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Properties

Label 16.8.63152272493...1552.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{28}\cdot 113^{7}$
Root discriminant $26.61$
Ramified primes $2, 113$
Class number $1$
Class group Trivial
Galois group $(C_2^3\times C_4).D_4$ (as 16T675)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![89, -390, 493, -128, 122, -106, -712, 642, 499, -622, -76, 266, -64, -28, 19, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 19*x^14 - 28*x^13 - 64*x^12 + 266*x^11 - 76*x^10 - 622*x^9 + 499*x^8 + 642*x^7 - 712*x^6 - 106*x^5 + 122*x^4 - 128*x^3 + 493*x^2 - 390*x + 89)
 
gp: K = bnfinit(x^16 - 6*x^15 + 19*x^14 - 28*x^13 - 64*x^12 + 266*x^11 - 76*x^10 - 622*x^9 + 499*x^8 + 642*x^7 - 712*x^6 - 106*x^5 + 122*x^4 - 128*x^3 + 493*x^2 - 390*x + 89, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 19 x^{14} - 28 x^{13} - 64 x^{12} + 266 x^{11} - 76 x^{10} - 622 x^{9} + 499 x^{8} + 642 x^{7} - 712 x^{6} - 106 x^{5} + 122 x^{4} - 128 x^{3} + 493 x^{2} - 390 x + 89 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(63152272493220359831552=2^{28}\cdot 113^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.61$
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:  $2, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{53} a^{14} + \frac{21}{53} a^{13} + \frac{15}{53} a^{12} - \frac{7}{53} a^{11} - \frac{20}{53} a^{10} + \frac{13}{53} a^{9} - \frac{18}{53} a^{8} + \frac{2}{53} a^{7} + \frac{19}{53} a^{6} + \frac{13}{53} a^{5} + \frac{26}{53} a^{4} + \frac{10}{53} a^{3} + \frac{15}{53} a^{2} + \frac{26}{53} a - \frac{3}{53}$, $\frac{1}{9860345564167411} a^{15} - \frac{69362699817163}{9860345564167411} a^{14} + \frac{3441636426331092}{9860345564167411} a^{13} - \frac{3601251261365246}{9860345564167411} a^{12} + \frac{2582148770508232}{9860345564167411} a^{11} - \frac{3874120630900558}{9860345564167411} a^{10} + \frac{3110352085358265}{9860345564167411} a^{9} - \frac{17020942276405}{186044255927687} a^{8} - \frac{2909642880706356}{9860345564167411} a^{7} + \frac{2355787752796825}{9860345564167411} a^{6} - \frac{120736318135043}{9860345564167411} a^{5} + \frac{2947306463322386}{9860345564167411} a^{4} - \frac{4173695257093971}{9860345564167411} a^{3} - \frac{347716551348230}{9860345564167411} a^{2} - \frac{411739812445778}{9860345564167411} a - \frac{3496593800051518}{9860345564167411}$

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:  $11$
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:  \( 240620.117375 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^3\times C_4).D_4$ (as 16T675):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 31 conjugacy class representatives for $(C_2^3\times C_4).D_4$
Character table for $(C_2^3\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.7232.1, 8.8.5910106112.1

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 $16$ $16$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ $16$

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$2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$113$113.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
113.4.3.4$x^{4} + 3051$$4$$1$$3$$C_4$$[\ ]_{4}$
113.8.4.1$x^{8} + 127690 x^{4} - 1442897 x^{2} + 4076184025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$