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Properties

Label 16.0.94287402390...2976.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{28}\cdot 37^{8}$
Root discriminant $20.46$
Ramified primes $2, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $Q_8:C_2^2.D_6$ (as 16T754)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![73, 318, 200, -744, -700, 774, 1312, -104, -778, -14, 320, -36, -90, 22, 16, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 16*x^14 + 22*x^13 - 90*x^12 - 36*x^11 + 320*x^10 - 14*x^9 - 778*x^8 - 104*x^7 + 1312*x^6 + 774*x^5 - 700*x^4 - 744*x^3 + 200*x^2 + 318*x + 73)
 
gp: K = bnfinit(x^16 - 8*x^15 + 16*x^14 + 22*x^13 - 90*x^12 - 36*x^11 + 320*x^10 - 14*x^9 - 778*x^8 - 104*x^7 + 1312*x^6 + 774*x^5 - 700*x^4 - 744*x^3 + 200*x^2 + 318*x + 73, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 16 x^{14} + 22 x^{13} - 90 x^{12} - 36 x^{11} + 320 x^{10} - 14 x^{9} - 778 x^{8} - 104 x^{7} + 1312 x^{6} + 774 x^{5} - 700 x^{4} - 744 x^{3} + 200 x^{2} + 318 x + 73 \)

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:  \(942874023903914622976=2^{28}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.46$
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:  $2, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{23} a^{14} - \frac{10}{23} a^{13} - \frac{9}{23} a^{12} + \frac{7}{23} a^{11} + \frac{2}{23} a^{10} - \frac{10}{23} a^{9} - \frac{3}{23} a^{8} + \frac{5}{23} a^{7} - \frac{9}{23} a^{6} + \frac{11}{23} a^{5} - \frac{7}{23} a^{4} - \frac{6}{23} a^{3} - \frac{5}{23} a^{2} - \frac{4}{23} a - \frac{4}{23}$, $\frac{1}{20872301038178069} a^{15} - \frac{17899293822611}{20872301038178069} a^{14} - \frac{5606255067706655}{20872301038178069} a^{13} + \frac{1092104665819618}{20872301038178069} a^{12} - \frac{1016407846779708}{20872301038178069} a^{11} + \frac{8404446115243721}{20872301038178069} a^{10} - \frac{1313701144469900}{20872301038178069} a^{9} + \frac{2791328835725507}{20872301038178069} a^{8} + \frac{7977605641176510}{20872301038178069} a^{7} + \frac{7131468710549749}{20872301038178069} a^{6} + \frac{8414957674768763}{20872301038178069} a^{5} - \frac{1325757084315534}{20872301038178069} a^{4} + \frac{7889227488038257}{20872301038178069} a^{3} - \frac{4518985841077289}{20872301038178069} a^{2} + \frac{390370406011578}{907491349486003} a + \frac{5666938651498920}{20872301038178069}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{9583562}{430944077} a^{15} + \frac{80201879}{430944077} a^{14} - \frac{184262035}{430944077} a^{13} - \frac{137816723}{430944077} a^{12} + \frac{923542476}{430944077} a^{11} - \frac{45849664}{430944077} a^{10} - \frac{3061991161}{430944077} a^{9} + \frac{1458635449}{430944077} a^{8} + \frac{6820653844}{430944077} a^{7} - \frac{1983614665}{430944077} a^{6} - \frac{11705251547}{430944077} a^{5} - \frac{2205434885}{430944077} a^{4} + \frac{8085337436}{430944077} a^{3} + \frac{4251042914}{430944077} a^{2} - \frac{3696079233}{430944077} a - \frac{1722879136}{430944077} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 24183.7752159 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$Q_8:C_2^2.D_6$ (as 16T754):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 23 conjugacy class representatives for $Q_8:C_2^2.D_6$
Character table for $Q_8:C_2^2.D_6$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.592.1, 8.0.5607424.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$37$37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.4.3.3$x^{4} + 74$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.3.4$x^{4} + 296$$4$$1$$3$$C_4$$[\ ]_{4}$
37.4.2.2$x^{4} - 37 x^{2} + 6845$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$