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Properties

Label 16.16.7144015978...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{24}\cdot 5^{4}\cdot 13^{8}\cdot 17^{4}$
Root discriminant $30.97$
Ramified primes $2, 5, 13, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4^2:C_2^2$ (as 16T117)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, 650, 1955, -930, -6803, 502, 9648, -1516, -6373, 1810, 1948, -792, -232, 140, 0, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 140*x^13 - 232*x^12 - 792*x^11 + 1948*x^10 + 1810*x^9 - 6373*x^8 - 1516*x^7 + 9648*x^6 + 502*x^5 - 6803*x^4 - 930*x^3 + 1955*x^2 + 650*x + 25)
 
gp: K = bnfinit(x^16 - 8*x^15 + 140*x^13 - 232*x^12 - 792*x^11 + 1948*x^10 + 1810*x^9 - 6373*x^8 - 1516*x^7 + 9648*x^6 + 502*x^5 - 6803*x^4 - 930*x^3 + 1955*x^2 + 650*x + 25, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 140 x^{13} - 232 x^{12} - 792 x^{11} + 1948 x^{10} + 1810 x^{9} - 6373 x^{8} - 1516 x^{7} + 9648 x^{6} + 502 x^{5} - 6803 x^{4} - 930 x^{3} + 1955 x^{2} + 650 x + 25 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(714401597868117852160000=2^{24}\cdot 5^{4}\cdot 13^{8}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.97$
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:  $2, 5, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{105} a^{12} - \frac{2}{35} a^{11} + \frac{1}{15} a^{10} - \frac{1}{7} a^{9} + \frac{16}{105} a^{8} - \frac{8}{21} a^{7} + \frac{22}{105} a^{6} + \frac{2}{15} a^{5} - \frac{37}{105} a^{4} + \frac{1}{105} a^{3} + \frac{47}{105} a^{2} + \frac{5}{21} a + \frac{4}{21}$, $\frac{1}{105} a^{13} + \frac{2}{35} a^{11} - \frac{8}{105} a^{10} - \frac{4}{105} a^{9} - \frac{2}{15} a^{8} - \frac{43}{105} a^{7} - \frac{29}{105} a^{6} + \frac{4}{35} a^{5} + \frac{8}{35} a^{4} - \frac{52}{105} a^{3} - \frac{43}{105} a^{2} + \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{105} a^{14} - \frac{1}{15} a^{11} - \frac{11}{105} a^{10} + \frac{2}{35} a^{9} + \frac{1}{105} a^{8} - \frac{34}{105} a^{7} + \frac{4}{21} a^{6} + \frac{2}{21} a^{5} - \frac{1}{21} a^{4} - \frac{2}{15} a^{3} - \frac{2}{5} a^{2} + \frac{8}{21} a + \frac{4}{21}$, $\frac{1}{558705} a^{15} + \frac{379}{79815} a^{14} - \frac{1334}{558705} a^{13} - \frac{31}{32865} a^{12} - \frac{56366}{558705} a^{11} - \frac{158}{37247} a^{10} + \frac{12923}{111741} a^{9} + \frac{4163}{111741} a^{8} - \frac{8422}{37247} a^{7} - \frac{107249}{558705} a^{6} - \frac{166}{1785} a^{5} + \frac{50243}{111741} a^{4} - \frac{11948}{32865} a^{3} + \frac{52747}{186235} a^{2} + \frac{8775}{37247} a - \frac{15973}{111741}$

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:  $15$
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:  \( 4978711.83037 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2:C_2^2$ (as 16T117):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $C_4^2:C_2^2$
Character table for $C_4^2:C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{26}) \), 4.4.183872.3, \(\Q(\sqrt{2}, \sqrt{13})\), 4.4.183872.2, 8.8.33808912384.1, 8.8.845222809600.2, 8.8.2924646400.1

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

Sibling fields

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$5$5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$13$13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$