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Properties

Label 16.0.48557483622...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{14}\cdot 1361^{2}$
Root discriminant $40.31$
Ramified primes $2, 5, 1361$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 16T1162

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![61, 56, 390, 180, 1110, -852, 2938, -2200, 2270, -1520, 858, -392, 185, -60, 20, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 20*x^14 - 60*x^13 + 185*x^12 - 392*x^11 + 858*x^10 - 1520*x^9 + 2270*x^8 - 2200*x^7 + 2938*x^6 - 852*x^5 + 1110*x^4 + 180*x^3 + 390*x^2 + 56*x + 61)
 
gp: K = bnfinit(x^16 - 4*x^15 + 20*x^14 - 60*x^13 + 185*x^12 - 392*x^11 + 858*x^10 - 1520*x^9 + 2270*x^8 - 2200*x^7 + 2938*x^6 - 852*x^5 + 1110*x^4 + 180*x^3 + 390*x^2 + 56*x + 61, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 20 x^{14} - 60 x^{13} + 185 x^{12} - 392 x^{11} + 858 x^{10} - 1520 x^{9} + 2270 x^{8} - 2200 x^{7} + 2938 x^{6} - 852 x^{5} + 1110 x^{4} + 180 x^{3} + 390 x^{2} + 56 x + 61 \)

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:  \(48557483622400000000000000=2^{32}\cdot 5^{14}\cdot 1361^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{5} + \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{6} + \frac{1}{5} a$, $\frac{1}{25} a^{12} + \frac{2}{25} a^{11} + \frac{1}{25} a^{10} + \frac{2}{25} a^{7} + \frac{9}{25} a^{6} - \frac{3}{25} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{6}{25} a^{2} + \frac{2}{25} a + \frac{1}{25}$, $\frac{1}{25} a^{13} + \frac{2}{25} a^{11} - \frac{2}{25} a^{10} + \frac{2}{25} a^{8} + \frac{1}{5} a^{7} - \frac{11}{25} a^{6} + \frac{11}{25} a^{5} + \frac{2}{5} a^{4} - \frac{9}{25} a^{3} - \frac{2}{5} a^{2} + \frac{2}{25} a - \frac{2}{25}$, $\frac{1}{25} a^{14} - \frac{1}{25} a^{11} - \frac{2}{25} a^{10} + \frac{2}{25} a^{9} - \frac{1}{5} a^{7} - \frac{12}{25} a^{6} + \frac{11}{25} a^{5} + \frac{6}{25} a^{4} - \frac{1}{5} a^{3} + \frac{9}{25} a - \frac{7}{25}$, $\frac{1}{281181015046065125} a^{15} + \frac{908820472349649}{56236203009213025} a^{14} + \frac{192010106186446}{56236203009213025} a^{13} - \frac{396206486482587}{56236203009213025} a^{12} + \frac{858693675150738}{11247240601842605} a^{11} - \frac{4163351003461352}{281181015046065125} a^{10} + \frac{614042923502974}{56236203009213025} a^{9} - \frac{1105259598490326}{56236203009213025} a^{8} + \frac{10458468552495547}{56236203009213025} a^{7} - \frac{633373079389663}{2249448120368521} a^{6} - \frac{118799325475128732}{281181015046065125} a^{5} + \frac{4464717826836941}{11247240601842605} a^{4} - \frac{10455751356756452}{56236203009213025} a^{3} - \frac{16655030868961911}{56236203009213025} a^{2} + \frac{5511532525686936}{11247240601842605} a + \frac{41681926742807146}{281181015046065125}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 335112.050124 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1162:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 49 conjugacy class representatives for t16n1162
Character table for t16n1162 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.0.5120000000.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$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