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Properties

Label 16.16.7808552801...8576.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{44}\cdot 3^{12}\cdot 17^{4}$
Root discriminant $31.14$
Ramified primes $2, 3, 17$
Class number $1$ (GRH)
Class group Trivial (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, 4, -56, 60, 468, -1280, 492, 1668, -1691, -368, 1024, -196, -210, 84, 8, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 8*x^14 + 84*x^13 - 210*x^12 - 196*x^11 + 1024*x^10 - 368*x^9 - 1691*x^8 + 1668*x^7 + 492*x^6 - 1280*x^5 + 468*x^4 + 60*x^3 - 56*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^16 - 8*x^15 + 8*x^14 + 84*x^13 - 210*x^12 - 196*x^11 + 1024*x^10 - 368*x^9 - 1691*x^8 + 1668*x^7 + 492*x^6 - 1280*x^5 + 468*x^4 + 60*x^3 - 56*x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 8 x^{14} + 84 x^{13} - 210 x^{12} - 196 x^{11} + 1024 x^{10} - 368 x^{9} - 1691 x^{8} + 1668 x^{7} + 492 x^{6} - 1280 x^{5} + 468 x^{4} + 60 x^{3} - 56 x^{2} + 4 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(780855280180961608728576=2^{44}\cdot 3^{12}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.14$
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:  $2, 3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{23} a^{13} + \frac{5}{23} a^{12} + \frac{1}{23} a^{11} - \frac{3}{23} a^{10} - \frac{10}{23} a^{9} - \frac{11}{23} a^{8} - \frac{7}{23} a^{7} + \frac{10}{23} a^{6} - \frac{7}{23} a^{5} + \frac{3}{23} a^{4} + \frac{1}{23} a^{3} + \frac{9}{23} a^{2} + \frac{5}{23} a - \frac{10}{23}$, $\frac{1}{1081} a^{14} - \frac{7}{1081} a^{13} - \frac{358}{1081} a^{12} + \frac{77}{1081} a^{11} - \frac{250}{1081} a^{10} + \frac{17}{1081} a^{9} - \frac{13}{1081} a^{8} + \frac{2}{1081} a^{7} - \frac{265}{1081} a^{6} - \frac{396}{1081} a^{5} + \frac{103}{1081} a^{4} + \frac{457}{1081} a^{3} - \frac{379}{1081} a^{2} - \frac{70}{1081} a - \frac{271}{1081}$, $\frac{1}{1081} a^{15} + \frac{16}{1081} a^{13} - \frac{314}{1081} a^{12} - \frac{369}{1081} a^{11} + \frac{241}{1081} a^{10} + \frac{200}{1081} a^{9} - \frac{418}{1081} a^{8} + \frac{31}{1081} a^{7} - \frac{183}{1081} a^{6} - \frac{225}{1081} a^{5} + \frac{285}{1081} a^{4} + \frac{3}{1081} a^{2} + \frac{273}{1081} a + \frac{359}{1081}$

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:  $15$
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:  \( 4986811.01116 \) (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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), 4.4.13824.1 x2, 4.4.27648.1 x2, 4.4.4352.1, \(\Q(\sqrt{2}, \sqrt{3})\), 4.4.9792.1, 8.8.1534132224.1, 8.8.3057647616.1, 8.8.883660161024.1

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
3Data not computed
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$