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Properties

Label 16.2.71235733773...6467.2
Degree $16$
Signature $[2, 7]$
Discriminant $-\,17^{15}\cdot 59^{7}$
Root discriminant $84.78$
Ramified primes $17, 59$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $D_{16}$ (as 16T56)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![594304, 1527452, 1414438, 378479, -241792, -222838, -69143, 4055, 17324, 6377, -81, -570, -137, -19, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 30*x^14 - 19*x^13 - 137*x^12 - 570*x^11 - 81*x^10 + 6377*x^9 + 17324*x^8 + 4055*x^7 - 69143*x^6 - 222838*x^5 - 241792*x^4 + 378479*x^3 + 1414438*x^2 + 1527452*x + 594304)
 
gp: K = bnfinit(x^16 - 8*x^15 + 30*x^14 - 19*x^13 - 137*x^12 - 570*x^11 - 81*x^10 + 6377*x^9 + 17324*x^8 + 4055*x^7 - 69143*x^6 - 222838*x^5 - 241792*x^4 + 378479*x^3 + 1414438*x^2 + 1527452*x + 594304, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 30 x^{14} - 19 x^{13} - 137 x^{12} - 570 x^{11} - 81 x^{10} + 6377 x^{9} + 17324 x^{8} + 4055 x^{7} - 69143 x^{6} - 222838 x^{5} - 241792 x^{4} + 378479 x^{3} + 1414438 x^{2} + 1527452 x + 594304 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-7123573377320035993002625946467=-\,17^{15}\cdot 59^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.78$
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:  $17, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{32} a^{13} + \frac{1}{32} a^{12} - \frac{3}{32} a^{10} + \frac{3}{32} a^{9} - \frac{1}{16} a^{8} + \frac{3}{32} a^{7} - \frac{1}{32} a^{6} + \frac{3}{32} a^{4} + \frac{13}{32} a^{3} + \frac{1}{16} a^{2} + \frac{3}{8} a$, $\frac{1}{704} a^{14} - \frac{1}{352} a^{13} + \frac{5}{704} a^{12} + \frac{37}{704} a^{11} + \frac{5}{176} a^{10} - \frac{51}{704} a^{9} + \frac{57}{704} a^{8} + \frac{83}{352} a^{7} - \frac{21}{704} a^{6} + \frac{1}{64} a^{5} - \frac{41}{176} a^{4} + \frac{83}{704} a^{3} + \frac{9}{32} a^{2} - \frac{85}{176} a + \frac{1}{11}$, $\frac{1}{45885735537756957546700991340544} a^{15} - \frac{863751553860377849903043911}{45885735537756957546700991340544} a^{14} - \frac{181573970853696033060066687081}{45885735537756957546700991340544} a^{13} - \frac{412036731088916270567315615823}{11471433884439239386675247835136} a^{12} - \frac{4719464156155634335188493731525}{45885735537756957546700991340544} a^{11} - \frac{3796840276918610775882933661375}{45885735537756957546700991340544} a^{10} - \frac{296259370590095333315122346821}{2867858471109809846668811958784} a^{9} + \frac{3180655329507494509436524378265}{45885735537756957546700991340544} a^{8} - \frac{5972365206100634920835857286907}{45885735537756957546700991340544} a^{7} - \frac{1981801314829564613765284050009}{11471433884439239386675247835136} a^{6} + \frac{10503185115799842451770793696645}{45885735537756957546700991340544} a^{5} - \frac{201094236389318961800124643999}{976292245484190586100021092352} a^{4} + \frac{12024586980628892456115110444431}{45885735537756957546700991340544} a^{3} - \frac{6256308450790737387377567714657}{22942867768878478773350495670272} a^{2} - \frac{682008231048290733202426939591}{11471433884439239386675247835136} a - \frac{25727628440298905409807842083}{358482308888726230833601494848}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $8$
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:  \( 5376866276.16 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{16}$ (as 16T56):

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 $D_{16}$
Character table for $D_{16}$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.2.289867.1, 8.2.84274946322067.2

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 16 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ $16$ $16$ $16$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ R

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
17Data not computed
59Data not computed