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Properties

Label 16.0.54168925842...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 31^{6}$
Root discriminant $14.82$
Ramified primes $5, 31$
Class number $1$
Class group Trivial
Galois group $C_2^3.C_4$ (as 16T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31, -124, 414, -929, 1526, -1932, 1901, -1522, 1012, -603, 356, -213, 126, -61, 24, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 24*x^14 - 61*x^13 + 126*x^12 - 213*x^11 + 356*x^10 - 603*x^9 + 1012*x^8 - 1522*x^7 + 1901*x^6 - 1932*x^5 + 1526*x^4 - 929*x^3 + 414*x^2 - 124*x + 31)
 
gp: K = bnfinit(x^16 - 6*x^15 + 24*x^14 - 61*x^13 + 126*x^12 - 213*x^11 + 356*x^10 - 603*x^9 + 1012*x^8 - 1522*x^7 + 1901*x^6 - 1932*x^5 + 1526*x^4 - 929*x^3 + 414*x^2 - 124*x + 31, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 24 x^{14} - 61 x^{13} + 126 x^{12} - 213 x^{11} + 356 x^{10} - 603 x^{9} + 1012 x^{8} - 1522 x^{7} + 1901 x^{6} - 1932 x^{5} + 1526 x^{4} - 929 x^{3} + 414 x^{2} - 124 x + 31 \)

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:  \(5416892584228515625=5^{14}\cdot 31^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.82$
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:  $5, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{41} a^{14} - \frac{11}{41} a^{13} + \frac{5}{41} a^{12} - \frac{10}{41} a^{11} + \frac{11}{41} a^{10} - \frac{20}{41} a^{9} + \frac{11}{41} a^{8} + \frac{2}{41} a^{7} - \frac{17}{41} a^{6} + \frac{14}{41} a^{5} + \frac{14}{41} a^{4} - \frac{4}{41} a^{3} + \frac{18}{41} a^{2} + \frac{15}{41} a - \frac{9}{41}$, $\frac{1}{1910291639} a^{15} + \frac{3381440}{1910291639} a^{14} + \frac{921525412}{1910291639} a^{13} - \frac{689178970}{1910291639} a^{12} - \frac{91112409}{1910291639} a^{11} - \frac{367942976}{1910291639} a^{10} - \frac{256327491}{1910291639} a^{9} - \frac{58512018}{1910291639} a^{8} - \frac{220551037}{1910291639} a^{7} - \frac{24967020}{1910291639} a^{6} - \frac{735290263}{1910291639} a^{5} - \frac{461930981}{1910291639} a^{4} - \frac{819233432}{1910291639} a^{3} - \frac{824922754}{1910291639} a^{2} + \frac{8110610}{46592479} a - \frac{243535233}{1910291639}$

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:  \( -\frac{519552898}{1910291639} a^{15} + \frac{2637555254}{1910291639} a^{14} - \frac{9418365380}{1910291639} a^{13} + \frac{19858805844}{1910291639} a^{12} - \frac{36166199227}{1910291639} a^{11} + \frac{54689811740}{1910291639} a^{10} - \frac{95138354282}{1910291639} a^{9} + \frac{167104672457}{1910291639} a^{8} - \frac{270232802221}{1910291639} a^{7} + \frac{358587856232}{1910291639} a^{6} - \frac{366196534525}{1910291639} a^{5} + \frac{292914137453}{1910291639} a^{4} - \frac{171337566531}{1910291639} a^{3} + \frac{76288864160}{1910291639} a^{2} - \frac{25988482774}{1910291639} a + \frac{4934395132}{1910291639} \) (order $10$)
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:  \( 2713.04336648 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.3875.1, \(\Q(\zeta_{5})\), 4.2.775.1, 8.0.75078125.1 x2, 8.0.15015625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
31Data not computed