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Properties

Label 16.0.35809920319...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 19^{4}\cdot 103^{4}$
Root discriminant $22.24$
Ramified primes $5, 19, 103$
Class number $5$
Class group $[5]$
Galois group $C_4\times S_4$ (as 16T181)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 5, -9, 28, 33, 74, 67, 201, 34, 57, 12, 15, -1, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 4*x^14 - x^13 + 15*x^12 + 12*x^11 + 57*x^10 + 34*x^9 + 201*x^8 + 67*x^7 + 74*x^6 + 33*x^5 + 28*x^4 - 9*x^3 + 5*x^2 - x + 1)
 
gp: K = bnfinit(x^16 + 4*x^14 - x^13 + 15*x^12 + 12*x^11 + 57*x^10 + 34*x^9 + 201*x^8 + 67*x^7 + 74*x^6 + 33*x^5 + 28*x^4 - 9*x^3 + 5*x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{16} + 4 x^{14} - x^{13} + 15 x^{12} + 12 x^{11} + 57 x^{10} + 34 x^{9} + 201 x^{8} + 67 x^{7} + 74 x^{6} + 33 x^{5} + 28 x^{4} - 9 x^{3} + 5 x^{2} - x + 1 \)

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:  \(3580992031933837890625=5^{12}\cdot 19^{4}\cdot 103^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.24$
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:  $5, 19, 103$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{42142} a^{13} + \frac{25}{21071} a^{12} + \frac{3892}{21071} a^{11} + \frac{199}{42142} a^{10} + \frac{5007}{21071} a^{9} + \frac{18579}{42142} a^{8} - \frac{7275}{21071} a^{7} + \frac{5816}{21071} a^{6} - \frac{2509}{42142} a^{5} - \frac{10357}{21071} a^{4} + \frac{2143}{42142} a^{3} + \frac{6432}{21071} a^{2} - \frac{758}{21071} a - \frac{14257}{42142}$, $\frac{1}{42142} a^{14} + \frac{2642}{21071} a^{12} - \frac{9723}{42142} a^{11} + \frac{32}{21071} a^{10} - \frac{18559}{42142} a^{9} - \frac{8188}{21071} a^{8} - \frac{9712}{21071} a^{7} + \frac{5879}{42142} a^{6} + \frac{10226}{21071} a^{5} - \frac{15707}{42142} a^{4} - \frac{5001}{21071} a^{3} - \frac{6293}{21071} a^{2} + \frac{19401}{42142} a - \frac{1782}{21071}$, $\frac{1}{42142} a^{15} + \frac{2273}{21071} a^{10} - \frac{5905}{21071} a^{5} + \frac{5163}{42142}$

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

Class group and class number

$C_{5}$, which has order $5$

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:  \( \frac{3324}{21071} a^{15} - \frac{831}{21071} a^{14} + \frac{12465}{21071} a^{13} - \frac{13161}{42142} a^{12} + \frac{2493}{1109} a^{11} + \frac{28254}{21071} a^{10} + \frac{167031}{21071} a^{9} + \frac{55677}{21071} a^{8} + \frac{1187657}{42142} a^{7} + \frac{27423}{21071} a^{6} + \frac{23268}{21071} a^{5} - \frac{7479}{21071} a^{4} + \frac{4155}{21071} a^{3} - \frac{259523}{42142} a^{2} + \frac{831}{21071} a \) (order $10$)
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:  \( 10491.0986114 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times S_4$ (as 16T181):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 20 conjugacy class representatives for $C_4\times S_4$
Character table for $C_4\times S_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.4.1957.1, 8.8.2393655625.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 sibling: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$103$103.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
103.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
103.8.4.1$x^{8} + 106090 x^{4} - 1092727 x^{2} + 2813772025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$