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Properties

Label 16.0.86380562306...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{12}$
Root discriminant $41.79$
Ramified primes $5, 29$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1711, 9889, 19169, 19024, 28871, 3012, 27841, -8493, 18077, -7367, 6096, -1822, 861, -154, 49, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 49*x^14 - 154*x^13 + 861*x^12 - 1822*x^11 + 6096*x^10 - 7367*x^9 + 18077*x^8 - 8493*x^7 + 27841*x^6 + 3012*x^5 + 28871*x^4 + 19024*x^3 + 19169*x^2 + 9889*x + 1711)
 
gp: K = bnfinit(x^16 - 4*x^15 + 49*x^14 - 154*x^13 + 861*x^12 - 1822*x^11 + 6096*x^10 - 7367*x^9 + 18077*x^8 - 8493*x^7 + 27841*x^6 + 3012*x^5 + 28871*x^4 + 19024*x^3 + 19169*x^2 + 9889*x + 1711, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 49 x^{14} - 154 x^{13} + 861 x^{12} - 1822 x^{11} + 6096 x^{10} - 7367 x^{9} + 18077 x^{8} - 8493 x^{7} + 27841 x^{6} + 3012 x^{5} + 28871 x^{4} + 19024 x^{3} + 19169 x^{2} + 9889 x + 1711 \)

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:  \(86380562306022715087890625=5^{12}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.79$
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:  $5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{10} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{5} - \frac{1}{10} a^{4} + \frac{1}{5} a^{3} - \frac{1}{2} a^{2} - \frac{3}{10} a + \frac{1}{10}$, $\frac{1}{10} a^{9} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{2} a^{5} - \frac{1}{10} a^{3} - \frac{3}{10} a^{2} - \frac{1}{2} a + \frac{1}{5}$, $\frac{1}{10} a^{10} - \frac{1}{2} a^{6} + \frac{1}{5} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{2}{5}$, $\frac{1}{10} a^{11} - \frac{1}{2} a^{7} + \frac{1}{5} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{2}{5} a$, $\frac{1}{260} a^{12} - \frac{3}{260} a^{11} + \frac{2}{65} a^{10} - \frac{2}{65} a^{9} - \frac{2}{65} a^{8} - \frac{7}{52} a^{7} + \frac{1}{52} a^{6} - \frac{27}{65} a^{5} + \frac{123}{260} a^{4} + \frac{37}{260} a^{3} - \frac{9}{20} a^{2} + \frac{37}{260} a - \frac{37}{260}$, $\frac{1}{260} a^{13} - \frac{1}{260} a^{11} - \frac{1}{26} a^{10} - \frac{3}{130} a^{9} - \frac{7}{260} a^{8} - \frac{12}{65} a^{7} - \frac{3}{52} a^{6} + \frac{33}{260} a^{5} - \frac{9}{65} a^{4} - \frac{29}{130} a^{3} - \frac{1}{130} a^{2} + \frac{12}{65} a - \frac{111}{260}$, $\frac{1}{76700} a^{14} - \frac{53}{38350} a^{13} + \frac{28}{19175} a^{12} + \frac{2981}{76700} a^{11} + \frac{1213}{38350} a^{10} + \frac{427}{76700} a^{9} + \frac{909}{38350} a^{8} + \frac{1449}{2950} a^{7} - \frac{8332}{19175} a^{6} + \frac{6888}{19175} a^{5} + \frac{549}{76700} a^{4} - \frac{7899}{76700} a^{3} - \frac{2179}{5900} a^{2} + \frac{7526}{19175} a - \frac{209}{1300}$, $\frac{1}{7214768850997992700} a^{15} + \frac{2916169809879}{721476885099799270} a^{14} - \frac{195629257669837}{138745554826884475} a^{13} + \frac{6670388589503619}{3607384425498996350} a^{12} - \frac{22816414749169283}{7214768850997992700} a^{11} - \frac{355871743196534217}{7214768850997992700} a^{10} + \frac{8312609796680099}{721476885099799270} a^{9} - \frac{15955560367654727}{1803692212749498175} a^{8} - \frac{3006393507569108629}{7214768850997992700} a^{7} + \frac{1997253586012386309}{7214768850997992700} a^{6} - \frac{2830227485797339309}{7214768850997992700} a^{5} - \frac{98453608758279249}{360738442549899635} a^{4} - \frac{436011585511245189}{1803692212749498175} a^{3} - \frac{3592287032334633383}{7214768850997992700} a^{2} - \frac{208067003590201498}{1803692212749498175} a - \frac{2159951693996749}{122284217813525300}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 249792.59056 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{29})\), 4.0.105125.1 x2, 4.0.3625.1 x2, 4.0.609725.2, 4.0.24389.1, 8.0.11051265625.4, 8.0.371764575625.1, 8.8.9294114390625.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.4.3.2$x^{4} - 116$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.2$x^{4} - 116$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.2$x^{4} - 116$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.2$x^{4} - 116$$4$$1$$3$$C_4$$[\ ]_{4}$