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Properties

Label 16.0.85030560000...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{12}\cdot 5^{12}$
Root discriminant $15.24$
Ramified primes $2, 3, 5$
Class number $2$
Class group $[2]$
Galois group $D_4\times C_2$ (as 16T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 18, -36, 37, 0, -18, 42, -60, 42, -18, 0, 37, -36, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 18*x^14 - 36*x^13 + 37*x^12 - 18*x^10 + 42*x^9 - 60*x^8 + 42*x^7 - 18*x^6 + 37*x^4 - 36*x^3 + 18*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^16 - 6*x^15 + 18*x^14 - 36*x^13 + 37*x^12 - 18*x^10 + 42*x^9 - 60*x^8 + 42*x^7 - 18*x^6 + 37*x^4 - 36*x^3 + 18*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 37 x^{12} - 18 x^{10} + 42 x^{9} - 60 x^{8} + 42 x^{7} - 18 x^{6} + 37 x^{4} - 36 x^{3} + 18 x^{2} - 6 x + 1 \)

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:  \(8503056000000000000=2^{16}\cdot 3^{12}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.24$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{6} + \frac{1}{4} a^{3} - \frac{3}{8}$, $\frac{1}{8} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{7} + \frac{1}{4} a^{4} - \frac{3}{8} a$, $\frac{1}{28336} a^{14} - \frac{73}{1288} a^{13} + \frac{1667}{28336} a^{12} + \frac{229}{1288} a^{11} - \frac{1095}{7084} a^{10} - \frac{1557}{14168} a^{9} - \frac{17}{1288} a^{8} - \frac{73}{3542} a^{7} + \frac{305}{1288} a^{6} - \frac{5099}{14168} a^{5} + \frac{2447}{7084} a^{4} - \frac{93}{1288} a^{3} + \frac{5209}{28336} a^{2} + \frac{249}{1288} a + \frac{10627}{28336}$, $\frac{1}{28336} a^{15} + \frac{1007}{28336} a^{13} + \frac{43}{1288} a^{12} - \frac{205}{1771} a^{11} + \frac{293}{2024} a^{10} + \frac{45}{184} a^{9} - \frac{1543}{7084} a^{8} + \frac{177}{1288} a^{7} + \frac{2733}{14168} a^{6} - \frac{260}{1771} a^{5} + \frac{235}{1288} a^{4} + \frac{13393}{28336} a^{3} + \frac{39}{92} a^{2} - \frac{4201}{28336} a + \frac{117}{644}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{1259}{28336} a^{15} - \frac{225}{7084} a^{14} - \frac{10585}{28336} a^{13} + \frac{18915}{14168} a^{12} - \frac{5755}{1771} a^{11} + \frac{30075}{14168} a^{10} + \frac{61115}{14168} a^{9} + \frac{4605}{7084} a^{8} + \frac{61115}{14168} a^{7} - \frac{60365}{14168} a^{6} - \frac{757}{1771} a^{5} + \frac{1065}{14168} a^{4} + \frac{15595}{28336} a^{3} + \frac{405}{77} a^{2} + \frac{6495}{28336} a - \frac{51}{1012} \) (order $12$)
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:  \( 4348.50756134 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4$ (as 16T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $D_4\times C_2$
Character table for $D_4\times C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{15})\), \(\Q(i, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{-5})\), 4.2.54000.1 x2, 4.2.54000.2 x2, 4.0.13500.1 x2, 4.0.13500.2 x2, 8.0.12960000.1, 8.0.2916000000.1, 8.0.2916000000.2, 8.0.2916000000.4 x2, 8.0.182250000.1 x2, 8.4.2916000000.1 x2, 8.0.2916000000.3 x2

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

Sibling fields

Degree 8 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 R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$