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Properties

Label 16.0.579121000000000000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{12}\cdot 5^{12}\cdot 761^{2}$
Root discriminant $12.89$
Ramified primes $2, 5, 761$
Class number $1$
Class group Trivial
Galois group 16T1385

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 42, -103, 186, -287, 370, -403, 403, -345, 258, -175, 102, -49, 20, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 20*x^14 - 49*x^13 + 102*x^12 - 175*x^11 + 258*x^10 - 345*x^9 + 403*x^8 - 403*x^7 + 370*x^6 - 287*x^5 + 186*x^4 - 103*x^3 + 42*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^16 - 6*x^15 + 20*x^14 - 49*x^13 + 102*x^12 - 175*x^11 + 258*x^10 - 345*x^9 + 403*x^8 - 403*x^7 + 370*x^6 - 287*x^5 + 186*x^4 - 103*x^3 + 42*x^2 - 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 20 x^{14} - 49 x^{13} + 102 x^{12} - 175 x^{11} + 258 x^{10} - 345 x^{9} + 403 x^{8} - 403 x^{7} + 370 x^{6} - 287 x^{5} + 186 x^{4} - 103 x^{3} + 42 x^{2} - 10 x + 1 \)

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:  \(579121000000000000=2^{12}\cdot 5^{12}\cdot 761^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.89$
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:  $2, 5, 761$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{10} a^{14} - \frac{1}{5} a^{13} - \frac{1}{5} a^{12} + \frac{1}{10} a^{11} - \frac{1}{10} a^{10} - \frac{3}{10} a^{9} - \frac{1}{2} a^{8} + \frac{1}{5} a^{7} + \frac{1}{10} a^{6} + \frac{3}{10} a^{5} + \frac{3}{10} a^{4} + \frac{3}{10} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{10}$, $\frac{1}{710} a^{15} - \frac{12}{355} a^{14} + \frac{97}{710} a^{13} - \frac{2}{71} a^{12} - \frac{124}{355} a^{11} + \frac{29}{710} a^{10} - \frac{132}{355} a^{9} - \frac{104}{355} a^{8} + \frac{121}{355} a^{7} - \frac{72}{355} a^{6} - \frac{233}{710} a^{5} + \frac{1}{355} a^{4} + \frac{15}{71} a^{3} + \frac{37}{710} a^{2} + \frac{43}{355} a + \frac{217}{710}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{86211}{710} a^{15} + \frac{240681}{355} a^{14} - \frac{761909}{355} a^{13} + \frac{3590029}{710} a^{12} - \frac{7299199}{710} a^{11} + \frac{12048633}{710} a^{10} - \frac{3445485}{142} a^{9} + \frac{11285869}{355} a^{8} - \frac{25346641}{710} a^{7} + \frac{24191367}{710} a^{6} - \frac{21825983}{710} a^{5} + \frac{15653407}{710} a^{4} - \frac{4758488}{355} a^{3} + \frac{2457528}{355} a^{2} - \frac{1573189}{710} a + \frac{20711}{71} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 954.558183076 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1385:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 71 conjugacy class representatives for t16n1385 are not computed
Character table for t16n1385 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.11890625.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.12.5$x^{8} + 6 x^{6} + 8 x^{5} + 80$$2$$4$$12$$C_8$$[3]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$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}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
761Data not computed