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Properties

Label 16.0.65790008098...0000.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{12}\cdot 89^{4}$
Root discriminant $41.08$
Ramified primes $2, 5, 89$
Class number $72$ (GRH)
Class group $[2, 6, 6]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T217)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1681, 12464, 52714, -112808, 312484, -220324, 182566, -95716, 55650, -22856, 9712, -3068, 1015, -244, 58, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 58*x^14 - 244*x^13 + 1015*x^12 - 3068*x^11 + 9712*x^10 - 22856*x^9 + 55650*x^8 - 95716*x^7 + 182566*x^6 - 220324*x^5 + 312484*x^4 - 112808*x^3 + 52714*x^2 + 12464*x + 1681)
 
gp: K = bnfinit(x^16 - 8*x^15 + 58*x^14 - 244*x^13 + 1015*x^12 - 3068*x^11 + 9712*x^10 - 22856*x^9 + 55650*x^8 - 95716*x^7 + 182566*x^6 - 220324*x^5 + 312484*x^4 - 112808*x^3 + 52714*x^2 + 12464*x + 1681, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 58 x^{14} - 244 x^{13} + 1015 x^{12} - 3068 x^{11} + 9712 x^{10} - 22856 x^{9} + 55650 x^{8} - 95716 x^{7} + 182566 x^{6} - 220324 x^{5} + 312484 x^{4} - 112808 x^{3} + 52714 x^{2} + 12464 x + 1681 \)

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:  \(65790008098816000000000000=2^{32}\cdot 5^{12}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.08$
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:  $2, 5, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{257396041177288495396870638491286499} a^{15} - \frac{13832448325738241557848832315980485}{257396041177288495396870638491286499} a^{14} + \frac{25245754339565766040018101585947809}{257396041177288495396870638491286499} a^{13} - \frac{79919830447582505980022008477489039}{257396041177288495396870638491286499} a^{12} - \frac{104712553024269814177115817891712382}{257396041177288495396870638491286499} a^{11} + \frac{13160296867144286065517941589967505}{257396041177288495396870638491286499} a^{10} - \frac{46579497880318804130776259482824274}{257396041177288495396870638491286499} a^{9} + \frac{11408015021068672508449742679219850}{257396041177288495396870638491286499} a^{8} + \frac{85660075664642994840285037735510292}{257396041177288495396870638491286499} a^{7} - \frac{18618273627044311631561125708356515}{257396041177288495396870638491286499} a^{6} + \frac{87941913959562373153661983617308587}{257396041177288495396870638491286499} a^{5} - \frac{127241160150641882039418139940938367}{257396041177288495396870638491286499} a^{4} - \frac{68051240848375233157542077131648186}{257396041177288495396870638491286499} a^{3} - \frac{99682973913339295322881037278008463}{257396041177288495396870638491286499} a^{2} - \frac{15361186085966382242695906171532742}{36770863025326927913838662641612357} a - \frac{1138021360224955532137465186289131}{6277952223836304765777332646128939}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{6}$, which has order $72$ (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:  \( 18891.2708489 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T217):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.4.91136000000.1, 8.0.8111104000000.27, 8.4.227840000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
5Data not computed
89Data not computed