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Properties

Label 16.8.68014001055...0625.1
Degree $16$
Signature $[8, 4]$
Discriminant $3^{8}\cdot 5^{10}\cdot 101^{6}$
Root discriminant $26.73$
Ramified primes $3, 5, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4.D_4$ (as 16T339)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![619, 604, -1613, -1348, 2613, 2066, -3319, -2944, 1721, 1774, -517, -532, 118, 78, -18, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 18*x^14 + 78*x^13 + 118*x^12 - 532*x^11 - 517*x^10 + 1774*x^9 + 1721*x^8 - 2944*x^7 - 3319*x^6 + 2066*x^5 + 2613*x^4 - 1348*x^3 - 1613*x^2 + 604*x + 619)
 
gp: K = bnfinit(x^16 - 4*x^15 - 18*x^14 + 78*x^13 + 118*x^12 - 532*x^11 - 517*x^10 + 1774*x^9 + 1721*x^8 - 2944*x^7 - 3319*x^6 + 2066*x^5 + 2613*x^4 - 1348*x^3 - 1613*x^2 + 604*x + 619, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 18 x^{14} + 78 x^{13} + 118 x^{12} - 532 x^{11} - 517 x^{10} + 1774 x^{9} + 1721 x^{8} - 2944 x^{7} - 3319 x^{6} + 2066 x^{5} + 2613 x^{4} - 1348 x^{3} - 1613 x^{2} + 604 x + 619 \)

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:  \(68014001055597275390625=3^{8}\cdot 5^{10}\cdot 101^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.73$
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:  $3, 5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{12} + \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{10} a^{8} - \frac{1}{2} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{3}{10} a^{3} - \frac{1}{10} a^{2} - \frac{1}{2} a + \frac{2}{5}$, $\frac{1}{10} a^{13} - \frac{1}{5} a^{11} + \frac{1}{10} a^{10} + \frac{1}{10} a^{9} - \frac{1}{5} a^{8} - \frac{1}{10} a^{7} - \frac{1}{10} a^{6} - \frac{1}{10} a^{4} - \frac{1}{2} a^{3} - \frac{3}{10} a^{2} - \frac{1}{10} a - \frac{3}{10}$, $\frac{1}{710} a^{14} - \frac{6}{355} a^{13} - \frac{7}{355} a^{12} + \frac{8}{355} a^{11} + \frac{63}{355} a^{9} - \frac{17}{355} a^{8} + \frac{181}{710} a^{7} + \frac{32}{355} a^{6} - \frac{3}{142} a^{5} - \frac{141}{710} a^{4} + \frac{243}{710} a^{3} + \frac{111}{355} a^{2} + \frac{9}{710} a - \frac{156}{355}$, $\frac{1}{11848642070590270} a^{15} - \frac{2328308579587}{5924321035295135} a^{14} - \frac{52096550789187}{1077149279144570} a^{13} - \frac{161398288342147}{11848642070590270} a^{12} - \frac{109220412079239}{1184864207059027} a^{11} + \frac{661030762687817}{11848642070590270} a^{10} + \frac{723781306260831}{5924321035295135} a^{9} + \frac{593534204805031}{5924321035295135} a^{8} + \frac{351383527755852}{5924321035295135} a^{7} + \frac{930574301753049}{2369728414118054} a^{6} - \frac{675584877889864}{5924321035295135} a^{5} + \frac{4995546384469453}{11848642070590270} a^{4} - \frac{4148811008186291}{11848642070590270} a^{3} - \frac{1997044194879744}{5924321035295135} a^{2} - \frac{2516632223010183}{11848642070590270} a - \frac{731240314960349}{11848642070590270}$

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:  $11$
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:  \( 254356.288762 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.D_4$ (as 16T339):

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 $C_2^4.D_4$
Character table for $C_2^4.D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.4.31878125.1, 8.8.52158988125.1, 8.4.260794940625.2

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
3Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
101Data not computed