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Properties

Label 16.0.55086646241...625.10
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 41^{12}$
Root discriminant $54.18$
Ramified primes $5, 41$
Class number $40$ (GRH)
Class group $[2, 2, 10]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![395254161, 0, -237776760, 0, 69284846, 0, -11934110, 0, 1286011, 0, -88710, 0, 3851, 0, -95, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 95*x^14 + 3851*x^12 - 88710*x^10 + 1286011*x^8 - 11934110*x^6 + 69284846*x^4 - 237776760*x^2 + 395254161)
 
gp: K = bnfinit(x^16 - 95*x^14 + 3851*x^12 - 88710*x^10 + 1286011*x^8 - 11934110*x^6 + 69284846*x^4 - 237776760*x^2 + 395254161, 1)
 

Normalized defining polynomial

\( x^{16} - 95 x^{14} + 3851 x^{12} - 88710 x^{10} + 1286011 x^{8} - 11934110 x^{6} + 69284846 x^{4} - 237776760 x^{2} + 395254161 \)

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:  \(5508664624112838398681640625=5^{12}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.18$
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:  $5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2946} a^{12} + \frac{29}{982} a^{10} + \frac{229}{982} a^{8} + \frac{142}{491} a^{6} + \frac{554}{1473} a^{4} - \frac{1}{2} a^{3} - \frac{327}{982} a^{2} - \frac{35}{982}$, $\frac{1}{138462} a^{13} + \frac{1762}{69231} a^{11} - \frac{753}{46154} a^{9} - \frac{8205}{23077} a^{7} + \frac{3500}{69231} a^{5} - \frac{14975}{69231} a^{3} - \frac{1}{2} a^{2} + \frac{15677}{46154} a - \frac{1}{2}$, $\frac{1}{202532060325554464662} a^{14} - \frac{11314902561097805}{202532060325554464662} a^{12} + \frac{10406618124445862993}{202532060325554464662} a^{10} + \frac{1887595230102650797}{67510686775184821554} a^{8} - \frac{1}{2} a^{7} - \frac{14627763792420253991}{202532060325554464662} a^{6} - \frac{1}{2} a^{5} - \frac{23843498725062876400}{101266030162777232331} a^{4} + \frac{88173534414176293613}{202532060325554464662} a^{2} - \frac{5001967119295389}{10187216957172902}$, $\frac{1}{28557020505903179517342} a^{15} - \frac{40031528049991276}{14278510252951589758671} a^{13} - \frac{2257182478202128632055}{28557020505903179517342} a^{11} - \frac{520185882744188117921}{9519006835301059839114} a^{9} + \frac{5482398048567584132822}{14278510252951589758671} a^{7} - \frac{1}{2} a^{6} - \frac{14199838144197245346485}{28557020505903179517342} a^{5} + \frac{1134520431276002142953}{28557020505903179517342} a^{3} - \frac{1}{2} a^{2} - \frac{610778287890328423}{1436397590961379182} a - \frac{1}{2}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{10}$, which has order $40$ (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:  \( 478806.025823 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{205}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{5}) \), 4.0.8615125.2, \(\Q(\sqrt{5}, \sqrt{41})\), 4.0.8615125.1, 4.0.8405.1 x2, 4.0.1025.1 x2, 8.0.74220378765625.1, 8.0.1766100625.1, 8.8.74220378765625.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$41$41.8.6.1$x^{8} - 9881 x^{4} + 34857216$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41.8.6.1$x^{8} - 9881 x^{4} + 34857216$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$