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Properties

Label 16.0.37682348496...4208.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 17^{11}$
Root discriminant $39.67$
Ramified primes $2, 17$
Class number $8$
Class group $[8]$
Galois group 16T1230

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![272, 0, 0, 0, 3808, 0, 816, 0, 920, 0, -112, 0, -8, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^14 - 8*x^12 - 112*x^10 + 920*x^8 + 816*x^6 + 3808*x^4 + 272)
 
gp: K = bnfinit(x^16 - 4*x^14 - 8*x^12 - 112*x^10 + 920*x^8 + 816*x^6 + 3808*x^4 + 272, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{14} - 8 x^{12} - 112 x^{10} + 920 x^{8} + 816 x^{6} + 3808 x^{4} + 272 \)

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:  \(37682348496175843883614208=2^{40}\cdot 17^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.67$
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:  $2, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{16} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4}$, $\frac{1}{16} a^{9} - \frac{1}{4} a^{4} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{48} a^{10} - \frac{1}{48} a^{8} - \frac{1}{8} a^{7} + \frac{1}{12} a^{6} + \frac{1}{12} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{12}$, $\frac{1}{48} a^{11} - \frac{1}{48} a^{9} + \frac{1}{12} a^{7} + \frac{1}{12} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{12} a - \frac{1}{2}$, $\frac{1}{576} a^{12} + \frac{1}{96} a^{8} + \frac{7}{72} a^{6} + \frac{19}{144} a^{4} + \frac{11}{36} a^{2} - \frac{23}{72}$, $\frac{1}{576} a^{13} + \frac{1}{96} a^{9} + \frac{7}{72} a^{7} - \frac{17}{144} a^{5} + \frac{11}{36} a^{3} + \frac{13}{72} a$, $\frac{1}{42759360} a^{14} - \frac{1301}{6108480} a^{12} + \frac{46243}{7126560} a^{10} - \frac{112793}{21379680} a^{8} - \frac{245641}{3563280} a^{6} + \frac{28929}{395920} a^{4} + \frac{52469}{5344920} a^{2} - \frac{1036607}{5344920}$, $\frac{1}{85518720} a^{15} + \frac{1163}{1527120} a^{13} - \frac{102227}{14253120} a^{11} - \frac{195227}{10689840} a^{9} - \frac{196151}{7126560} a^{7} - \frac{114311}{1781640} a^{5} - \frac{1}{4} a^{4} - \frac{2323051}{10689840} a^{3} - \frac{227177}{668115} a - \frac{1}{2}$

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

Class group and class number

$C_{8}$, which has order $8$

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:  \( \frac{9631}{8551872} a^{14} - \frac{2713}{610848} a^{12} - \frac{13973}{1425312} a^{10} - \frac{266713}{2137968} a^{8} + \frac{244817}{237552} a^{6} + \frac{372283}{356328} a^{4} + \frac{4034831}{1068984} a^{2} + \frac{25367}{534492} \) (order $4$)
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:  \( 1422169.22961 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1230:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 58 conjugacy class representatives for t16n1230 are not computed
Character table for t16n1230 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{17})\), 8.0.23262937088.3

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$2.8.24.79$x^{8} + 16 x^{7} + 16 x^{4} + 80$$8$$1$$24$$Z_8 : Z_8^\times$$[2, 2, 3, 4]^{2}$
2.8.16.65$x^{8} + 4 x^{6} + 28 x^{4} + 20$$8$$1$$16$$QD_{16}$$[2, 2, 5/2]^{2}$
$17$17.8.7.2$x^{8} - 153$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$