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Properties

Label 16.0.65790008098...0000.1
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 $112$ (GRH)
Class group $[2, 56]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T210)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5351, -14976, 37368, -54320, 59523, -43844, 27006, -9824, 4132, -1148, 652, -216, 94, -12, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 4*x^14 - 12*x^13 + 94*x^12 - 216*x^11 + 652*x^10 - 1148*x^9 + 4132*x^8 - 9824*x^7 + 27006*x^6 - 43844*x^5 + 59523*x^4 - 54320*x^3 + 37368*x^2 - 14976*x + 5351)
 
gp: K = bnfinit(x^16 + 4*x^14 - 12*x^13 + 94*x^12 - 216*x^11 + 652*x^10 - 1148*x^9 + 4132*x^8 - 9824*x^7 + 27006*x^6 - 43844*x^5 + 59523*x^4 - 54320*x^3 + 37368*x^2 - 14976*x + 5351, 1)
 

Normalized defining polynomial

\( x^{16} + 4 x^{14} - 12 x^{13} + 94 x^{12} - 216 x^{11} + 652 x^{10} - 1148 x^{9} + 4132 x^{8} - 9824 x^{7} + 27006 x^{6} - 43844 x^{5} + 59523 x^{4} - 54320 x^{3} + 37368 x^{2} - 14976 x + 5351 \)

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 a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{2}{9} a^{9} + \frac{1}{9} a^{8} + \frac{2}{9} a^{7} + \frac{2}{9} a^{6} + \frac{1}{9} a^{5} + \frac{4}{9} a^{4} - \frac{1}{3} a^{3} - \frac{4}{9} a^{2} - \frac{1}{3} a + \frac{4}{9}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{2}{9} a^{10} - \frac{1}{9} a^{9} - \frac{2}{9} a^{8} - \frac{1}{3} a^{7} - \frac{4}{9} a^{6} - \frac{4}{9} a^{4} - \frac{4}{9} a^{3} - \frac{2}{9} a^{2} + \frac{4}{9} a - \frac{4}{9}$, $\frac{1}{6210897937201928645780694035571} a^{15} - \frac{17313402975683628870627834022}{2070299312400642881926898011857} a^{14} - \frac{301096397605336394921522064973}{6210897937201928645780694035571} a^{13} - \frac{20737875154927148652815183968}{6210897937201928645780694035571} a^{12} - \frac{45559329429424757628422451215}{6210897937201928645780694035571} a^{11} - \frac{1007661978456631343658574292347}{6210897937201928645780694035571} a^{10} + \frac{320769017753819039844400572251}{2070299312400642881926898011857} a^{9} + \frac{2195107914501583290688727552251}{6210897937201928645780694035571} a^{8} - \frac{2372375315199094496777441374481}{6210897937201928645780694035571} a^{7} - \frac{690448747826136651147478924933}{6210897937201928645780694035571} a^{6} + \frac{299820812464124250271160524986}{690099770800214293975632670619} a^{5} - \frac{287052431749028644326940309462}{690099770800214293975632670619} a^{4} - \frac{81725823709694596918437458998}{270039040747909941120899740677} a^{3} + \frac{307922200176681287572371864515}{2070299312400642881926898011857} a^{2} + \frac{311603408140730581510365872021}{6210897937201928645780694035571} a + \frac{2216105546875031369880429169363}{6210897937201928645780694035571}$

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

Class group and class number

$C_{2}\times C_{56}$, which has order $112$ (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:  \( 15950.2679788 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T210):

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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.0.712000.3, 4.0.11125.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.3645440000.1, 8.0.506944000000.11, 8.0.91136000000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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