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Properties

Label 16.0.27712902066...5776.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 3^{8}\cdot 7^{4}$
Root discriminant $18.95$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group $(C_2 \times Q_8):C_2$ (as 16T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2401, 0, -2744, 0, 3038, 0, -1484, 0, 667, 0, -4, 0, -22, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 - 22*x^12 - 4*x^10 + 667*x^8 - 1484*x^6 + 3038*x^4 - 2744*x^2 + 2401)
 
gp: K = bnfinit(x^16 - 4*x^14 - 22*x^12 - 4*x^10 + 667*x^8 - 1484*x^6 + 3038*x^4 - 2744*x^2 + 2401, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} - 22 x^{12} - 4 x^{10} + 667 x^{8} - 1484 x^{6} + 3038 x^{4} - 2744 x^{2} + 2401 \)

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:  \(277129020662429515776=2^{44}\cdot 3^{8}\cdot 7^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.95$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6} a^{8} - \frac{1}{3} a^{6} - \frac{1}{6} a^{4} + \frac{1}{3} a^{2} + \frac{1}{6}$, $\frac{1}{6} a^{9} - \frac{1}{3} a^{7} - \frac{1}{6} a^{5} + \frac{1}{3} a^{3} + \frac{1}{6} a$, $\frac{1}{84} a^{10} - \frac{1}{12} a^{9} + \frac{1}{28} a^{8} - \frac{1}{3} a^{7} - \frac{29}{84} a^{6} + \frac{1}{12} a^{5} - \frac{13}{28} a^{4} + \frac{1}{3} a^{3} + \frac{23}{84} a^{2} + \frac{5}{12} a - \frac{1}{12}$, $\frac{1}{84} a^{11} - \frac{1}{21} a^{9} - \frac{1}{12} a^{8} + \frac{9}{28} a^{7} - \frac{1}{3} a^{6} - \frac{8}{21} a^{5} + \frac{1}{12} a^{4} - \frac{11}{28} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{5}{12}$, $\frac{1}{588} a^{12} + \frac{1}{196} a^{10} + \frac{4}{49} a^{8} + \frac{185}{588} a^{6} + \frac{33}{98} a^{4} - \frac{1}{4} a^{2} - \frac{1}{12}$, $\frac{1}{588} a^{13} + \frac{1}{196} a^{11} + \frac{4}{49} a^{9} + \frac{185}{588} a^{7} + \frac{33}{98} a^{5} - \frac{1}{4} a^{3} - \frac{1}{12} a$, $\frac{1}{939983268} a^{14} - \frac{280555}{469991634} a^{12} + \frac{1248081}{313327756} a^{10} - \frac{14472835}{313327756} a^{8} - \frac{26764091}{313327756} a^{6} - \frac{62960125}{134283324} a^{4} - \frac{147671}{4795833} a^{2} + \frac{53031}{913492}$, $\frac{1}{939983268} a^{15} - \frac{280555}{469991634} a^{13} + \frac{1248081}{313327756} a^{11} - \frac{14472835}{313327756} a^{9} - \frac{26764091}{313327756} a^{7} - \frac{62960125}{134283324} a^{5} - \frac{147671}{4795833} a^{3} + \frac{53031}{913492} a$

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:  \( \frac{261199}{234995817} a^{14} - \frac{1485417}{313327756} a^{12} - \frac{8146501}{313327756} a^{10} + \frac{1109229}{156663878} a^{8} + \frac{760223227}{939983268} a^{6} - \frac{8417368}{4795833} a^{4} + \frac{6355081}{2740476} a^{2} - \frac{4983617}{2740476} \) (order $24$)
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:  \( 28493.4786755 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.C_2^3$ (as 16T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 17 conjugacy class representatives for $(C_2 \times Q_8):C_2$
Character table for $(C_2 \times Q_8):C_2$

Intermediate fields

\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\zeta_{12})\), \(\Q(\zeta_{24})\)

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 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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$