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Properties

Label 16.0.96148443603515625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{14}\cdot 7^{4}$
Root discriminant $11.52$
Ramified primes $3, 5, 7$
Class number $1$
Class group Trivial
Galois group $(C_8:C_2):C_2$ (as 16T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31, -55, 186, -260, 472, -550, 673, -655, 600, -470, 337, -205, 112, -50, 19, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 19*x^14 - 50*x^13 + 112*x^12 - 205*x^11 + 337*x^10 - 470*x^9 + 600*x^8 - 655*x^7 + 673*x^6 - 550*x^5 + 472*x^4 - 260*x^3 + 186*x^2 - 55*x + 31)
 
gp: K = bnfinit(x^16 - 5*x^15 + 19*x^14 - 50*x^13 + 112*x^12 - 205*x^11 + 337*x^10 - 470*x^9 + 600*x^8 - 655*x^7 + 673*x^6 - 550*x^5 + 472*x^4 - 260*x^3 + 186*x^2 - 55*x + 31, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 19 x^{14} - 50 x^{13} + 112 x^{12} - 205 x^{11} + 337 x^{10} - 470 x^{9} + 600 x^{8} - 655 x^{7} + 673 x^{6} - 550 x^{5} + 472 x^{4} - 260 x^{3} + 186 x^{2} - 55 x + 31 \)

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:  \(96148443603515625=3^{8}\cdot 5^{14}\cdot 7^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.52$
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:  $3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{44812979} a^{15} + \frac{14626779}{44812979} a^{14} + \frac{18493359}{44812979} a^{13} + \frac{11393787}{44812979} a^{12} + \frac{11543768}{44812979} a^{11} + \frac{21473568}{44812979} a^{10} + \frac{462339}{44812979} a^{9} - \frac{14721668}{44812979} a^{8} + \frac{8185872}{44812979} a^{7} + \frac{6222570}{44812979} a^{6} - \frac{3731964}{44812979} a^{5} + \frac{13582595}{44812979} a^{4} - \frac{10266601}{44812979} a^{3} - \frac{3685003}{44812979} a^{2} + \frac{8644643}{44812979} a + \frac{10257195}{44812979}$

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:  \( -\frac{9202800}{44812979} a^{15} + \frac{22393882}{44812979} a^{14} - \frac{42184958}{44812979} a^{13} - \frac{35163009}{44812979} a^{12} + \frac{273134244}{44812979} a^{11} - \frac{900861095}{44812979} a^{10} + \frac{1792274094}{44812979} a^{9} - \frac{3185334985}{44812979} a^{8} + \frac{4351991271}{44812979} a^{7} - \frac{5462530666}{44812979} a^{6} + \frac{5318377038}{44812979} a^{5} - \frac{5454613242}{44812979} a^{4} + \frac{3302755617}{44812979} a^{3} - \frac{2878719443}{44812979} a^{2} + \frac{933636594}{44812979} a - \frac{586302526}{44812979} \) (order $30$)
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:  \( 733.048403988 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}:C_2$ (as 16T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $(C_8:C_2):C_2$
Character table for $(C_8:C_2):C_2$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\zeta_{15})\)

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

Sibling fields

Galois closure: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
3Data not computed
5Data not computed
$7$7.8.0.1$x^{8} - x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
7.8.4.2$x^{8} + 49 x^{4} - 1029 x^{2} + 12005$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$