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Properties

Label 16.8.18678870765...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{16}\cdot 5^{14}\cdot 1361^{5}$
Root discriminant $77.98$
Ramified primes $2, 5, 1361$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1281

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![70561, -268022, -337540, 573390, -179065, -238766, 240112, -98840, 18520, 2130, -4508, 2226, -385, 50, 0, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 50*x^13 - 385*x^12 + 2226*x^11 - 4508*x^10 + 2130*x^9 + 18520*x^8 - 98840*x^7 + 240112*x^6 - 238766*x^5 - 179065*x^4 + 573390*x^3 - 337540*x^2 - 268022*x + 70561)
 
gp: K = bnfinit(x^16 - 8*x^15 + 50*x^13 - 385*x^12 + 2226*x^11 - 4508*x^10 + 2130*x^9 + 18520*x^8 - 98840*x^7 + 240112*x^6 - 238766*x^5 - 179065*x^4 + 573390*x^3 - 337540*x^2 - 268022*x + 70561, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 50 x^{13} - 385 x^{12} + 2226 x^{11} - 4508 x^{10} + 2130 x^{9} + 18520 x^{8} - 98840 x^{7} + 240112 x^{6} - 238766 x^{5} - 179065 x^{4} + 573390 x^{3} - 337540 x^{2} - 268022 x + 70561 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1867887076585120400000000000000=2^{16}\cdot 5^{14}\cdot 1361^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.98$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{10} a^{8} + \frac{1}{10} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{3} - \frac{3}{10} a^{2} - \frac{1}{10} a - \frac{2}{5}$, $\frac{1}{10} a^{9} + \frac{1}{10} a^{7} + \frac{1}{10} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{2} a^{3} + \frac{1}{5} a^{2} - \frac{3}{10} a - \frac{1}{10}$, $\frac{1}{20} a^{10} - \frac{1}{20} a^{9} - \frac{1}{20} a^{7} - \frac{1}{20} a^{6} + \frac{1}{10} a^{5} - \frac{7}{20} a^{4} + \frac{1}{4} a^{3} - \frac{1}{10} a^{2} + \frac{3}{20} a + \frac{1}{4}$, $\frac{1}{20} a^{11} - \frac{1}{20} a^{9} - \frac{1}{20} a^{8} - \frac{1}{10} a^{7} + \frac{1}{20} a^{6} - \frac{1}{4} a^{5} - \frac{1}{10} a^{4} + \frac{3}{20} a^{3} + \frac{1}{20} a^{2} + \frac{2}{5} a + \frac{1}{4}$, $\frac{1}{200} a^{12} + \frac{1}{50} a^{11} + \frac{1}{50} a^{10} + \frac{17}{100} a^{7} + \frac{23}{100} a^{6} + \frac{9}{50} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{21}{50} a^{2} + \frac{43}{100} a + \frac{61}{200}$, $\frac{1}{400} a^{13} - \frac{1}{400} a^{12} + \frac{1}{100} a^{11} - \frac{1}{40} a^{10} + \frac{1}{40} a^{9} + \frac{7}{200} a^{8} + \frac{33}{200} a^{7} - \frac{11}{100} a^{6} + \frac{1}{4} a^{5} - \frac{19}{40} a^{4} - \frac{3}{200} a^{3} - \frac{27}{200} a^{2} - \frac{99}{400} a + \frac{21}{80}$, $\frac{1}{400} a^{14} - \frac{1}{400} a^{12} - \frac{1}{200} a^{11} + \frac{1}{100} a^{10} - \frac{1}{25} a^{9} - \frac{1}{20} a^{8} - \frac{27}{200} a^{7} - \frac{11}{50} a^{6} - \frac{67}{200} a^{5} + \frac{13}{50} a^{4} + \frac{9}{20} a^{3} + \frac{131}{400} a^{2} + \frac{81}{200} a + \frac{181}{400}$, $\frac{1}{702353714757890209465758557735200} a^{15} - \frac{95077054192736223344448204659}{140470742951578041893151711547040} a^{14} + \frac{486474551915256498041389032621}{702353714757890209465758557735200} a^{13} - \frac{1226273050982555879306218932081}{702353714757890209465758557735200} a^{12} + \frac{8631859379351919200595761619947}{351176857378945104732879278867600} a^{11} - \frac{173505547803392340596471236449}{43897107172368138091609909858450} a^{10} - \frac{1549003905368238930776284544311}{35117685737894510473287927886760} a^{9} - \frac{3885193650613409678901533697453}{351176857378945104732879278867600} a^{8} - \frac{49065515861246145379271471426537}{351176857378945104732879278867600} a^{7} + \frac{44337237788742517545866107940043}{351176857378945104732879278867600} a^{6} - \frac{41717163409129574895971947854369}{351176857378945104732879278867600} a^{5} + \frac{4631399976967098204570746491677}{17558842868947255236643963943380} a^{4} + \frac{2271088373471259512500608469199}{702353714757890209465758557735200} a^{3} + \frac{213063699136776877633698079493061}{702353714757890209465758557735200} a^{2} - \frac{336146899201655294792909533859499}{702353714757890209465758557735200} a + \frac{45624451679946891041357659809183}{702353714757890209465758557735200}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 4037848194.05 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1281:

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 t16n1281
Character table for t16n1281 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.7409284000000.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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