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Properties

Label 16.8.29875440679...3521.1
Degree $16$
Signature $[8, 4]$
Discriminant $31^{12}\cdot 41^{14}$
Root discriminant $338.62$
Ramified primes $31, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T565)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-686260787, 2013183756, 2366911254, -9056589516, 2955391761, 1608355766, -1104467286, 215410535, -6534866, -3538362, 765081, -90470, -1604, 1877, -132, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 - 132*x^14 + 1877*x^13 - 1604*x^12 - 90470*x^11 + 765081*x^10 - 3538362*x^9 - 6534866*x^8 + 215410535*x^7 - 1104467286*x^6 + 1608355766*x^5 + 2955391761*x^4 - 9056589516*x^3 + 2366911254*x^2 + 2013183756*x - 686260787)
 
gp: K = bnfinit(x^16 - 7*x^15 - 132*x^14 + 1877*x^13 - 1604*x^12 - 90470*x^11 + 765081*x^10 - 3538362*x^9 - 6534866*x^8 + 215410535*x^7 - 1104467286*x^6 + 1608355766*x^5 + 2955391761*x^4 - 9056589516*x^3 + 2366911254*x^2 + 2013183756*x - 686260787, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} - 132 x^{14} + 1877 x^{13} - 1604 x^{12} - 90470 x^{11} + 765081 x^{10} - 3538362 x^{9} - 6534866 x^{8} + 215410535 x^{7} - 1104467286 x^{6} + 1608355766 x^{5} + 2955391761 x^{4} - 9056589516 x^{3} + 2366911254 x^{2} + 2013183756 x - 686260787 \)

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:  \(29875440679295555535575643982688502833521=31^{12}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $338.62$
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:  $31, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{47797458824031419062928831453720089493461693435662247091592187815672308762} a^{15} - \frac{11190709505622082729662083017685420231628672104575268078499468054686532439}{47797458824031419062928831453720089493461693435662247091592187815672308762} a^{14} - \frac{88774665233586333403327449043423627687248851180389366131324359805376597}{446705222654499243578774125735701770966931714351983617678431661828713166} a^{13} + \frac{7365728473873499987210734941736419126470572782988218604086578882539611357}{47797458824031419062928831453720089493461693435662247091592187815672308762} a^{12} - \frac{22221188045730768293226754431372753806753086514523945421397513882422752825}{47797458824031419062928831453720089493461693435662247091592187815672308762} a^{11} - \frac{18028566090789355801694322855813346906868410426364805415461331271648705275}{47797458824031419062928831453720089493461693435662247091592187815672308762} a^{10} - \frac{17555020010227608444431864807945515296214141584221034669365938762229623963}{47797458824031419062928831453720089493461693435662247091592187815672308762} a^{9} - \frac{10001572921352925641926138105233026089826085861294229240838299529273081127}{47797458824031419062928831453720089493461693435662247091592187815672308762} a^{8} + \frac{13228213879285369097916854740598749355945743923856197029345156977076960743}{47797458824031419062928831453720089493461693435662247091592187815672308762} a^{7} + \frac{7819717699320726035319718255328421062095708107415972108060780728087638745}{47797458824031419062928831453720089493461693435662247091592187815672308762} a^{6} + \frac{424237605697983631194656586556731081884362003831013425668695801239188871}{47797458824031419062928831453720089493461693435662247091592187815672308762} a^{5} - \frac{18891992843689447350500294030719510644884821274318686866927243648221341629}{47797458824031419062928831453720089493461693435662247091592187815672308762} a^{4} + \frac{11889293729243481946594051129763436566544951792066510525547406742986319769}{47797458824031419062928831453720089493461693435662247091592187815672308762} a^{3} + \frac{23011171986089005543794930591679441828314950068037148626996806462483844647}{47797458824031419062928831453720089493461693435662247091592187815672308762} a^{2} - \frac{19030825135390327529234984347682336811822341995758086755267306702378689657}{47797458824031419062928831453720089493461693435662247091592187815672308762} a - \frac{3786375476452252523939910471579652581402615440088402873513000995076850828}{23898729412015709531464415726860044746730846717831123545796093907836154381}$

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:  \( 139132652452000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T565):

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 $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.179859661768855001.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{12}$

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
$31$31.4.3.2$x^{4} - 31$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.4.3.2$x^{4} - 31$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.4.3.2$x^{4} - 31$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.4.3.2$x^{4} - 31$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$