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Properties

Label 16.4.51399544780...7056.3
Degree $16$
Signature $[4, 6]$
Discriminant $2^{24}\cdot 3^{12}\cdot 7^{8}$
Root discriminant $17.06$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2\wr C_2$ (as 16T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 30, -66, -42, 154, -16, -186, 10, 112, -24, -62, 18, 27, -4, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - 4*x^13 + 27*x^12 + 18*x^11 - 62*x^10 - 24*x^9 + 112*x^8 + 10*x^7 - 186*x^6 - 16*x^5 + 154*x^4 - 42*x^3 - 66*x^2 + 30*x - 3)
 
gp: K = bnfinit(x^16 - 8*x^14 - 4*x^13 + 27*x^12 + 18*x^11 - 62*x^10 - 24*x^9 + 112*x^8 + 10*x^7 - 186*x^6 - 16*x^5 + 154*x^4 - 42*x^3 - 66*x^2 + 30*x - 3, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{14} - 4 x^{13} + 27 x^{12} + 18 x^{11} - 62 x^{10} - 24 x^{9} + 112 x^{8} + 10 x^{7} - 186 x^{6} - 16 x^{5} + 154 x^{4} - 42 x^{3} - 66 x^{2} + 30 x - 3 \)

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:  \(51399544780206637056=2^{24}\cdot 3^{12}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.06$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{2}{7} a^{11} - \frac{3}{7} a^{10} - \frac{3}{7} a^{9} - \frac{1}{7} a^{8} + \frac{3}{7} a^{7} + \frac{1}{7} a^{6} + \frac{3}{7} a^{4} - \frac{3}{7} a^{3} - \frac{1}{7} a^{2} + \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{13} + \frac{3}{7} a^{10} - \frac{2}{7} a^{9} - \frac{2}{7} a^{8} + \frac{2}{7} a^{7} - \frac{2}{7} a^{6} + \frac{3}{7} a^{5} - \frac{2}{7} a^{4} - \frac{2}{7} a^{3} + \frac{3}{7} a^{2} + \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{14} + \frac{3}{7} a^{11} - \frac{2}{7} a^{10} - \frac{2}{7} a^{9} + \frac{2}{7} a^{8} - \frac{2}{7} a^{7} + \frac{3}{7} a^{6} - \frac{2}{7} a^{5} - \frac{2}{7} a^{4} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} - \frac{1}{7} a$, $\frac{1}{7913077627} a^{15} + \frac{159674771}{7913077627} a^{14} + \frac{41762464}{1130439661} a^{13} - \frac{51811276}{7913077627} a^{12} - \frac{1667878080}{7913077627} a^{11} - \frac{1102380815}{7913077627} a^{10} - \frac{453888665}{1130439661} a^{9} - \frac{1338955078}{7913077627} a^{8} - \frac{3168314104}{7913077627} a^{7} - \frac{593590934}{7913077627} a^{6} - \frac{96341221}{608698279} a^{5} - \frac{193089705}{7913077627} a^{4} - \frac{3027520434}{7913077627} a^{3} - \frac{682235363}{7913077627} a^{2} - \frac{1070908733}{7913077627} a - \frac{1725374876}{7913077627}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 5981.48582831 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\wr C_2$ (as 16T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2\wr C_2$
Character table for $C_2^2\wr C_2$

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{3}) \), 4.2.21168.1, 4.2.1323.1, 4.2.9408.2, 4.2.9408.1, 4.2.84672.5, \(\Q(\sqrt{3}, \sqrt{7})\), 4.2.1728.1, 8.4.7169347584.1, 8.4.448084224.2, 8.4.796594176.1

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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{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$$[\ ]_{4}^{2}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$