + \frac{101414783630}{7006426781121} a^{11} - \frac{762101914969}{21019280343363} a^{10} + \frac{565225384646}{21019280343363} a^{9} - \frac{3089770037890}{21019280343363} a^{8} - \frac{3259784574550}{7006426781121} a^{7} - \frac{738489528997}{2335475593707} a^{6} + \frac{10306006866244}{21019280343363} a^{5} + \frac{4556245676714}{21019280343363} a^{4} + \frac{792048343757}{7006426781121} a^{3} + \frac{32028582113}{778491864569} a^{2} + \frac{6900498401545}{21019280343363} a + \frac{593312265635}{21019280343363}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 10253.3634365 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\wr C_2$ (as 16T28):

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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 4.4.5125.1, 4.4.9225.1, 8.8.2127515625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5Data not computed
$41$41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.8.4.1$x^{8} + 57154 x^{4} - 68921 x^{2} + 816644929$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$