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Properties

Label 16.0.27173700865...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{12}\cdot 19^{8}$
Root discriminant $29.15$
Ramified primes $2, 5, 19$
Class number $16$ (GRH)
Class group $[2, 8]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![390625, 0, 140625, 0, 35000, 0, 6975, 0, 1111, 0, 279, 0, 56, 0, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 9*x^14 + 56*x^12 + 279*x^10 + 1111*x^8 + 6975*x^6 + 35000*x^4 + 140625*x^2 + 390625)
 
gp: K = bnfinit(x^16 + 9*x^14 + 56*x^12 + 279*x^10 + 1111*x^8 + 6975*x^6 + 35000*x^4 + 140625*x^2 + 390625, 1)
 

Normalized defining polynomial

\( x^{16} + 9 x^{14} + 56 x^{12} + 279 x^{10} + 1111 x^{8} + 6975 x^{6} + 35000 x^{4} + 140625 x^{2} + 390625 \)

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:  \(271737008656000000000000=2^{16}\cdot 5^{12}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.15$
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:  $2, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(380=2^{2}\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{380}(1,·)$, $\chi_{380}(267,·)$, $\chi_{380}(77,·)$, $\chi_{380}(341,·)$, $\chi_{380}(343,·)$, $\chi_{380}(153,·)$, $\chi_{380}(37,·)$, $\chi_{380}(227,·)$, $\chi_{380}(229,·)$, $\chi_{380}(39,·)$, $\chi_{380}(303,·)$, $\chi_{380}(113,·)$, $\chi_{380}(379,·)$, $\chi_{380}(151,·)$, $\chi_{380}(189,·)$, $\chi_{380}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{7} + \frac{1}{5} a^{5} - \frac{1}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{27775} a^{10} - \frac{6}{25} a^{8} - \frac{4}{25} a^{6} - \frac{11}{25} a^{4} + \frac{1}{25} a^{2} + \frac{279}{1111}$, $\frac{1}{138875} a^{11} - \frac{6}{125} a^{9} + \frac{46}{125} a^{7} - \frac{61}{125} a^{5} + \frac{51}{125} a^{3} - \frac{1943}{5555} a$, $\frac{1}{694375} a^{12} + \frac{9}{694375} a^{10} + \frac{121}{625} a^{8} - \frac{11}{625} a^{6} + \frac{1}{625} a^{4} + \frac{279}{27775} a^{2} + \frac{56}{1111}$, $\frac{1}{3471875} a^{13} + \frac{9}{3471875} a^{11} + \frac{121}{3125} a^{9} - \frac{11}{3125} a^{7} + \frac{1}{3125} a^{5} + \frac{279}{138875} a^{3} + \frac{56}{5555} a$, $\frac{1}{17359375} a^{14} + \frac{9}{17359375} a^{12} + \frac{56}{17359375} a^{10} - \frac{3136}{15625} a^{8} + \frac{1}{15625} a^{6} + \frac{279}{694375} a^{4} + \frac{56}{27775} a^{2} + \frac{9}{1111}$, $\frac{1}{86796875} a^{15} + \frac{9}{86796875} a^{13} + \frac{56}{86796875} a^{11} - \frac{3136}{78125} a^{9} - \frac{31249}{78125} a^{7} + \frac{1389029}{3471875} a^{5} - \frac{55494}{138875} a^{3} + \frac{2231}{5555} a$

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

Class group and class number

$C_{2}\times C_{8}$, which has order $16$ (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:  \( \frac{9}{86796875} a^{15} + \frac{56}{86796875} a^{13} + \frac{279}{86796875} a^{11} + \frac{1}{78125} a^{9} + \frac{284}{78125} a^{7} + \frac{56}{138875} a^{5} + \frac{9}{5555} a^{3} + \frac{5}{1111} a \) (order $20$)
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:  \( 123279.066823 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-95}) \), \(\Q(\sqrt{95}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{19}) \), \(\Q(i, \sqrt{95})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{19})\), \(\Q(\sqrt{5}, \sqrt{-19})\), \(\Q(\sqrt{-5}, \sqrt{19})\), \(\Q(\sqrt{5}, \sqrt{19})\), \(\Q(\sqrt{-5}, \sqrt{-19})\), \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), 4.4.45125.1, 4.0.722000.3, 8.0.20851360000.1, \(\Q(\zeta_{20})\), 8.0.521284000000.2, 8.0.2036265625.1, 8.0.521284000000.1, 8.0.521284000000.3, 8.8.521284000000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$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}$
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$