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Properties

Label 16.0.13724372348...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{14}\cdot 1361^{4}$
Root discriminant $49.67$
Ramified primes $2, 5, 1361$
Class number $32$ (GRH)
Class group $[4, 8]$ (GRH)
Galois group 16T1263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![91181, -278608, 419722, -394094, 277076, -157598, 83260, -38012, 13029, -2526, 140, 44, 56, -22, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 18*x^14 - 22*x^13 + 56*x^12 + 44*x^11 + 140*x^10 - 2526*x^9 + 13029*x^8 - 38012*x^7 + 83260*x^6 - 157598*x^5 + 277076*x^4 - 394094*x^3 + 419722*x^2 - 278608*x + 91181)
 
gp: K = bnfinit(x^16 - 6*x^15 + 18*x^14 - 22*x^13 + 56*x^12 + 44*x^11 + 140*x^10 - 2526*x^9 + 13029*x^8 - 38012*x^7 + 83260*x^6 - 157598*x^5 + 277076*x^4 - 394094*x^3 + 419722*x^2 - 278608*x + 91181, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 18 x^{14} - 22 x^{13} + 56 x^{12} + 44 x^{11} + 140 x^{10} - 2526 x^{9} + 13029 x^{8} - 38012 x^{7} + 83260 x^{6} - 157598 x^{5} + 277076 x^{4} - 394094 x^{3} + 419722 x^{2} - 278608 x + 91181 \)

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:  \(1372437234816400000000000000=2^{16}\cdot 5^{14}\cdot 1361^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.67$
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:  $2, 5, 1361$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{11} + \frac{1}{16} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{3}{16} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} + \frac{1}{16} a^{2} - \frac{1}{8} a - \frac{7}{16}$, $\frac{1}{16} a^{13} + \frac{1}{16} a^{11} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{3}{16} a^{7} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{16} a^{3} - \frac{7}{16} a - \frac{1}{8}$, $\frac{1}{32} a^{14} - \frac{1}{32} a^{13} - \frac{3}{32} a^{11} + \frac{1}{32} a^{10} - \frac{1}{16} a^{9} - \frac{3}{32} a^{8} + \frac{1}{32} a^{7} + \frac{5}{32} a^{6} - \frac{1}{16} a^{5} - \frac{7}{32} a^{4} + \frac{13}{32} a^{3} - \frac{3}{8} a^{2} - \frac{13}{32} a - \frac{3}{32}$, $\frac{1}{93754163170324827415525783208288} a^{15} + \frac{32881359608556611550741464103}{2130776435689200623080131436552} a^{14} - \frac{2167312446143444626259252114825}{93754163170324827415525783208288} a^{13} + \frac{2839278772388033452618091176015}{93754163170324827415525783208288} a^{12} + \frac{271636062907207913642000795671}{4261552871378401246160262873104} a^{11} + \frac{191351872593411340587888325711}{8523105742756802492320525746208} a^{10} + \frac{1872662101174269058249744027855}{93754163170324827415525783208288} a^{9} - \frac{2826214633692109354093312852649}{46877081585162413707762891604144} a^{8} + \frac{118298208650148239776374128993}{46877081585162413707762891604144} a^{7} - \frac{20404747411729230705837089284687}{93754163170324827415525783208288} a^{6} + \frac{22156064206039389480702285203675}{93754163170324827415525783208288} a^{5} + \frac{23725841348387365933302599819}{46877081585162413707762891604144} a^{4} + \frac{1337142425512740296026895110445}{93754163170324827415525783208288} a^{3} + \frac{14157837441612153368601903582393}{93754163170324827415525783208288} a^{2} - \frac{1844637950417289713964447993529}{23438540792581206853881445802072} a + \frac{2051395968660829728263109561763}{93754163170324827415525783208288}$

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

Class group and class number

$C_{4}\times C_{8}$, which has order $32$ (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:  \( 906591.467306 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1263:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1263
Character table for t16n1263 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.0.5444000000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
1361Data not computed