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Properties

Label 16.4.16226472272...9456.4
Degree $16$
Signature $[4, 6]$
Discriminant $2^{48}\cdot 7^{8}$
Root discriminant $21.17$
Ramified primes $2, 7$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times D_8):C_2$ (as 16T126)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14, 168, -216, -336, 192, 400, 104, -120, -254, -192, 12, 96, 36, -16, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - 16*x^13 + 36*x^12 + 96*x^11 + 12*x^10 - 192*x^9 - 254*x^8 - 120*x^7 + 104*x^6 + 400*x^5 + 192*x^4 - 336*x^3 - 216*x^2 + 168*x + 14)
 
gp: K = bnfinit(x^16 - 8*x^14 - 16*x^13 + 36*x^12 + 96*x^11 + 12*x^10 - 192*x^9 - 254*x^8 - 120*x^7 + 104*x^6 + 400*x^5 + 192*x^4 - 336*x^3 - 216*x^2 + 168*x + 14, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{14} - 16 x^{13} + 36 x^{12} + 96 x^{11} + 12 x^{10} - 192 x^{9} - 254 x^{8} - 120 x^{7} + 104 x^{6} + 400 x^{5} + 192 x^{4} - 336 x^{3} - 216 x^{2} + 168 x + 14 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1622647227216566419456=2^{48}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.17$
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:  $2, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{31} a^{13} + \frac{1}{31} a^{12} - \frac{2}{31} a^{11} + \frac{11}{31} a^{10} - \frac{1}{31} a^{9} + \frac{9}{31} a^{8} + \frac{1}{31} a^{7} - \frac{15}{31} a^{6} + \frac{14}{31} a^{5} - \frac{2}{31} a^{4} - \frac{2}{31} a^{3} + \frac{11}{31} a^{2} - \frac{10}{31} a + \frac{2}{31}$, $\frac{1}{31} a^{14} - \frac{3}{31} a^{12} + \frac{13}{31} a^{11} - \frac{12}{31} a^{10} + \frac{10}{31} a^{9} - \frac{8}{31} a^{8} + \frac{15}{31} a^{7} - \frac{2}{31} a^{6} + \frac{15}{31} a^{5} + \frac{13}{31} a^{3} + \frac{10}{31} a^{2} + \frac{12}{31} a - \frac{2}{31}$, $\frac{1}{1014548459395509553} a^{15} + \frac{6777071828803574}{1014548459395509553} a^{14} + \frac{1197372559293971}{1014548459395509553} a^{13} - \frac{454927187673753319}{1014548459395509553} a^{12} + \frac{447931591759632875}{1014548459395509553} a^{11} + \frac{445321389841428465}{1014548459395509553} a^{10} - \frac{20778621561757469}{1014548459395509553} a^{9} - \frac{172402674920606608}{1014548459395509553} a^{8} + \frac{10903078563921903}{27420228632311069} a^{7} - \frac{258129826970146270}{1014548459395509553} a^{6} - \frac{292739062483702507}{1014548459395509553} a^{5} + \frac{483759553930430695}{1014548459395509553} a^{4} - \frac{470179660677462825}{1014548459395509553} a^{3} - \frac{2219070365070621}{32727369657919663} a^{2} - \frac{497191789977766925}{1014548459395509553} a - \frac{39565114398543825}{1014548459395509553}$

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:  $9$
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:  \( 26399.5973815 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times D_8):C_2$ (as 16T126):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 19 conjugacy class representatives for $(C_2\times D_8):C_2$
Character table for $(C_2\times D_8):C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.2.50176.1, 4.2.1792.1, 4.4.7168.1, 8.2.1438646272.4, 8.2.359661568.1, 8.4.10070523904.2

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$