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Properties

Label 16.4.12478436359...5625.4
Degree $16$
Signature $[4, 6]$
Discriminant $5^{12}\cdot 59^{10}$
Root discriminant $42.76$
Ramified primes $5, 59$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.D_4$ (as 16T330)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, 108, 360, 609, 206, -113, -134, 220, -636, -15, 94, 26, -174, 78, -15, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 15*x^14 + 78*x^13 - 174*x^12 + 26*x^11 + 94*x^10 - 15*x^9 - 636*x^8 + 220*x^7 - 134*x^6 - 113*x^5 + 206*x^4 + 609*x^3 + 360*x^2 + 108*x + 81)
 
gp: K = bnfinit(x^16 - x^15 - 15*x^14 + 78*x^13 - 174*x^12 + 26*x^11 + 94*x^10 - 15*x^9 - 636*x^8 + 220*x^7 - 134*x^6 - 113*x^5 + 206*x^4 + 609*x^3 + 360*x^2 + 108*x + 81, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 15 x^{14} + 78 x^{13} - 174 x^{12} + 26 x^{11} + 94 x^{10} - 15 x^{9} - 636 x^{8} + 220 x^{7} - 134 x^{6} - 113 x^{5} + 206 x^{4} + 609 x^{3} + 360 x^{2} + 108 x + 81 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(124784363598789404541015625=5^{12}\cdot 59^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.76$
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:  $5, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{2}{9} a^{9} - \frac{1}{9} a^{8} - \frac{2}{9} a^{7} + \frac{2}{9} a^{6} + \frac{2}{9} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{4}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{11} - \frac{2}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{4}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{2}{9} a^{7} + \frac{1}{3} a^{6} + \frac{2}{9} a^{5} + \frac{4}{9} a^{4} - \frac{4}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{64961333529125245916337} a^{15} - \frac{1757440933045249155949}{64961333529125245916337} a^{14} + \frac{195154858794280735196}{7217925947680582879593} a^{13} - \frac{828245363174166651442}{21653777843041748638779} a^{12} + \frac{154508028520943195155}{2405975315893527626531} a^{11} + \frac{1850507592543263783795}{64961333529125245916337} a^{10} + \frac{6614091898973809102612}{64961333529125245916337} a^{9} + \frac{2131457975907048401531}{21653777843041748638779} a^{8} - \frac{2092575580041123085196}{7217925947680582879593} a^{7} + \frac{27391851225495242779243}{64961333529125245916337} a^{6} - \frac{1833066595839376864046}{64961333529125245916337} a^{5} + \frac{13095918151050960616234}{64961333529125245916337} a^{4} - \frac{4642583975449376483281}{64961333529125245916337} a^{3} - \frac{3193597001828325116353}{7217925947680582879593} a^{2} - \frac{2671767490490991775078}{7217925947680582879593} a - \frac{997092451809658767467}{2405975315893527626531}$

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:  $9$
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:  \( 3723658.84124 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.D_4$ (as 16T330):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4.D_4$
Character table for $C_2^4.D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.1475.1, 8.4.37866753125.1, 8.2.3209046875.2, 8.2.2234138434375.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ R

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$59$59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.4.3.1$x^{4} + 177$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.3.1$x^{4} + 177$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$