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Properties

Label 16.0.81811674303...1296.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{34}\cdot 3^{10}\cdot 73^{8}$
Root discriminant $74.05$
Ramified primes $2, 3, 73$
Class number $1088$ (GRH)
Class group $[2, 2, 2, 136]$ (GRH)
Galois group $C_2.C_2\wr C_2^2$ (as 16T394)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2916, 0, 16524, 0, 30942, 0, 27558, 0, 13141, 0, 3474, 0, 500, 0, 36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 36*x^14 + 500*x^12 + 3474*x^10 + 13141*x^8 + 27558*x^6 + 30942*x^4 + 16524*x^2 + 2916)
 
gp: K = bnfinit(x^16 + 36*x^14 + 500*x^12 + 3474*x^10 + 13141*x^8 + 27558*x^6 + 30942*x^4 + 16524*x^2 + 2916, 1)
 

Normalized defining polynomial

\( x^{16} + 36 x^{14} + 500 x^{12} + 3474 x^{10} + 13141 x^{8} + 27558 x^{6} + 30942 x^{4} + 16524 x^{2} + 2916 \)

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:  \(818116743035720881057925431296=2^{34}\cdot 3^{10}\cdot 73^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.05$
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:  $2, 3, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{6} - \frac{1}{3} a^{2}$, $\frac{1}{18} a^{11} + \frac{5}{18} a^{7} - \frac{4}{9} a^{3}$, $\frac{1}{54} a^{12} - \frac{13}{54} a^{8} - \frac{4}{27} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{54} a^{13} - \frac{13}{54} a^{9} - \frac{4}{27} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{2455434} a^{14} + \frac{296}{136413} a^{12} - \frac{41215}{2455434} a^{10} + \frac{23740}{136413} a^{8} + \frac{581792}{1227717} a^{6} - \frac{28577}{136413} a^{4} + \frac{6028}{15157} a^{2} - \frac{5207}{15157}$, $\frac{1}{7366302} a^{15} + \frac{296}{409239} a^{13} - \frac{41215}{7366302} a^{11} - \frac{88933}{818478} a^{9} - \frac{645925}{3683151} a^{7} + \frac{352085}{818478} a^{5} + \frac{6028}{45471} a^{3} - \frac{6788}{15157} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{136}$, which has order $1088$ (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:  \( 348640.934607 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2.C_2\wr C_2^2$ (as 16T394):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 17 conjugacy class representatives for $C_2.C_2\wr C_2^2$
Character table for $C_2.C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{73}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{219}) \), 4.4.63948.1 x2, 4.4.10512.1 x2, \(\Q(\sqrt{3}, \sqrt{73})\), 8.8.588865925376.1

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.18.5$x^{8} + 2 x^{6} + 14 x^{4} + 20 x^{2} + 4$$4$$2$$18$$Q_8:C_2$$[2, 3, 7/2]^{2}$
2.8.16.21$x^{8} + 12 x^{7} + 20 x^{5} + 16 x^{4} + 40 x + 20$$4$$2$$16$$Q_8:C_2$$[2, 3, 3]^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$73$73.8.4.1$x^{8} + 138554 x^{4} - 389017 x^{2} + 4799302729$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
73.8.4.1$x^{8} + 138554 x^{4} - 389017 x^{2} + 4799302729$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$