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Properties

Label 16.8.12478436359...5625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{12}\cdot 59^{10}$
Root discriminant $42.76$
Ramified primes $5, 59$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\wr C_4$ (as 16T158)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25775, -1100, -50900, 63150, 57685, -72935, -27190, 33470, 4391, -7523, 398, 744, -215, 14, 18, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 18*x^14 + 14*x^13 - 215*x^12 + 744*x^11 + 398*x^10 - 7523*x^9 + 4391*x^8 + 33470*x^7 - 27190*x^6 - 72935*x^5 + 57685*x^4 + 63150*x^3 - 50900*x^2 - 1100*x + 25775)
 
gp: K = bnfinit(x^16 - 8*x^15 + 18*x^14 + 14*x^13 - 215*x^12 + 744*x^11 + 398*x^10 - 7523*x^9 + 4391*x^8 + 33470*x^7 - 27190*x^6 - 72935*x^5 + 57685*x^4 + 63150*x^3 - 50900*x^2 - 1100*x + 25775, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 18 x^{14} + 14 x^{13} - 215 x^{12} + 744 x^{11} + 398 x^{10} - 7523 x^{9} + 4391 x^{8} + 33470 x^{7} - 27190 x^{6} - 72935 x^{5} + 57685 x^{4} + 63150 x^{3} - 50900 x^{2} - 1100 x + 25775 \)

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:  \(124784363598789404541015625=5^{12}\cdot 59^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.76$
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:  $5, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{4}$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{5}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{6}$, $\frac{1}{15} a^{12} + \frac{1}{15} a^{10} - \frac{1}{15} a^{9} + \frac{1}{15} a^{8} + \frac{1}{3} a^{7} + \frac{1}{15} a^{6} + \frac{1}{3} a^{5} + \frac{7}{15} a^{4} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{15} a^{13} + \frac{1}{15} a^{11} - \frac{1}{15} a^{10} + \frac{1}{15} a^{9} - \frac{1}{15} a^{8} - \frac{1}{3} a^{7} - \frac{1}{15} a^{6} + \frac{1}{15} a^{5} - \frac{2}{5} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{50546090325} a^{14} - \frac{7}{50546090325} a^{13} + \frac{414537702}{16848696775} a^{12} + \frac{529507904}{10109218065} a^{11} + \frac{792958688}{10109218065} a^{10} - \frac{1971337696}{50546090325} a^{9} + \frac{3923551162}{50546090325} a^{8} + \frac{15147538469}{50546090325} a^{7} - \frac{3007023154}{10109218065} a^{6} - \frac{325114561}{3369739355} a^{5} + \frac{2326858204}{10109218065} a^{4} - \frac{173550541}{10109218065} a^{3} - \frac{30632909}{673947871} a^{2} - \frac{540517358}{2021843613} a + \frac{289844399}{2021843613}$, $\frac{1}{1899976989226425} a^{15} + \frac{18787}{1899976989226425} a^{14} + \frac{29486462837798}{1899976989226425} a^{13} - \frac{13874596721637}{633325663075475} a^{12} - \frac{18168110411011}{379995397845285} a^{11} - \frac{52199952392401}{1899976989226425} a^{10} - \frac{135131770252802}{1899976989226425} a^{9} - \frac{145295538694678}{1899976989226425} a^{8} - \frac{510711995526164}{1899976989226425} a^{7} - \frac{44160790839693}{126665132615095} a^{6} - \frac{36834270514433}{75999079569057} a^{5} + \frac{6487092288233}{379995397845285} a^{4} - \frac{36935973735704}{379995397845285} a^{3} + \frac{25375802387540}{75999079569057} a^{2} + \frac{29221254668548}{75999079569057} a - \frac{10515670300042}{25333026523019}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 19039970.8352 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T158):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.435125.1, 4.2.1475.1, 4.2.7375.1, 8.6.11170692171875.1 x2, 8.4.37866753125.1 x2, 8.4.189333765625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ R

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$59$59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.8.6.1$x^{8} - 59 x^{4} + 55696$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$