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Properties

Label 16.0.38416000000000000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 7^{4}$
Root discriminant $10.88$
Ramified primes $2, 5, 7$
Class number $1$
Class group Trivial
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, -4, 14, -46, 127, -260, 400, -480, 478, -422, 344, -256, 164, -84, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 164*x^12 - 256*x^11 + 344*x^10 - 422*x^9 + 478*x^8 - 480*x^7 + 400*x^6 - 260*x^5 + 127*x^4 - 46*x^3 + 14*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 164*x^12 - 256*x^11 + 344*x^10 - 422*x^9 + 478*x^8 - 480*x^7 + 400*x^6 - 260*x^5 + 127*x^4 - 46*x^3 + 14*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 84 x^{13} + 164 x^{12} - 256 x^{11} + 344 x^{10} - 422 x^{9} + 478 x^{8} - 480 x^{7} + 400 x^{6} - 260 x^{5} + 127 x^{4} - 46 x^{3} + 14 x^{2} - 4 x + 1 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(38416000000000000=2^{16}\cdot 5^{12}\cdot 7^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $10.88$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11} a^{14} + \frac{4}{11} a^{13} + \frac{4}{11} a^{12} + \frac{1}{11} a^{11} + \frac{4}{11} a^{10} + \frac{2}{11} a^{9} - \frac{2}{11} a^{8} - \frac{4}{11} a^{7} - \frac{1}{11} a^{6} - \frac{1}{11} a^{5} + \frac{2}{11} a^{4} + \frac{5}{11} a^{3} + \frac{2}{11} a^{2} + \frac{5}{11} a - \frac{1}{11}$, $\frac{1}{1661} a^{15} + \frac{68}{1661} a^{14} + \frac{821}{1661} a^{13} - \frac{51}{1661} a^{12} + \frac{365}{1661} a^{11} - \frac{149}{1661} a^{10} - \frac{259}{1661} a^{9} - \frac{57}{151} a^{8} - \frac{213}{1661} a^{7} - \frac{813}{1661} a^{6} - \frac{535}{1661} a^{5} + \frac{1}{1661} a^{4} - \frac{250}{1661} a^{3} + \frac{584}{1661} a^{2} + \frac{69}{151} a - \frac{757}{1661}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{1000}{1661} a^{15} - \frac{4480}{1661} a^{14} + \frac{564}{151} a^{13} + \frac{9551}{1661} a^{12} - \frac{56290}{1661} a^{11} + \frac{129142}{1661} a^{10} - \frac{197996}{1661} a^{9} + \frac{255444}{1661} a^{8} - \frac{321720}{1661} a^{7} + \frac{34756}{151} a^{6} - \frac{379470}{1661} a^{5} + \frac{274612}{1661} a^{4} - \frac{127388}{1661} a^{3} + \frac{40400}{1661} a^{2} - \frac{12006}{1661} a + \frac{6456}{1661} \) (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:  \( 175.447877376 \)
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.2.14000.1, 8.4.196000000.1

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 R ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
5Data not computed
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$