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Properties

Label 16.8.43538996073...0625.2
Degree $16$
Signature $[8, 4]$
Discriminant $5^{14}\cdot 61^{10}$
Root discriminant $53.39$
Ramified primes $5, 61$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times OD_{16}).C_2$ (as 16T123)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2404, -2590, -10577, -31470, -19337, 14670, 16361, 4600, 470, -2180, -1334, 400, 273, -20, -27, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 27*x^14 - 20*x^13 + 273*x^12 + 400*x^11 - 1334*x^10 - 2180*x^9 + 470*x^8 + 4600*x^7 + 16361*x^6 + 14670*x^5 - 19337*x^4 - 31470*x^3 - 10577*x^2 - 2590*x - 2404)
 
gp: K = bnfinit(x^16 - 27*x^14 - 20*x^13 + 273*x^12 + 400*x^11 - 1334*x^10 - 2180*x^9 + 470*x^8 + 4600*x^7 + 16361*x^6 + 14670*x^5 - 19337*x^4 - 31470*x^3 - 10577*x^2 - 2590*x - 2404, 1)
 

Normalized defining polynomial

\( x^{16} - 27 x^{14} - 20 x^{13} + 273 x^{12} + 400 x^{11} - 1334 x^{10} - 2180 x^{9} + 470 x^{8} + 4600 x^{7} + 16361 x^{6} + 14670 x^{5} - 19337 x^{4} - 31470 x^{3} - 10577 x^{2} - 2590 x - 2404 \)

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:  \(4353899607317398687744140625=5^{14}\cdot 61^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.39$
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:  $5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{3065401494923930055966789088} a^{15} + \frac{580984918042700592530708459}{3065401494923930055966789088} a^{14} - \frac{129600412897469288494246849}{1532700747461965027983394544} a^{13} + \frac{378226367057425674791034571}{1532700747461965027983394544} a^{12} - \frac{161768448156467464179359405}{3065401494923930055966789088} a^{11} + \frac{280674090748846438344629665}{3065401494923930055966789088} a^{10} - \frac{99811487968493417280358555}{3065401494923930055966789088} a^{9} + \frac{645383850589371663782493203}{3065401494923930055966789088} a^{8} - \frac{239636748594083455300382217}{3065401494923930055966789088} a^{7} + \frac{491425883669854081255451621}{3065401494923930055966789088} a^{6} + \frac{36110292906747795022180079}{95793796716372814248962159} a^{5} - \frac{183670143820635200828323329}{1532700747461965027983394544} a^{4} + \frac{311885303527827204390975265}{3065401494923930055966789088} a^{3} + \frac{1397234172371545525906059149}{3065401494923930055966789088} a^{2} - \frac{80351540569460580964323793}{1532700747461965027983394544} a - \frac{130711518398152515800196349}{766350373730982513991697272}$

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:  $11$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 123590007.283 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).C_2$ (as 16T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $(C_2\times OD_{16}).C_2$
Character table for $(C_2\times OD_{16}).C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{305}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{61}) \), 4.4.465125.1 x2, 4.4.7625.1 x2, \(\Q(\sqrt{5}, \sqrt{61})\), 8.8.216341265625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$61$61.8.4.1$x^{8} + 14884 x^{4} - 226981 x^{2} + 55383364$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
61.8.6.2$x^{8} + 183 x^{4} + 14884$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$