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Properties

Label 16.0.44012651103...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 11^{8}\cdot 29^{2}$
Root discriminant $16.89$
Ramified primes $5, 11, 29$
Class number $2$
Class group $[2]$
Galois group $C_2^4.C_2^3.C_2$ (as 16T547)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, 128, 256, 0, 16, -56, 100, -6, 107, 3, 25, 7, 1, 0, 4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 4*x^14 + x^12 + 7*x^11 + 25*x^10 + 3*x^9 + 107*x^8 - 6*x^7 + 100*x^6 - 56*x^5 + 16*x^4 + 256*x^2 + 128*x + 256)
 
gp: K = bnfinit(x^16 - x^15 + 4*x^14 + x^12 + 7*x^11 + 25*x^10 + 3*x^9 + 107*x^8 - 6*x^7 + 100*x^6 - 56*x^5 + 16*x^4 + 256*x^2 + 128*x + 256, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 4 x^{14} + x^{12} + 7 x^{11} + 25 x^{10} + 3 x^{9} + 107 x^{8} - 6 x^{7} + 100 x^{6} - 56 x^{5} + 16 x^{4} + 256 x^{2} + 128 x + 256 \)

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:  \(44012651103759765625=5^{12}\cdot 11^{8}\cdot 29^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.89$
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:  $5, 11, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{2} a^{8} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{3}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{1}{4} a^{9} - \frac{7}{16} a^{8} + \frac{7}{16} a^{7} + \frac{5}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{12} - \frac{1}{8} a^{10} - \frac{7}{32} a^{9} + \frac{7}{32} a^{8} - \frac{11}{32} a^{7} - \frac{1}{32} a^{6} + \frac{15}{32} a^{5} + \frac{3}{16} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{2393024} a^{14} - \frac{30433}{2393024} a^{13} - \frac{4751}{299128} a^{12} - \frac{13651}{598256} a^{11} - \frac{23127}{2393024} a^{10} - \frac{156521}{2393024} a^{9} - \frac{786467}{2393024} a^{8} + \frac{593335}{2393024} a^{7} - \frac{259225}{2393024} a^{6} - \frac{51305}{1196512} a^{5} + \frac{97909}{299128} a^{4} - \frac{146599}{299128} a^{3} + \frac{617}{74782} a^{2} - \frac{23475}{74782} a - \frac{2}{37391}$, $\frac{1}{90934912} a^{15} - \frac{7}{90934912} a^{14} + \frac{203475}{45467456} a^{13} - \frac{388201}{22733728} a^{12} - \frac{3038799}{90934912} a^{11} - \frac{867823}{90934912} a^{10} + \frac{6501547}{90934912} a^{9} - \frac{11074111}{90934912} a^{8} - \frac{1318955}{90934912} a^{7} + \frac{2947659}{22733728} a^{6} - \frac{854361}{22733728} a^{5} - \frac{5492639}{11366864} a^{4} + \frac{50297}{1420858} a^{3} - \frac{103935}{1420858} a^{2} + \frac{407565}{1420858} a - \frac{67817}{710429}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 628.035700482 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T547):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.275.1, 4.4.15125.1, 4.2.1375.1, 8.0.6634203125.1, 8.0.2193125.1, 8.4.228765625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
5Data not computed
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$