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Properties

Label 16.0.51399544780...7056.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{12}\cdot 7^{8}$
Root discriminant $17.06$
Ramified primes $2, 3, 7$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2:D_4$ (as 16T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 7, -20, 115, 0, 235, -132, 196, -148, 67, -56, 59, -50, 28, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 28*x^14 - 50*x^13 + 59*x^12 - 56*x^11 + 67*x^10 - 148*x^9 + 196*x^8 - 132*x^7 + 235*x^6 + 115*x^4 - 20*x^3 + 7*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^16 - 8*x^15 + 28*x^14 - 50*x^13 + 59*x^12 - 56*x^11 + 67*x^10 - 148*x^9 + 196*x^8 - 132*x^7 + 235*x^6 + 115*x^4 - 20*x^3 + 7*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 28 x^{14} - 50 x^{13} + 59 x^{12} - 56 x^{11} + 67 x^{10} - 148 x^{9} + 196 x^{8} - 132 x^{7} + 235 x^{6} + 115 x^{4} - 20 x^{3} + 7 x^{2} - 6 x + 1 \)

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:  \(51399544780206637056=2^{24}\cdot 3^{12}\cdot 7^{8}\)
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{34} a^{12} - \frac{3}{17} a^{11} - \frac{7}{34} a^{10} + \frac{5}{17} a^{9} + \frac{13}{34} a^{8} + \frac{6}{17} a^{7} - \frac{5}{34} a^{6} + \frac{1}{17} a^{5} + \frac{1}{17} a^{4} - \frac{6}{17} a^{3} - \frac{1}{17} a^{2} + \frac{5}{17} a - \frac{1}{34}$, $\frac{1}{34} a^{13} - \frac{9}{34} a^{11} + \frac{1}{17} a^{10} + \frac{5}{34} a^{9} - \frac{6}{17} a^{8} - \frac{1}{34} a^{7} + \frac{3}{17} a^{6} + \frac{7}{17} a^{5} - \frac{3}{17} a^{3} - \frac{1}{17} a^{2} - \frac{9}{34} a - \frac{3}{17}$, $\frac{1}{34} a^{14} + \frac{8}{17} a^{11} + \frac{5}{17} a^{10} + \frac{5}{17} a^{9} + \frac{7}{17} a^{8} + \frac{6}{17} a^{7} + \frac{3}{34} a^{6} - \frac{8}{17} a^{5} + \frac{6}{17} a^{4} - \frac{4}{17} a^{3} + \frac{7}{34} a^{2} + \frac{8}{17} a - \frac{9}{34}$, $\frac{1}{1470780931562} a^{15} - \frac{7206039503}{1470780931562} a^{14} - \frac{12293952497}{1470780931562} a^{13} - \frac{15694774821}{1470780931562} a^{12} + \frac{648367908517}{1470780931562} a^{11} + \frac{34331913375}{1470780931562} a^{10} - \frac{154353741407}{1470780931562} a^{9} + \frac{328388364389}{1470780931562} a^{8} - \frac{205589700363}{735390465781} a^{7} + \frac{237310935392}{735390465781} a^{6} - \frac{152778918549}{735390465781} a^{5} + \frac{208716996403}{735390465781} a^{4} - \frac{228584054923}{1470780931562} a^{3} - \frac{364483963899}{1470780931562} a^{2} - \frac{54424213360}{735390465781} a - \frac{113507479377}{735390465781}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{20195477}{47743327} a^{15} + \frac{319653395}{95486654} a^{14} - \frac{554497825}{47743327} a^{13} + \frac{984056721}{47743327} a^{12} - \frac{1182305811}{47743327} a^{11} + \frac{1157190632}{47743327} a^{10} - \frac{1388586573}{47743327} a^{9} + \frac{2977715798}{47743327} a^{8} - \frac{3832646183}{47743327} a^{7} + \frac{5376805601}{95486654} a^{6} - \frac{4980622584}{47743327} a^{5} - \frac{192504334}{47743327} a^{4} - \frac{2837390938}{47743327} a^{3} + \frac{29359445}{95486654} a^{2} - \frac{303094367}{47743327} a + \frac{14470383}{5616862} \) (order $6$)
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:  \( 3374.2130113 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:D_4$ (as 16T43):

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 $C_2^2:D_4$
Character table for $C_2^2:D_4$

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), 4.2.84672.1 x2, 4.0.12096.2 x2, 4.0.3528.1, \(\Q(\sqrt{-3}, \sqrt{-7})\), 4.0.392.1, 8.0.12446784.1, 8.0.7169347584.6, 8.0.112021056.1

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 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 ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
3Data not computed
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$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}$