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Properties

Label 16.8.33304100518...4128.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{39}\cdot 7^{12}\cdot 17^{8}\cdot 89^{4}$
Root discriminant $295.23$
Ramified primes $2, 7, 17, 89$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 16T1581

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![314703872, 0, 1730871296, 0, -155595776, 0, -88510464, 0, 5439392, 0, 173264, 0, -4242, 0, -56, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 56*x^14 - 4242*x^12 + 173264*x^10 + 5439392*x^8 - 88510464*x^6 - 155595776*x^4 + 1730871296*x^2 + 314703872)
 
gp: K = bnfinit(x^16 - 56*x^14 - 4242*x^12 + 173264*x^10 + 5439392*x^8 - 88510464*x^6 - 155595776*x^4 + 1730871296*x^2 + 314703872, 1)
 

Normalized defining polynomial

\( x^{16} - 56 x^{14} - 4242 x^{12} + 173264 x^{10} + 5439392 x^{8} - 88510464 x^{6} - 155595776 x^{4} + 1730871296 x^{2} + 314703872 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3330410051873424453724414210438886064128=2^{39}\cdot 7^{12}\cdot 17^{8}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $295.23$
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:  $2, 7, 17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{14} a^{4}$, $\frac{1}{28} a^{5} - \frac{1}{2} a$, $\frac{1}{56} a^{6} + \frac{1}{4} a^{2}$, $\frac{1}{112} a^{7} + \frac{1}{8} a^{3}$, $\frac{1}{1568} a^{8} + \frac{1}{112} a^{4}$, $\frac{1}{3136} a^{9} + \frac{1}{224} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{12544} a^{10} - \frac{1}{3136} a^{8} - \frac{1}{128} a^{6} - \frac{5}{224} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{25088} a^{11} - \frac{1}{6272} a^{9} - \frac{1}{256} a^{7} - \frac{5}{448} a^{5} + \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{2107392} a^{12} - \frac{1}{25088} a^{10} - \frac{1}{6272} a^{9} - \frac{1}{21504} a^{8} - \frac{31}{5376} a^{6} + \frac{1}{64} a^{5} + \frac{11}{672} a^{4} + \frac{1}{4} a^{3} - \frac{1}{24} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{4214784} a^{13} - \frac{1}{50176} a^{11} - \frac{1}{43008} a^{9} - \frac{1}{3136} a^{8} - \frac{31}{10752} a^{7} + \frac{11}{1344} a^{5} + \frac{1}{32} a^{4} + \frac{23}{48} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a$, $\frac{1}{2818947363720585216} a^{14} + \frac{6004642165}{704736840930146304} a^{12} + \frac{4493976830921}{201353383122898944} a^{10} - \frac{1}{6272} a^{9} - \frac{15040399313087}{50338345780724736} a^{8} - \frac{154836763887}{74908252649888} a^{6} + \frac{1}{64} a^{5} + \frac{376511435215}{11827618839456} a^{4} + \frac{1}{4} a^{3} - \frac{24824316849}{1337647368748} a^{2} - \frac{1}{2} a - \frac{183927147178}{1003235526561}$, $\frac{1}{11275789454882340864} a^{15} - \frac{54734533337}{469824560620097536} a^{13} + \frac{12519861043409}{805413532491595776} a^{11} - \frac{6349758208889}{100676691561449472} a^{9} + \frac{6650684774509}{7191192254389248} a^{7} + \frac{45726144803}{11827618839456} a^{5} - \frac{15866302483883}{32103536849952} a^{3} - \frac{71062569296}{334411842187} a$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 221387435824000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1581:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 94 conjugacy class representatives for t16n1581 are not computed
Character table for t16n1581 is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.113288.1, 4.4.205768.1, 4.4.80661056.1, 8.8.6506205955035136.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.4.11.16$x^{4} + 14$$4$$1$$11$$D_{4}$$[2, 3, 4]$
2.4.11.15$x^{4} + 30$$4$$1$$11$$D_{4}$$[2, 3, 4]$
2.4.11.4$x^{4} + 12 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
7Data not computed
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$89$89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.8.4.1$x^{8} + 427734 x^{4} - 704969 x^{2} + 45739093689$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$