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Properties

Label 16.0.16710207732...3216.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{12}\cdot 37^{4}$
Root discriminant $15.90$
Ramified primes $2, 3, 37$
Class number $1$
Class group Trivial
Galois group $C_2^4.C_2^3$ (as 16T373)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![169, 0, -322, 0, 469, 0, -430, 0, 280, 0, -142, 0, 49, 0, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 10*x^14 + 49*x^12 - 142*x^10 + 280*x^8 - 430*x^6 + 469*x^4 - 322*x^2 + 169)
 
gp: K = bnfinit(x^16 - 10*x^14 + 49*x^12 - 142*x^10 + 280*x^8 - 430*x^6 + 469*x^4 - 322*x^2 + 169, 1)
 

Normalized defining polynomial

\( x^{16} - 10 x^{14} + 49 x^{12} - 142 x^{10} + 280 x^{8} - 430 x^{6} + 469 x^{4} - 322 x^{2} + 169 \)

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:  \(16710207732203913216=2^{24}\cdot 3^{12}\cdot 37^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.90$
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:  $2, 3, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{6} a^{8} + \frac{1}{6} a^{6} + \frac{1}{6} a^{2} + \frac{1}{6}$, $\frac{1}{12} a^{9} - \frac{1}{6} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{12} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a - \frac{1}{4}$, $\frac{1}{12} a^{10} - \frac{1}{4} a^{7} + \frac{1}{6} a^{6} + \frac{1}{12} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{3}$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{8} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{12} a^{5} - \frac{1}{2} a^{4} - \frac{1}{12} a^{2} - \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{24} a^{12} + \frac{1}{12} a^{6} - \frac{1}{2} a^{4} + \frac{1}{24}$, $\frac{1}{24} a^{13} + \frac{1}{12} a^{7} - \frac{1}{2} a^{5} + \frac{1}{24} a$, $\frac{1}{720} a^{14} + \frac{13}{720} a^{12} - \frac{1}{60} a^{10} + \frac{1}{360} a^{8} - \frac{1}{4} a^{7} - \frac{77}{360} a^{6} + \frac{7}{30} a^{4} - \frac{1}{2} a^{3} - \frac{287}{720} a^{2} - \frac{1}{4} a + \frac{157}{720}$, $\frac{1}{18720} a^{15} - \frac{1}{1440} a^{14} - \frac{257}{18720} a^{13} + \frac{17}{1440} a^{12} - \frac{3}{130} a^{11} - \frac{1}{30} a^{10} + \frac{241}{9360} a^{9} - \frac{1}{720} a^{8} + \frac{1453}{9360} a^{7} - \frac{133}{720} a^{6} - \frac{17}{520} a^{5} + \frac{11}{120} a^{4} + \frac{8113}{18720} a^{3} - \frac{433}{1440} a^{2} + \frac{8167}{18720} a + \frac{473}{1440}$

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:  \( -\frac{17}{720} a^{14} + \frac{139}{720} a^{12} - \frac{4}{5} a^{10} + \frac{643}{360} a^{8} - \frac{971}{360} a^{6} + \frac{59}{20} a^{4} - \frac{1001}{720} a^{2} - \frac{29}{720} \) (order $12$)
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:  \( 9743.09601137 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T373):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), 4.2.15984.2, 4.2.15984.3, \(\Q(\zeta_{12})\), 8.0.255488256.8, 8.0.110481408.2, 8.0.12275712.1

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.12.20$x^{8} + 8 x^{6} + 12 x^{4} + 80$$4$$2$$12$$C_2^3: C_4$$[2, 2, 2]^{4}$
2.8.12.20$x^{8} + 8 x^{6} + 12 x^{4} + 80$$4$$2$$12$$C_2^3: C_4$$[2, 2, 2]^{4}$
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$37$37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$