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Properties

Label 16.0.22178360058...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{12}\cdot 61^{4}$
Root discriminant $16.19$
Ramified primes $3, 5, 61$
Class number $1$
Class group Trivial
Galois group $C_2^4:C_2^2.C_2$ (as 16T317)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![961, -403, 2127, 86, 2443, 20, 2701, -627, 2018, -564, 854, -202, 198, -33, 23, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 23*x^14 - 33*x^13 + 198*x^12 - 202*x^11 + 854*x^10 - 564*x^9 + 2018*x^8 - 627*x^7 + 2701*x^6 + 20*x^5 + 2443*x^4 + 86*x^3 + 2127*x^2 - 403*x + 961)
 
gp: K = bnfinit(x^16 - 2*x^15 + 23*x^14 - 33*x^13 + 198*x^12 - 202*x^11 + 854*x^10 - 564*x^9 + 2018*x^8 - 627*x^7 + 2701*x^6 + 20*x^5 + 2443*x^4 + 86*x^3 + 2127*x^2 - 403*x + 961, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 23 x^{14} - 33 x^{13} + 198 x^{12} - 202 x^{11} + 854 x^{10} - 564 x^{9} + 2018 x^{8} - 627 x^{7} + 2701 x^{6} + 20 x^{5} + 2443 x^{4} + 86 x^{3} + 2127 x^{2} - 403 x + 961 \)

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:  \(22178360058837890625=3^{8}\cdot 5^{12}\cdot 61^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.19$
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:  $3, 5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{31} a^{14} + \frac{1}{31} a^{13} - \frac{5}{31} a^{12} + \frac{14}{31} a^{11} - \frac{8}{31} a^{10} - \frac{9}{31} a^{9} - \frac{10}{31} a^{8} - \frac{5}{31} a^{7} - \frac{12}{31} a^{6} - \frac{12}{31} a^{5} - \frac{1}{31} a^{4} - \frac{14}{31} a^{3} + \frac{14}{31} a^{2} + \frac{4}{31} a$, $\frac{1}{458940543697806301} a^{15} + \frac{2057895900735708}{458940543697806301} a^{14} - \frac{35325285620741523}{458940543697806301} a^{13} + \frac{40408582463431605}{458940543697806301} a^{12} + \frac{161775156646237305}{458940543697806301} a^{11} + \frac{12475068457112351}{458940543697806301} a^{10} - \frac{23511083983977446}{458940543697806301} a^{9} - \frac{163015288276160101}{458940543697806301} a^{8} + \frac{72290959756952370}{458940543697806301} a^{7} + \frac{12902031664585781}{458940543697806301} a^{6} + \frac{75202449264263000}{458940543697806301} a^{5} + \frac{694699213100055}{458940543697806301} a^{4} - \frac{113632058562000738}{458940543697806301} a^{3} + \frac{52351746744678087}{458940543697806301} a^{2} - \frac{60225851081529634}{458940543697806301} a + \frac{20849661404732}{14804533667671171}$

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:  \( -\frac{5208856525147705}{458940543697806301} a^{15} - \frac{765005871130132}{458940543697806301} a^{14} - \frac{99321946429059167}{458940543697806301} a^{13} - \frac{58479074293319719}{458940543697806301} a^{12} - \frac{726099432834564984}{458940543697806301} a^{11} - \frac{655202294549778626}{458940543697806301} a^{10} - \frac{2751883776062449464}{458940543697806301} a^{9} - \frac{3108789335285151327}{458940543697806301} a^{8} - \frac{6088109079260837715}{458940543697806301} a^{7} - \frac{7509494898150799099}{458940543697806301} a^{6} - \frac{8551604409594837162}{458940543697806301} a^{5} - \frac{9184642793772590733}{458940543697806301} a^{4} - \frac{8069536060687769920}{458940543697806301} a^{3} - \frac{5637873311545676152}{458940543697806301} a^{2} - \frac{3782003918261348927}{458940543697806301} a - \frac{73553134945315591}{14804533667671171} \) (order $30$)
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:  \( 11835.103803 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_2^2.C_2$ (as 16T317):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $C_2^4:C_2^2.C_2$
Character table for $C_2^4:C_2^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{5})\), \(\Q(\zeta_{15})^+\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\zeta_{15})\)

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
61Data not computed