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Properties

Label 16.12.2111361753...2288.2
Degree $16$
Signature $[12, 2]$
Discriminant $2^{16}\cdot 17^{15}\cdot 103^{4}$
Root discriminant $90.74$
Ramified primes $2, 17, 103$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times OD_{16}).D_4$ (as 16T591)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1913364977, 0, -1226039694, 0, 214259364, 0, -3153551, 0, -1622378, 0, 65892, 0, 2516, 0, -119, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 119*x^14 + 2516*x^12 + 65892*x^10 - 1622378*x^8 - 3153551*x^6 + 214259364*x^4 - 1226039694*x^2 + 1913364977)
 
gp: K = bnfinit(x^16 - 119*x^14 + 2516*x^12 + 65892*x^10 - 1622378*x^8 - 3153551*x^6 + 214259364*x^4 - 1226039694*x^2 + 1913364977, 1)
 

Normalized defining polynomial

\( x^{16} - 119 x^{14} + 2516 x^{12} + 65892 x^{10} - 1622378 x^{8} - 3153551 x^{6} + 214259364 x^{4} - 1226039694 x^{2} + 1913364977 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(21113617530364765644381463052288=2^{16}\cdot 17^{15}\cdot 103^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.74$
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:  $2, 17, 103$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{206} a^{10} - \frac{8}{103} a^{8} - \frac{59}{206} a^{6} - \frac{14}{103} a^{4} - \frac{25}{206} a^{2} - \frac{1}{2}$, $\frac{1}{206} a^{11} - \frac{8}{103} a^{9} - \frac{59}{206} a^{7} - \frac{14}{103} a^{5} - \frac{25}{206} a^{3} - \frac{1}{2} a$, $\frac{1}{42436} a^{12} + \frac{87}{42436} a^{10} + \frac{9829}{42436} a^{8} - \frac{20525}{42436} a^{6} - \frac{4969}{42436} a^{4} - \frac{19}{103} a^{2} + \frac{1}{4}$, $\frac{1}{42436} a^{13} + \frac{87}{42436} a^{11} + \frac{9829}{42436} a^{9} - \frac{20525}{42436} a^{7} - \frac{4969}{42436} a^{5} - \frac{19}{103} a^{3} + \frac{1}{4} a$, $\frac{1}{6068720158151111687062311544} a^{14} + \frac{772701190816212175930}{758590019768888960882788943} a^{12} + \frac{2876327137750500732881471}{1517180039537777921765577886} a^{10} - \frac{348887192727304798917807176}{758590019768888960882788943} a^{8} + \frac{809470115521948772821945687}{3034360079075555843531155772} a^{6} - \frac{3747145847280134208064179}{58919613185933123175362248} a^{4} + \frac{116449012404799232302617}{572035079475078865780216} a^{2} - \frac{1833833502350671345153}{5553738635680377337672}$, $\frac{1}{6068720158151111687062311544} a^{15} + \frac{772701190816212175930}{758590019768888960882788943} a^{13} + \frac{2876327137750500732881471}{1517180039537777921765577886} a^{11} - \frac{348887192727304798917807176}{758590019768888960882788943} a^{9} + \frac{809470115521948772821945687}{3034360079075555843531155772} a^{7} - \frac{3747145847280134208064179}{58919613185933123175362248} a^{5} + \frac{116449012404799232302617}{572035079475078865780216} a^{3} - \frac{1833833502350671345153}{5553738635680377337672} a$

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

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 5678560818.23 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).D_4$ (as 16T591):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $(C_2\times OD_{16}).D_4$
Character table for $(C_2\times OD_{16}).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
2.8.8.6$x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$$2$$4$$8$$(C_8:C_2):C_2$$[2, 2, 2]^{4}$
17Data not computed
103Data not computed