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Properties

Label 16.0.25657264090...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{12}\cdot 5^{12}\cdot 37^{6}$
Root discriminant $21.78$
Ramified primes $2, 5, 37$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $Q_8:C_2^2.D_6$ (as 16T754)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![36, -24, 364, -312, 104, 604, -444, 46, 540, -434, 102, 48, -45, 19, -2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 2*x^14 + 19*x^13 - 45*x^12 + 48*x^11 + 102*x^10 - 434*x^9 + 540*x^8 + 46*x^7 - 444*x^6 + 604*x^5 + 104*x^4 - 312*x^3 + 364*x^2 - 24*x + 36)
 
gp: K = bnfinit(x^16 - 2*x^15 - 2*x^14 + 19*x^13 - 45*x^12 + 48*x^11 + 102*x^10 - 434*x^9 + 540*x^8 + 46*x^7 - 444*x^6 + 604*x^5 + 104*x^4 - 312*x^3 + 364*x^2 - 24*x + 36, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 2 x^{14} + 19 x^{13} - 45 x^{12} + 48 x^{11} + 102 x^{10} - 434 x^{9} + 540 x^{8} + 46 x^{7} - 444 x^{6} + 604 x^{5} + 104 x^{4} - 312 x^{3} + 364 x^{2} - 24 x + 36 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2565726409000000000000=2^{12}\cdot 5^{12}\cdot 37^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.78$
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:  $2, 5, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6}$, $\frac{1}{18} a^{14} + \frac{1}{6} a^{13} - \frac{1}{9} a^{12} + \frac{1}{6} a^{10} - \frac{1}{2} a^{9} + \frac{1}{6} a^{8} + \frac{2}{9} a^{7} - \frac{7}{18} a^{6} - \frac{2}{9} a^{5} - \frac{4}{9} a^{4} - \frac{1}{3} a^{3} - \frac{2}{9} a^{2} - \frac{4}{9} a + \frac{1}{3}$, $\frac{1}{212127216633333728532} a^{15} - \frac{3498730021093885}{453263283404559249} a^{14} - \frac{11606332152464856554}{53031804158333432133} a^{13} + \frac{10034786942817073325}{70709072211111242844} a^{12} - \frac{2266891500173409179}{70709072211111242844} a^{11} + \frac{327371432550139716}{1964140894753090079} a^{10} + \frac{2914899052840167455}{35354536105555621422} a^{9} - \frac{24004029758103402560}{53031804158333432133} a^{8} - \frac{45209342471720856629}{106063608316666864266} a^{7} - \frac{1626461383331277463}{4079369550641033241} a^{6} + \frac{37233581640209334821}{106063608316666864266} a^{5} - \frac{823473382865043449}{1964140894753090079} a^{4} + \frac{20998426845558046184}{53031804158333432133} a^{3} - \frac{14624631306406773611}{53031804158333432133} a^{2} + \frac{1330163950055359624}{17677268052777810711} a - \frac{2019861771986584247}{5892422684259270237}$

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:  $7$
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:  \( 6174.30151656 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_8:C_2^2.D_6$ (as 16T754):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 23 conjugacy class representatives for $Q_8:C_2^2.D_6$
Character table for $Q_8:C_2^2.D_6$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.3700.1, 8.0.13690000.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.12.12.27$x^{12} - 18 x^{10} + 171 x^{8} + 116 x^{6} - 313 x^{4} + 190 x^{2} + 877$$6$$2$$12$12T30$[4/3, 4/3]_{3}^{4}$
5Data not computed
$37$37.4.2.2$x^{4} - 37 x^{2} + 6845$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.8.4.1$x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$