Properties

Label 16.0.60516574977...0000.6
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 7^{8}$
Root discriminant $30.65$
Ramified primes $2, 3, 5, 7$
Class number $32$ (GRH)
Class group $[4, 8]$ (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, -171875, 0, 60000, 0, -19525, 0, 6191, 0, -781, 0, 96, 0, -11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 11*x^14 + 96*x^12 - 781*x^10 + 6191*x^8 - 19525*x^6 + 60000*x^4 - 171875*x^2 + 390625)
 
gp: K = bnfinit(x^16 - 11*x^14 + 96*x^12 - 781*x^10 + 6191*x^8 - 19525*x^6 + 60000*x^4 - 171875*x^2 + 390625, 1)
 

Normalized defining polynomial

\( x^{16} - 11 x^{14} + 96 x^{12} - 781 x^{10} + 6191 x^{8} - 19525 x^{6} + 60000 x^{4} - 171875 x^{2} + 390625 \)

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:  \(605165749776000000000000=2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 7^{8}\)
magma: Discriminant(K);
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.65$
magma: Abs(Discriminant(K))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(K));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(420=2^{2}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{420}(1,·)$, $\chi_{420}(209,·)$, $\chi_{420}(337,·)$, $\chi_{420}(83,·)$, $\chi_{420}(251,·)$, $\chi_{420}(419,·)$, $\chi_{420}(293,·)$, $\chi_{420}(167,·)$, $\chi_{420}(41,·)$, $\chi_{420}(43,·)$, $\chi_{420}(211,·)$, $\chi_{420}(169,·)$, $\chi_{420}(377,·)$, $\chi_{420}(379,·)$, $\chi_{420}(253,·)$, $\chi_{420}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{7} + \frac{1}{5} a^{5} - \frac{1}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{154775} a^{10} + \frac{4}{25} a^{8} + \frac{6}{25} a^{6} + \frac{9}{25} a^{4} + \frac{1}{25} a^{2} - \frac{781}{6191}$, $\frac{1}{773875} a^{11} + \frac{4}{125} a^{9} + \frac{56}{125} a^{7} + \frac{34}{125} a^{5} - \frac{24}{125} a^{3} + \frac{1082}{6191} a$, $\frac{1}{3869375} a^{12} - \frac{11}{3869375} a^{10} - \frac{44}{625} a^{8} - \frac{241}{625} a^{6} + \frac{1}{625} a^{4} - \frac{781}{154775} a^{2} + \frac{96}{6191}$, $\frac{1}{19346875} a^{13} - \frac{11}{19346875} a^{11} - \frac{44}{3125} a^{9} + \frac{1009}{3125} a^{7} - \frac{624}{3125} a^{5} + \frac{153994}{773875} a^{3} - \frac{1219}{6191} a$, $\frac{1}{96734375} a^{14} - \frac{11}{96734375} a^{12} + \frac{96}{96734375} a^{10} - \frac{2741}{15625} a^{8} + \frac{1}{15625} a^{6} - \frac{781}{3869375} a^{4} + \frac{96}{154775} a^{2} - \frac{11}{6191}$, $\frac{1}{483671875} a^{15} - \frac{11}{483671875} a^{13} + \frac{96}{483671875} a^{11} - \frac{2741}{78125} a^{9} + \frac{15626}{78125} a^{7} - \frac{3870156}{19346875} a^{5} + \frac{154871}{773875} a^{3} - \frac{6202}{30955} a$

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

Class group and class number

$C_{4}\times C_{8}$, which has order $32$ (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:  \( -\frac{3311}{483671875} a^{15} + \frac{28896}{483671875} a^{13} - \frac{235081}{483671875} a^{11} + \frac{301}{78125} a^{9} - \frac{2286}{78125} a^{7} + \frac{28896}{773875} a^{5} - \frac{3311}{30955} a^{3} + \frac{1505}{6191} a \) (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:  \( 107520.786508 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{21}) \), \(\Q(i, \sqrt{105})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{21})\), \(\Q(\sqrt{5}, \sqrt{-21})\), \(\Q(\sqrt{-5}, \sqrt{21})\), \(\Q(\sqrt{5}, \sqrt{21})\), \(\Q(\sqrt{-5}, \sqrt{-21})\), 4.4.882000.1, 4.0.55125.1, \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), 8.0.31116960000.8, 8.0.777924000000.8, \(\Q(\zeta_{20})\), 8.0.777924000000.10, 8.0.777924000000.6, 8.8.777924000000.2, 8.0.3038765625.3

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 R R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{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}$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$