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Properties

Label 16.4.17415183620...4081.1
Degree $16$
Signature $[4, 6]$
Discriminant $3^{8}\cdot 61^{12}$
Root discriminant $37.81$
Ramified primes $3, 61$
Class number $3$ (GRH)
Class group $[3]$ (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![-675, 8505, -15471, 27603, -26109, 20160, -13789, 6869, -3132, 1080, -244, 0, 57, -35, 15, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 15*x^14 - 35*x^13 + 57*x^12 - 244*x^10 + 1080*x^9 - 3132*x^8 + 6869*x^7 - 13789*x^6 + 20160*x^5 - 26109*x^4 + 27603*x^3 - 15471*x^2 + 8505*x - 675)
 
gp: K = bnfinit(x^16 - 4*x^15 + 15*x^14 - 35*x^13 + 57*x^12 - 244*x^10 + 1080*x^9 - 3132*x^8 + 6869*x^7 - 13789*x^6 + 20160*x^5 - 26109*x^4 + 27603*x^3 - 15471*x^2 + 8505*x - 675, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 15 x^{14} - 35 x^{13} + 57 x^{12} - 244 x^{10} + 1080 x^{9} - 3132 x^{8} + 6869 x^{7} - 13789 x^{6} + 20160 x^{5} - 26109 x^{4} + 27603 x^{3} - 15471 x^{2} + 8505 x - 675 \)

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:  \(17415183620366462784744081=3^{8}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.81$
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:  $3, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{15} a^{11} - \frac{1}{15} a^{10} - \frac{7}{15} a^{9} - \frac{2}{15} a^{8} + \frac{7}{15} a^{7} + \frac{2}{5} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{4}{15} a^{3} + \frac{4}{15} a^{2} - \frac{1}{5} a$, $\frac{1}{15} a^{12} + \frac{2}{15} a^{10} + \frac{2}{5} a^{9} - \frac{1}{3} a^{8} - \frac{2}{15} a^{7} - \frac{4}{15} a^{6} - \frac{1}{3} a^{5} - \frac{1}{15} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{45} a^{13} + \frac{1}{45} a^{12} - \frac{1}{45} a^{11} - \frac{4}{45} a^{10} + \frac{7}{45} a^{9} - \frac{1}{45} a^{8} - \frac{4}{15} a^{7} + \frac{2}{5} a^{6} - \frac{7}{15} a^{5} - \frac{13}{45} a^{4} - \frac{1}{15} a^{3} + \frac{2}{15} a^{2} - \frac{1}{5} a$, $\frac{1}{45} a^{14} + \frac{1}{45} a^{12} - \frac{1}{45} a^{10} - \frac{11}{45} a^{9} - \frac{17}{45} a^{8} - \frac{1}{15} a^{6} + \frac{8}{45} a^{5} - \frac{8}{45} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{5} a$, $\frac{1}{1166351478580629507887775} a^{15} - \frac{11487103270529227236022}{1166351478580629507887775} a^{14} + \frac{708572154644064435644}{129594608731181056431975} a^{13} + \frac{20985150661091455122397}{1166351478580629507887775} a^{12} - \frac{5698901977276599671308}{388783826193543169295925} a^{11} + \frac{16113342183447029314343}{129594608731181056431975} a^{10} + \frac{61496191255535287829893}{233270295716125901577555} a^{9} - \frac{31709388596869466472121}{77756765238708633859185} a^{8} + \frac{124027626051285508225576}{388783826193543169295925} a^{7} - \frac{7485585924160884216533}{233270295716125901577555} a^{6} - \frac{440297028880214166804934}{1166351478580629507887775} a^{5} + \frac{24202989305738739067664}{388783826193543169295925} a^{4} - \frac{32010566963405366833603}{77756765238708633859185} a^{3} - \frac{1569066955075208808811}{43198202910393685477325} a^{2} - \frac{6911176507454271836978}{25918921746236211286395} a + \frac{383594983016797775856}{1727928116415747419093}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 3657121.45053 \) (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

\(\Q(\sqrt{61}) \), 4.2.11163.1, 8.2.1391050107747.2, 8.4.68412300381.1, 8.2.22804100127.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
61Data not computed