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Properties

Label 16.4.46864773471...0000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{24}\cdot 5^{4}\cdot 41^{6}\cdot 97^{2}$
Root discriminant $30.16$
Ramified primes $2, 5, 41, 97$
Class number $2$
Class group $[2]$
Galois group 16T1116

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2209, 9400, 13194, -10276, -12691, 11700, 3754, -5540, 383, 1132, -350, -56, 29, 0, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 2*x^14 + 29*x^12 - 56*x^11 - 350*x^10 + 1132*x^9 + 383*x^8 - 5540*x^7 + 3754*x^6 + 11700*x^5 - 12691*x^4 - 10276*x^3 + 13194*x^2 + 9400*x + 2209)
 
gp: K = bnfinit(x^16 - 4*x^15 + 2*x^14 + 29*x^12 - 56*x^11 - 350*x^10 + 1132*x^9 + 383*x^8 - 5540*x^7 + 3754*x^6 + 11700*x^5 - 12691*x^4 - 10276*x^3 + 13194*x^2 + 9400*x + 2209, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 2 x^{14} + 29 x^{12} - 56 x^{11} - 350 x^{10} + 1132 x^{9} + 383 x^{8} - 5540 x^{7} + 3754 x^{6} + 11700 x^{5} - 12691 x^{4} - 10276 x^{3} + 13194 x^{2} + 9400 x + 2209 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(468647734710831677440000=2^{24}\cdot 5^{4}\cdot 41^{6}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.16$
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:  $2, 5, 41, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{140} a^{14} + \frac{2}{35} a^{13} - \frac{1}{35} a^{12} + \frac{11}{140} a^{11} + \frac{9}{140} a^{10} + \frac{3}{28} a^{9} + \frac{8}{35} a^{8} + \frac{3}{20} a^{7} - \frac{1}{35} a^{6} - \frac{13}{28} a^{5} - \frac{13}{140} a^{4} - \frac{61}{140} a^{3} - \frac{11}{70} a^{2} - \frac{12}{35} a - \frac{13}{140}$, $\frac{1}{2300070527723024682288017180} a^{15} - \frac{7953813336232919293958803}{2300070527723024682288017180} a^{14} - \frac{36746004959739278266834488}{575017631930756170572004295} a^{13} - \frac{47922413761371586455239}{4893767080261754643165994} a^{12} - \frac{13867598111648626454420681}{135298266336648510722824540} a^{11} - \frac{62190130499527140930316956}{575017631930756170572004295} a^{10} + \frac{224155646876494347440076937}{2300070527723024682288017180} a^{9} + \frac{284298525894355240763751617}{1150035263861512341144008590} a^{8} - \frac{79899635497051171601236099}{460014105544604936457603436} a^{7} + \frac{5920789652441312296827793}{33824566584162127680706135} a^{6} - \frac{478238600444144568627395249}{1150035263861512341144008590} a^{5} - \frac{1045559224942078490798186973}{2300070527723024682288017180} a^{4} - \frac{939016488184662972794630421}{2300070527723024682288017180} a^{3} - \frac{3910608725588861150425491}{26745006136314240491721130} a^{2} + \frac{97400994678728677611505923}{230007052772302468228801718} a + \frac{7389560364125789771285227}{24468835401308773215829970}$

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

Class group and class number

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

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 315177.150295 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1116:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 76 conjugacy class representatives for t16n1116 are not computed
Character table for t16n1116 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.2624.1, 8.4.27383140352.1, 8.4.684578508800.2, 8.4.172134400.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.12.1$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$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.8.6.2$x^{8} + 943 x^{4} + 242064$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$97$$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$