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Properties

Label 16.0.49496507867...1201.2
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 16573^{4}$
Root discriminant $19.65$
Ramified primes $3, 16573$
Class number $1$
Class group Trivial
Galois group 16T1869

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2749, 3971, 4632, 1586, -1505, -2053, -1277, 51, 554, -279, 548, -325, 156, -75, 20, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 20*x^14 - 75*x^13 + 156*x^12 - 325*x^11 + 548*x^10 - 279*x^9 + 554*x^8 + 51*x^7 - 1277*x^6 - 2053*x^5 - 1505*x^4 + 1586*x^3 + 4632*x^2 + 3971*x + 2749)
 
gp: K = bnfinit(x^16 - 5*x^15 + 20*x^14 - 75*x^13 + 156*x^12 - 325*x^11 + 548*x^10 - 279*x^9 + 554*x^8 + 51*x^7 - 1277*x^6 - 2053*x^5 - 1505*x^4 + 1586*x^3 + 4632*x^2 + 3971*x + 2749, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 20 x^{14} - 75 x^{13} + 156 x^{12} - 325 x^{11} + 548 x^{10} - 279 x^{9} + 554 x^{8} + 51 x^{7} - 1277 x^{6} - 2053 x^{5} - 1505 x^{4} + 1586 x^{3} + 4632 x^{2} + 3971 x + 2749 \)

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:  \(494965078673757801201=3^{8}\cdot 16573^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.65$
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:  $3, 16573$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{154112218762813875088205251} a^{15} - \frac{56685692438190127494016900}{154112218762813875088205251} a^{14} - \frac{70936488392963073237204836}{154112218762813875088205251} a^{13} + \frac{22949210674136677808609907}{154112218762813875088205251} a^{12} + \frac{26090359383115270573317477}{154112218762813875088205251} a^{11} - \frac{69377420631373626738236117}{154112218762813875088205251} a^{10} + \frac{40024808740709177815334223}{154112218762813875088205251} a^{9} - \frac{64295704288118369067919947}{154112218762813875088205251} a^{8} - \frac{23851533711644860583407248}{154112218762813875088205251} a^{7} + \frac{2663683782309710719860814}{154112218762813875088205251} a^{6} + \frac{71779534984817229732035346}{154112218762813875088205251} a^{5} + \frac{10204621449063271451372708}{154112218762813875088205251} a^{4} - \frac{18529240998202866166270600}{154112218762813875088205251} a^{3} - \frac{43997858430307000819095362}{154112218762813875088205251} a^{2} - \frac{1000316680161390058031390}{4165195101697672299681223} a + \frac{66923564218954135168939670}{154112218762813875088205251}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{122930989654}{443433910836079} a^{15} + \frac{333657084389}{443433910836079} a^{14} - \frac{899889892129}{443433910836079} a^{13} + \frac{3642647496784}{443433910836079} a^{12} - \frac{24554869838}{443433910836079} a^{11} + \frac{372094970759}{443433910836079} a^{10} - \frac{3638843866552}{443433910836079} a^{9} - \frac{69169346021228}{443433910836079} a^{8} + \frac{4450999869338}{443433910836079} a^{7} - \frac{74407615218177}{443433910836079} a^{6} + \frac{317938339814038}{443433910836079} a^{5} + \frac{178085477828676}{443433910836079} a^{4} + \frac{186343532366066}{443433910836079} a^{3} - \frac{278410530133433}{443433910836079} a^{2} - \frac{17633896818624}{11984700292867} a - \frac{149802315217443}{443433910836079} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 12143.537393 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1869:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 77 conjugacy class representatives for t16n1869 are not computed
Character table for t16n1869 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 8.0.1342413.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
16573Data not computed