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Properties

Label 16.0.39413667678...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 71^{8}$
Root discriminant $34.45$
Ramified primes $5, 71$
Class number $14$ (GRH)
Class group $[14]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, 0, -775, 0, 9375, 0, -8825, 0, 21060, 0, 4495, 0, 475, 0, 25, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 25*x^14 + 475*x^12 + 4495*x^10 + 21060*x^8 - 8825*x^6 + 9375*x^4 - 775*x^2 + 25)
 
gp: K = bnfinit(x^16 + 25*x^14 + 475*x^12 + 4495*x^10 + 21060*x^8 - 8825*x^6 + 9375*x^4 - 775*x^2 + 25, 1)
 

Normalized defining polynomial

\( x^{16} + 25 x^{14} + 475 x^{12} + 4495 x^{10} + 21060 x^{8} - 8825 x^{6} + 9375 x^{4} - 775 x^{2} + 25 \)

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:  \(3941366767857427978515625=5^{14}\cdot 71^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.45$
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:  $5, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{10} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{20} a^{10} - \frac{1}{4}$, $\frac{1}{40} a^{11} - \frac{1}{40} a^{10} - \frac{1}{2} a^{4} - \frac{1}{8} a - \frac{3}{8}$, $\frac{1}{80} a^{12} - \frac{1}{80} a^{10} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{16} a^{2} - \frac{1}{2} a + \frac{1}{16}$, $\frac{1}{160} a^{13} - \frac{1}{160} a^{12} - \frac{1}{160} a^{11} + \frac{1}{160} a^{10} + \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{3}{8} a^{4} - \frac{1}{32} a^{3} - \frac{15}{32} a^{2} + \frac{1}{32} a - \frac{1}{32}$, $\frac{1}{7708593265600} a^{14} + \frac{3984946913}{770859326560} a^{12} + \frac{31402310577}{1541718653120} a^{10} + \frac{10653854421}{385429663280} a^{8} + \frac{21241758857}{192714831640} a^{6} - \frac{74548196057}{308343730624} a^{4} - \frac{1}{2} a^{3} + \frac{49115233559}{154171865312} a^{2} - \frac{1}{2} a - \frac{73659708865}{308343730624}$, $\frac{1}{15417186531200} a^{15} - \frac{1}{15417186531200} a^{14} + \frac{3984946913}{1541718653120} a^{13} - \frac{3984946913}{1541718653120} a^{12} + \frac{31402310577}{3083437306240} a^{11} - \frac{31402310577}{3083437306240} a^{10} - \frac{27889111907}{770859326560} a^{9} + \frac{27889111907}{770859326560} a^{8} - \frac{75115656963}{385429663280} a^{7} + \frac{75115656963}{385429663280} a^{6} - \frac{74548196057}{616687461248} a^{5} - \frac{233795534567}{616687461248} a^{4} - \frac{105056631753}{308343730624} a^{3} - \frac{49115233559}{308343730624} a^{2} + \frac{234684021759}{616687461248} a - \frac{234684021759}{616687461248}$

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

Class group and class number

$C_{14}$, which has order $14$ (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:  \( -\frac{34262239}{9635741582} a^{14} - \frac{34374884743}{385429663280} a^{12} - \frac{653823527577}{385429663280} a^{10} - \frac{388405051507}{24089353955} a^{8} - \frac{1469026558175}{19271483164} a^{6} + \frac{480524539077}{19271483164} a^{4} - \frac{2467041119961}{77085932656} a^{2} + \frac{81124781657}{77085932656} \) (order $10$)
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:  \( 144974.394976 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{-71}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-355}) \), \(\Q(\zeta_{5})\), 4.4.630125.1, \(\Q(\sqrt{5}, \sqrt{-71})\), 8.0.393828125.1 x2, 8.4.1985287578125.1 x2, 8.0.397057515625.2

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$71$71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$