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Properties

Label 16.0.39071704248...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{4}\cdot 5^{10}\cdot 251^{6}$
Root discriminant $25.82$
Ramified primes $2, 5, 251$
Class number $8$
Class group $[2, 4]$
Galois group 16T1046

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4331, -20125, 38418, -34763, 18332, -2254, -2558, 2481, -983, 443, -42, 4, 44, -14, 8, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 8*x^14 - 14*x^13 + 44*x^12 + 4*x^11 - 42*x^10 + 443*x^9 - 983*x^8 + 2481*x^7 - 2558*x^6 - 2254*x^5 + 18332*x^4 - 34763*x^3 + 38418*x^2 - 20125*x + 4331)
 
gp: K = bnfinit(x^16 - 2*x^15 + 8*x^14 - 14*x^13 + 44*x^12 + 4*x^11 - 42*x^10 + 443*x^9 - 983*x^8 + 2481*x^7 - 2558*x^6 - 2254*x^5 + 18332*x^4 - 34763*x^3 + 38418*x^2 - 20125*x + 4331, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 8 x^{14} - 14 x^{13} + 44 x^{12} + 4 x^{11} - 42 x^{10} + 443 x^{9} - 983 x^{8} + 2481 x^{7} - 2558 x^{6} - 2254 x^{5} + 18332 x^{4} - 34763 x^{3} + 38418 x^{2} - 20125 x + 4331 \)

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:  \(39071704248281406250000=2^{4}\cdot 5^{10}\cdot 251^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.82$
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:  $2, 5, 251$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{14} - \frac{2}{9} a^{12} + \frac{1}{9} a^{10} - \frac{1}{3} a^{9} - \frac{4}{9} a^{8} - \frac{1}{3} a^{7} - \frac{4}{9} a^{6} + \frac{1}{9} a^{5} + \frac{4}{9} a^{4} + \frac{1}{3} a^{3} + \frac{1}{9} a^{2} + \frac{1}{3} a + \frac{2}{9}$, $\frac{1}{7242911381931781800011028225273} a^{15} - \frac{94038974639075642399742797}{12214015821132852951114718761} a^{14} + \frac{176388774737566862583634229368}{7242911381931781800011028225273} a^{13} - \frac{1080329521237480925168346543949}{7242911381931781800011028225273} a^{12} - \frac{813115883896999234156532645837}{7242911381931781800011028225273} a^{11} + \frac{549421767529396651653013269419}{7242911381931781800011028225273} a^{10} - \frac{654442376317396470410585508811}{7242911381931781800011028225273} a^{9} + \frac{2885924742661691551722633216814}{7242911381931781800011028225273} a^{8} - \frac{2599547058191529401992861478410}{7242911381931781800011028225273} a^{7} + \frac{982948601257418128027424563463}{7242911381931781800011028225273} a^{6} - \frac{1142348545097430028440007908782}{2414303793977260600003676075091} a^{5} - \frac{1050888424229113548102679039924}{7242911381931781800011028225273} a^{4} - \frac{2251117878019069106414209313900}{7242911381931781800011028225273} a^{3} + \frac{3201182947346895397643107376576}{7242911381931781800011028225273} a^{2} + \frac{1179839856616173115111924576595}{7242911381931781800011028225273} a - \frac{8608112381728840667473543094}{118736252162816095082148003693}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$

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:  \( 5392.63795631 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1046:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 768
The 40 conjugacy class representatives for t16n1046
Character table for t16n1046 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.1255.1, 8.4.39375625.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.12.0.1$x^{12} - 26 x^{10} + 275 x^{8} - 1500 x^{6} + 4375 x^{4} - 6250 x^{2} + 7221$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
251Data not computed