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Properties

Label 16.16.1837669247...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{34}\cdot 5^{8}\cdot 7^{8}\cdot 41^{6}$
Root discriminant $103.88$
Ramified primes $2, 5, 7, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1228

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6461764, 0, -10549956, 0, 6764918, 0, -2204898, 0, 400103, 0, -41778, 0, 2485, 0, -78, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 78*x^14 + 2485*x^12 - 41778*x^10 + 400103*x^8 - 2204898*x^6 + 6764918*x^4 - 10549956*x^2 + 6461764)
 
gp: K = bnfinit(x^16 - 78*x^14 + 2485*x^12 - 41778*x^10 + 400103*x^8 - 2204898*x^6 + 6764918*x^4 - 10549956*x^2 + 6461764, 1)
 

Normalized defining polynomial

\( x^{16} - 78 x^{14} + 2485 x^{12} - 41778 x^{10} + 400103 x^{8} - 2204898 x^{6} + 6764918 x^{4} - 10549956 x^{2} + 6461764 \)

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:  \(183766924754340714800742400000000=2^{34}\cdot 5^{8}\cdot 7^{8}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $103.88$
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:  $2, 5, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{4} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{6} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{7} - \frac{1}{3} a^{3}$, $\frac{1}{246} a^{12} + \frac{2}{123} a^{10} + \frac{25}{246} a^{8} - \frac{20}{123} a^{6} - \frac{19}{82} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{246} a^{13} + \frac{2}{123} a^{11} + \frac{25}{246} a^{9} - \frac{20}{123} a^{7} - \frac{19}{82} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{14634435302646} a^{14} - \frac{2148409207}{4878145100882} a^{12} + \frac{249602376925}{4878145100882} a^{10} + \frac{1309219403801}{14634435302646} a^{8} + \frac{4167818777149}{14634435302646} a^{6} - \frac{5635352013137}{14634435302646} a^{4} - \frac{18847084645}{178468723203} a^{2} + \frac{19040206375}{178468723203}$, $\frac{1}{453667494382026} a^{15} + \frac{826408813993}{453667494382026} a^{13} + \frac{18714658599877}{453667494382026} a^{11} + \frac{41643150847679}{453667494382026} a^{9} - \frac{4837302528255}{151222498127342} a^{7} - \frac{160427224604539}{453667494382026} a^{5} - \frac{316294956650}{5532530419293} a^{3} - \frac{271271278413}{1844176806431} a$

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:  \( 212590057627 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1228:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 61 conjugacy class representatives for t16n1228 are not computed
Character table for t16n1228 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{35}) \), 4.4.16400.1, 4.4.50225.1, \(\Q(\sqrt{5}, \sqrt{7})\), 8.8.645772960000.1

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.16.30$x^{8} + 8 x^{7} + 20$$4$$2$$16$$C_2^3: C_4$$[2, 2, 3]^{4}$
2.8.18.20$x^{8} + 4 x^{7} + 2 x^{6} + 10 x^{4} + 20 x^{2} + 4$$4$$2$$18$$C_2^3: C_4$$[2, 2, 3, 7/2]^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7Data not computed
$41$41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.8.6.1$x^{8} - 9881 x^{4} + 34857216$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$