Properties

Label 16.0.12213047191...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{8}$
Root discriminant $18.01$
Ramified primes $5, 29$
Class number $2$
Class group $[2]$
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 6*x^14 - 12*x^13 + 42*x^12 + 69*x^11 + 147*x^10 + 168*x^9 + 456*x^8 + 168*x^7 + 147*x^6 + 69*x^5 + 42*x^4 - 12*x^3 + 6*x^2 - x + 1)
 
gp: K = bnfinit(x^16 - x^15 + 6*x^14 - 12*x^13 + 42*x^12 + 69*x^11 + 147*x^10 + 168*x^9 + 456*x^8 + 168*x^7 + 147*x^6 + 69*x^5 + 42*x^4 - 12*x^3 + 6*x^2 - x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 6, -12, 42, 69, 147, 168, 456, 168, 147, 69, 42, -12, 6, -1, 1]);
 

Normalized defining polynomial

\( x^{16} - x^{15} + 6 x^{14} - 12 x^{13} + 42 x^{12} + 69 x^{11} + 147 x^{10} + 168 x^{9} + 456 x^{8} + 168 x^{7} + 147 x^{6} + 69 x^{5} + 42 x^{4} - 12 x^{3} + 6 x^{2} - x + 1 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(122130471914306640625=5^{12}\cdot 29^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $18.01$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $5, 29$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $16$
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{6} a^{5} - \frac{1}{6}$, $\frac{1}{6} a^{6} - \frac{1}{6} a$, $\frac{1}{6} a^{7} - \frac{1}{6} a^{2}$, $\frac{1}{6} a^{8} - \frac{1}{6} a^{3}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{4}$, $\frac{1}{72} a^{10} + \frac{1}{18} a^{5} + \frac{31}{72}$, $\frac{1}{72} a^{11} + \frac{1}{18} a^{6} + \frac{31}{72} a$, $\frac{1}{360} a^{12} + \frac{1}{15} a^{9} - \frac{1}{18} a^{7} - \frac{1}{15} a^{6} + \frac{1}{3} a^{4} + \frac{2}{5} a^{3} + \frac{11}{72} a^{2} - \frac{1}{3} a - \frac{2}{5}$, $\frac{1}{360} a^{13} - \frac{1}{360} a^{10} - \frac{1}{18} a^{8} - \frac{1}{15} a^{7} + \frac{1}{18} a^{5} + \frac{2}{5} a^{4} + \frac{11}{72} a^{3} - \frac{1}{3} a^{2} - \frac{2}{5} a - \frac{11}{72}$, $\frac{1}{360} a^{14} - \frac{1}{360} a^{11} - \frac{1}{18} a^{9} - \frac{1}{15} a^{8} + \frac{1}{18} a^{6} + \frac{1}{15} a^{5} + \frac{11}{72} a^{4} - \frac{1}{3} a^{3} - \frac{2}{5} a^{2} - \frac{11}{72} a + \frac{1}{3}$, $\frac{1}{10800} a^{15} - \frac{1}{900} a^{14} - \frac{1}{900} a^{13} + \frac{7}{1800} a^{11} + \frac{19}{3600} a^{10} + \frac{1}{45} a^{9} + \frac{11}{225} a^{8} + \frac{3}{50} a^{7} - \frac{2}{45} a^{6} + \frac{1}{400} a^{5} - \frac{19}{900} a^{4} - \frac{23}{180} a^{3} + \frac{23}{50} a^{2} - \frac{107}{1800} a + \frac{2291}{10800}$

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

Class group and class number

$C_{2}$, which has order $2$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{7}{48} a^{15} - \frac{7}{40} a^{14} + \frac{7}{8} a^{13} - \frac{341}{180} a^{12} + \frac{63}{10} a^{11} + \frac{147}{16} a^{10} + \frac{91}{5} a^{9} + \frac{91}{5} a^{8} + \frac{518}{9} a^{7} + \frac{63}{10} a^{6} + \frac{259}{80} a^{5} + \frac{7}{8} a^{4} - \frac{7}{40} a^{3} - \frac{1379}{180} a^{2} + \frac{7}{240} \) (order $10$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 3793.72993285 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$C_2^2:C_4$ (as 16T10):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

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, 4.4.4205.1 x2, 4.4.725.1 x2, \(\Q(\zeta_{5})\), 4.0.105125.2, 8.0.11051265625.4, 8.8.442050625.1, 8.0.11051265625.1, 8.0.13140625.1 x2, 8.0.11051265625.5 x2

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

Sibling fields

Degree 8 siblings: 8.0.13140625.1, 8.0.11051265625.5

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

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$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$