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Properties

Label 16.16.8422523572...0625.1
Degree $16$
Signature $[16, 0]$
Discriminant $5^{14}\cdot 53^{14}$
Root discriminant $131.93$
Ramified primes $5, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1985409, 9245136, 12749774, -23145964, -42119744, -10828192, 16208101, 12490108, 2257212, -670628, -274489, 602, 9846, 464, -156, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 156*x^14 + 464*x^13 + 9846*x^12 + 602*x^11 - 274489*x^10 - 670628*x^9 + 2257212*x^8 + 12490108*x^7 + 16208101*x^6 - 10828192*x^5 - 42119744*x^4 - 23145964*x^3 + 12749774*x^2 + 9245136*x - 1985409)
 
gp: K = bnfinit(x^16 - 6*x^15 - 156*x^14 + 464*x^13 + 9846*x^12 + 602*x^11 - 274489*x^10 - 670628*x^9 + 2257212*x^8 + 12490108*x^7 + 16208101*x^6 - 10828192*x^5 - 42119744*x^4 - 23145964*x^3 + 12749774*x^2 + 9245136*x - 1985409, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 156 x^{14} + 464 x^{13} + 9846 x^{12} + 602 x^{11} - 274489 x^{10} - 670628 x^{9} + 2257212 x^{8} + 12490108 x^{7} + 16208101 x^{6} - 10828192 x^{5} - 42119744 x^{4} - 23145964 x^{3} + 12749774 x^{2} + 9245136 x - 1985409 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8422523572126121633218804931640625=5^{14}\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $131.93$
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:  $5, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} + \frac{1}{6} a$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{6} a$, $\frac{1}{12} a^{10} - \frac{1}{12} a^{9} - \frac{1}{12} a^{8} - \frac{1}{12} a^{7} + \frac{5}{12} a^{6} - \frac{1}{3} a^{5} + \frac{5}{12} a^{4} + \frac{5}{12} a^{3} + \frac{1}{12} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{12} a^{11} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{3} a^{3} - \frac{1}{4}$, $\frac{1}{12} a^{12} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{8} - \frac{1}{12} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{5}{12} a^{2} - \frac{1}{3} a$, $\frac{1}{4176} a^{14} - \frac{71}{2088} a^{13} + \frac{41}{4176} a^{12} + \frac{19}{2088} a^{11} + \frac{15}{464} a^{10} + \frac{61}{1044} a^{9} + \frac{101}{2088} a^{8} - \frac{11}{261} a^{7} - \frac{67}{232} a^{6} - \frac{367}{1044} a^{5} + \frac{719}{4176} a^{4} + \frac{133}{522} a^{3} + \frac{379}{4176} a^{2} + \frac{9}{116} a - \frac{159}{464}$, $\frac{1}{1419586515577513378707786149887008} a^{15} - \frac{104605743027876237741986765915}{1419586515577513378707786149887008} a^{14} + \frac{6095344387535693205374781431519}{157731835064168153189754016654112} a^{13} + \frac{8901015749800177717456781433223}{473195505192504459569262049962336} a^{12} + \frac{6372516284778968202995670374521}{1419586515577513378707786149887008} a^{11} + \frac{49528171728611221919296864951621}{1419586515577513378707786149887008} a^{10} + \frac{727002887844249747543273637}{13548517012898827795031267536} a^{9} + \frac{15519677415824686485969211035203}{236597752596252229784631024981168} a^{8} - \frac{118679495840459648610684880748171}{709793257788756689353893074943504} a^{7} + \frac{210026859503748391940952269940565}{709793257788756689353893074943504} a^{6} - \frac{221594826549298446515243226601879}{473195505192504459569262049962336} a^{5} - \frac{69543301951048995620223214874607}{157731835064168153189754016654112} a^{4} - \frac{43540329594429264969222176194441}{1419586515577513378707786149887008} a^{3} - \frac{121417803751670865450331330498927}{1419586515577513378707786149887008} a^{2} + \frac{38008901149544766115224628526111}{473195505192504459569262049962336} a + \frac{150603399845995052454367327}{715009610401440397775866912}$

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:  $15$
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:  \( 2292411816440 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{265}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{5}, \sqrt{53})\), 4.4.18609625.1, 4.4.18609625.2, 8.8.346318142640625.1, 8.8.91774307799765625.1 x2

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

Sibling fields

Degree 8 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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
53Data not computed