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Properties

Label 16.0.22012954217...5536.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{50}\cdot 3^{4}\cdot 17^{6}$
Root discriminant $33.22$
Ramified primes $2, 3, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T456)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![374544, 0, -499392, 0, 319464, 0, -131784, 0, 36466, 0, -6420, 0, 682, 0, -40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 40*x^14 + 682*x^12 - 6420*x^10 + 36466*x^8 - 131784*x^6 + 319464*x^4 - 499392*x^2 + 374544)
 
gp: K = bnfinit(x^16 - 40*x^14 + 682*x^12 - 6420*x^10 + 36466*x^8 - 131784*x^6 + 319464*x^4 - 499392*x^2 + 374544, 1)
 

Normalized defining polynomial

\( x^{16} - 40 x^{14} + 682 x^{12} - 6420 x^{10} + 36466 x^{8} - 131784 x^{6} + 319464 x^{4} - 499392 x^{2} + 374544 \)

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:  \(2201295421769100124225536=2^{50}\cdot 3^{4}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.22$
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:  $2, 3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{2}$, $\frac{1}{12} a^{11} + \frac{1}{6} a^{9} - \frac{1}{6} a^{7} - \frac{1}{6} a^{3}$, $\frac{1}{612} a^{12} - \frac{10}{153} a^{10} + \frac{35}{306} a^{8} - \frac{25}{51} a^{6} - \frac{127}{306} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{612} a^{13} + \frac{11}{612} a^{11} - \frac{67}{306} a^{9} + \frac{35}{102} a^{7} - \frac{127}{306} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2016003059352} a^{14} - \frac{185456953}{504000764838} a^{12} - \frac{20122803451}{1008001529676} a^{10} + \frac{13965670954}{84000127473} a^{8} + \frac{196377175133}{1008001529676} a^{6} + \frac{40698086261}{84000127473} a^{4} + \frac{30530033}{1647061323} a^{2} + \frac{238514419}{549020441}$, $\frac{1}{2016003059352} a^{15} - \frac{185456953}{504000764838} a^{13} - \frac{20122803451}{1008001529676} a^{11} + \frac{13965670954}{84000127473} a^{9} + \frac{196377175133}{1008001529676} a^{7} + \frac{40698086261}{84000127473} a^{5} + \frac{30530033}{1647061323} a^{3} + \frac{238514419}{549020441} a$

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:  \( \frac{196134493}{2016003059352} a^{14} - \frac{3795028379}{1008001529676} a^{12} + \frac{60528485921}{1008001529676} a^{10} - \frac{42666129437}{84000127473} a^{8} + \frac{2475892898501}{1008001529676} a^{6} - \frac{1202950568879}{168000254946} a^{4} + \frac{22594038706}{1647061323} a^{2} - \frac{7132155793}{549020441} \) (order $8$)
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:  \( 1821807.79264 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T456):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), 4.4.4352.1, 4.0.1088.2, \(\Q(\zeta_{8})\), 8.0.18939904.2

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.8.25.7$x^{8} + 10 x^{4} + 28 x^{2} + 18$$8$$1$$25$$C_2^3: C_4$$[2, 3, 7/2, 4, 17/4]$
2.8.25.3$x^{8} + 10 x^{4} + 12 x^{2} + 18$$8$$1$$25$$C_2^3: C_4$$[2, 3, 7/2, 4, 17/4]$
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.4.3.4$x^{4} + 459$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.3$x^{4} + 51$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$