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Properties

Label 16.0.78186208447...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 71^{6}$
Root discriminant $20.22$
Ramified primes $5, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^3.C_4$ (as 16T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![781, 1846, 3006, 3324, 2683, 1786, 754, 392, 78, 16, 6, -43, 23, -18, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 9*x^14 - 18*x^13 + 23*x^12 - 43*x^11 + 6*x^10 + 16*x^9 + 78*x^8 + 392*x^7 + 754*x^6 + 1786*x^5 + 2683*x^4 + 3324*x^3 + 3006*x^2 + 1846*x + 781)
 
gp: K = bnfinit(x^16 - 3*x^15 + 9*x^14 - 18*x^13 + 23*x^12 - 43*x^11 + 6*x^10 + 16*x^9 + 78*x^8 + 392*x^7 + 754*x^6 + 1786*x^5 + 2683*x^4 + 3324*x^3 + 3006*x^2 + 1846*x + 781, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 9 x^{14} - 18 x^{13} + 23 x^{12} - 43 x^{11} + 6 x^{10} + 16 x^{9} + 78 x^{8} + 392 x^{7} + 754 x^{6} + 1786 x^{5} + 2683 x^{4} + 3324 x^{3} + 3006 x^{2} + 1846 x + 781 \)

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:  \(781862084478759765625=5^{14}\cdot 71^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.22$
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:  $5, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} - \frac{5}{11} a^{11} + \frac{1}{11} a^{10} - \frac{3}{11} a^{9} - \frac{5}{11} a^{8} + \frac{1}{11} a^{7} - \frac{2}{11} a^{6} - \frac{2}{11} a^{5} - \frac{3}{11} a^{4} + \frac{1}{11} a^{3} + \frac{3}{11} a^{2} + \frac{2}{11} a$, $\frac{1}{121} a^{14} - \frac{1}{121} a^{13} + \frac{6}{121} a^{12} - \frac{5}{121} a^{11} + \frac{7}{121} a^{10} - \frac{24}{121} a^{9} - \frac{49}{121} a^{8} - \frac{58}{121} a^{7} + \frac{1}{11} a^{6} - \frac{12}{121} a^{5} - \frac{7}{121} a^{4} - \frac{31}{121} a^{3} - \frac{34}{121} a^{2} + \frac{20}{121} a + \frac{5}{11}$, $\frac{1}{17089894729930889119} a^{15} - \frac{40988714880351592}{17089894729930889119} a^{14} - \frac{534135089293337690}{17089894729930889119} a^{13} - \frac{25119240345141760}{141238799420916439} a^{12} + \frac{2470802874335418877}{17089894729930889119} a^{11} + \frac{4658771750170891564}{17089894729930889119} a^{10} - \frac{907794840114765298}{17089894729930889119} a^{9} + \frac{3827046300850604449}{17089894729930889119} a^{8} - \frac{518426130269390505}{17089894729930889119} a^{7} + \frac{5837390546586598262}{17089894729930889119} a^{6} + \frac{8132602394154368971}{17089894729930889119} a^{5} - \frac{3235331461685708189}{17089894729930889119} a^{4} - \frac{5724045834720207822}{17089894729930889119} a^{3} + \frac{3654551217200906890}{17089894729930889119} a^{2} - \frac{5882894502558712670}{17089894729930889119} a + \frac{102848262798304029}{1553626793630080829}$

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:  \( -\frac{81482896390942}{17089894729930889119} a^{15} + \frac{21690397837460048}{17089894729930889119} a^{14} - \frac{156826788645124381}{17089894729930889119} a^{13} + \frac{54855340919976686}{1553626793630080829} a^{12} - \frac{1682663990125282045}{17089894729930889119} a^{11} + \frac{3500834188345957459}{17089894729930889119} a^{10} - \frac{5771291024952579659}{17089894729930889119} a^{9} + \frac{7711554790514482706}{17089894729930889119} a^{8} - \frac{6635561522621844302}{17089894729930889119} a^{7} + \frac{3590548027888443140}{17089894729930889119} a^{6} + \frac{1506095017926403906}{17089894729930889119} a^{5} - \frac{5524663738756269576}{17089894729930889119} a^{4} + \frac{5096659284728456297}{17089894729930889119} a^{3} - \frac{21355212641558755474}{17089894729930889119} a^{2} + \frac{5626570474487287101}{17089894729930889119} a - \frac{469598433253968041}{1553626793630080829} \) (order $10$)
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:  \( 34534.2764565 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.1775.1, \(\Q(\zeta_{5})\), 4.2.8875.1, 8.0.393828125.1 x2, 8.0.78765625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
5Data not computed
71Data not computed