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Properties

Label 16.0.61522891577...6769.1
Degree $16$
Signature $[0, 8]$
Discriminant $71^{2}\cdot 73^{14}$
Root discriminant $72.75$
Ramified primes $71, 73$
Class number $89$ (GRH)
Class group $[89]$ (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T258)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![126736, 0, 115167, 0, 131445, 0, 59400, 0, 13094, 0, 3306, 0, 360, 0, 27, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 27*x^14 + 360*x^12 + 3306*x^10 + 13094*x^8 + 59400*x^6 + 131445*x^4 + 115167*x^2 + 126736)
 
gp: K = bnfinit(x^16 + 27*x^14 + 360*x^12 + 3306*x^10 + 13094*x^8 + 59400*x^6 + 131445*x^4 + 115167*x^2 + 126736, 1)
 

Normalized defining polynomial

\( x^{16} + 27 x^{14} + 360 x^{12} + 3306 x^{10} + 13094 x^{8} + 59400 x^{6} + 131445 x^{4} + 115167 x^{2} + 126736 \)

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:  \(615228915774362553504098556769=71^{2}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.75$
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:  $71, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{12} a^{8} + \frac{1}{4} a^{2} + \frac{1}{3}$, $\frac{1}{24} a^{9} - \frac{1}{24} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{3} + \frac{3}{8} a^{2} - \frac{5}{24} a - \frac{1}{6}$, $\frac{1}{24} a^{10} - \frac{1}{8} a^{7} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{24} a^{11} - \frac{1}{24} a^{8} + \frac{1}{8} a^{5} - \frac{1}{3} a^{3} - \frac{1}{8} a^{2} + \frac{1}{3}$, $\frac{1}{24} a^{12} - \frac{1}{24} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} + \frac{1}{6} a^{4} + \frac{3}{8} a^{2} - \frac{3}{8} a - \frac{1}{6}$, $\frac{1}{48} a^{13} - \frac{1}{48} a^{12} - \frac{1}{48} a^{11} - \frac{1}{48} a^{10} - \frac{1}{48} a^{9} - \frac{1}{48} a^{8} - \frac{1}{16} a^{7} + \frac{3}{16} a^{6} + \frac{1}{48} a^{5} - \frac{7}{48} a^{4} + \frac{17}{48} a^{3} + \frac{17}{48} a^{2} - \frac{11}{24} a + \frac{1}{6}$, $\frac{1}{728714427880704} a^{14} - \frac{778659524429}{60726202323392} a^{12} + \frac{2923316286451}{182178606970176} a^{10} - \frac{3435560953175}{121452404646784} a^{8} - \frac{1}{8} a^{7} - \frac{10876922350589}{182178606970176} a^{6} - \frac{1031341348201}{60726202323392} a^{4} - \frac{50853181419959}{728714427880704} a^{2} + \frac{1}{8} a + \frac{1010359977029}{15181550580848}$, $\frac{1}{129711168162765312} a^{15} - \frac{1}{1457428855761408} a^{14} + \frac{255750381301129}{32427792040691328} a^{13} + \frac{778659524429}{121452404646784} a^{12} - \frac{429750875267717}{32427792040691328} a^{11} + \frac{1555819667991}{121452404646784} a^{10} - \frac{94524864438263}{21618528027127552} a^{9} - \frac{20056418302171}{728714427880704} a^{8} + \frac{2721802182202051}{32427792040691328} a^{7} - \frac{80212381134499}{364357213940352} a^{6} - \frac{542039069664707}{32427792040691328} a^{5} - \frac{21740984523071}{121452404646784} a^{4} - \frac{49056898456397303}{129711168162765312} a^{3} - \frac{124743411614595}{485809618587136} a^{2} - \frac{101465106443695}{2702316003390944} a - \frac{40984956383207}{91089303485088}$

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

Class group and class number

$C_{89}$, which has order $89$ (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:  \( -1 \) (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:  \( 819937823.341 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T258):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.389017.1, 8.2.784365294855887.1, 8.0.11047398519097.1, 8.6.10744730066519.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$71$71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$73$73.8.7.3$x^{8} - 45625$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.7.3$x^{8} - 45625$$8$$1$$7$$C_8$$[\ ]_{8}$