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Properties

Label 16.0.37908698439...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 353^{4}$
Root discriminant $14.49$
Ramified primes $5, 353$
Class number $2$
Class group $[2]$
Galois group $C_2^4.(C_4\times S_3)$ (as 16T725)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 5, 20, -58, 15, 160, -240, 79, 50, -10, -15, -12, 10, 5, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 5*x^14 + 10*x^13 - 12*x^12 - 15*x^11 - 10*x^10 + 50*x^9 + 79*x^8 - 240*x^7 + 160*x^6 + 15*x^5 - 58*x^4 + 20*x^3 + 5*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^16 - 5*x^15 + 5*x^14 + 10*x^13 - 12*x^12 - 15*x^11 - 10*x^10 + 50*x^9 + 79*x^8 - 240*x^7 + 160*x^6 + 15*x^5 - 58*x^4 + 20*x^3 + 5*x^2 - 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 5 x^{14} + 10 x^{13} - 12 x^{12} - 15 x^{11} - 10 x^{10} + 50 x^{9} + 79 x^{8} - 240 x^{7} + 160 x^{6} + 15 x^{5} - 58 x^{4} + 20 x^{3} + 5 x^{2} - 5 x + 1 \)

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:  \(3790869843994140625=5^{12}\cdot 353^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.49$
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:  $5, 353$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{34} a^{12} + \frac{8}{17} a^{11} - \frac{6}{17} a^{10} + \frac{15}{34} a^{9} - \frac{2}{17} a^{8} + \frac{3}{17} a^{7} + \frac{13}{34} a^{6} - \frac{2}{17} a^{5} - \frac{5}{17} a^{4} + \frac{15}{34} a^{3} + \frac{6}{17} a^{2} + \frac{2}{17} a - \frac{9}{34}$, $\frac{1}{34} a^{13} + \frac{2}{17} a^{11} + \frac{3}{34} a^{10} - \frac{3}{17} a^{9} + \frac{1}{17} a^{8} - \frac{15}{34} a^{7} - \frac{4}{17} a^{6} - \frac{7}{17} a^{5} + \frac{5}{34} a^{4} + \frac{5}{17} a^{3} + \frac{8}{17} a^{2} - \frac{5}{34} a + \frac{4}{17}$, $\frac{1}{34} a^{14} + \frac{7}{34} a^{11} + \frac{4}{17} a^{10} + \frac{5}{17} a^{9} + \frac{1}{34} a^{8} + \frac{1}{17} a^{7} + \frac{1}{17} a^{6} - \frac{13}{34} a^{5} + \frac{8}{17} a^{4} - \frac{5}{17} a^{3} + \frac{15}{34} a^{2} - \frac{4}{17} a + \frac{1}{17}$, $\frac{1}{14926} a^{15} - \frac{52}{7463} a^{14} + \frac{6}{439} a^{13} + \frac{4}{7463} a^{12} + \frac{2232}{7463} a^{11} - \frac{2134}{7463} a^{10} - \frac{2093}{7463} a^{9} + \frac{1341}{7463} a^{8} - \frac{1239}{7463} a^{7} - \frac{818}{7463} a^{6} + \frac{2042}{7463} a^{5} + \frac{2643}{7463} a^{4} + \frac{2592}{7463} a^{3} + \frac{1534}{7463} a^{2} + \frac{2884}{7463} a - \frac{2971}{14926}$

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:  \( -\frac{24391}{14926} a^{15} + \frac{57570}{7463} a^{14} - \frac{44408}{7463} a^{13} - \frac{135757}{7463} a^{12} + \frac{107233}{7463} a^{11} + \frac{218103}{7463} a^{10} + \frac{185628}{7463} a^{9} - \frac{564751}{7463} a^{8} - \frac{1134248}{7463} a^{7} + \frac{2613094}{7463} a^{6} - \frac{1171807}{7463} a^{5} - \frac{548769}{7463} a^{4} + \frac{510353}{7463} a^{3} - \frac{2572}{439} a^{2} - \frac{76865}{7463} a + \frac{57879}{14926} \) (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:  \( 1185.47860255 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.(C_4\times S_3)$ (as 16T725):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 28 conjugacy class representatives for $C_2^4.(C_4\times S_3)$
Character table for $C_2^4.(C_4\times S_3)$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.4.1947015625.1

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 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
5Data not computed
353Data not computed