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Properties

Label 16.16.7585344425...0000.2
Degree $16$
Signature $[16, 0]$
Discriminant $2^{40}\cdot 5^{12}\cdot 41^{4}$
Root discriminant $47.86$
Ramified primes $2, 5, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4:C_2^2.C_2$ (as 16T317)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![316, 24528, 107488, 130176, -55468, -193432, -53392, 82048, 40510, -13752, -9504, 976, 1022, -24, -52, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 52*x^14 - 24*x^13 + 1022*x^12 + 976*x^11 - 9504*x^10 - 13752*x^9 + 40510*x^8 + 82048*x^7 - 53392*x^6 - 193432*x^5 - 55468*x^4 + 130176*x^3 + 107488*x^2 + 24528*x + 316)
 
gp: K = bnfinit(x^16 - 52*x^14 - 24*x^13 + 1022*x^12 + 976*x^11 - 9504*x^10 - 13752*x^9 + 40510*x^8 + 82048*x^7 - 53392*x^6 - 193432*x^5 - 55468*x^4 + 130176*x^3 + 107488*x^2 + 24528*x + 316, 1)
 

Normalized defining polynomial

\( x^{16} - 52 x^{14} - 24 x^{13} + 1022 x^{12} + 976 x^{11} - 9504 x^{10} - 13752 x^{9} + 40510 x^{8} + 82048 x^{7} - 53392 x^{6} - 193432 x^{5} - 55468 x^{4} + 130176 x^{3} + 107488 x^{2} + 24528 x + 316 \)

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:  \(758534442582016000000000000=2^{40}\cdot 5^{12}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.86$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{82} a^{14} + \frac{10}{41} a^{13} + \frac{1}{41} a^{12} - \frac{8}{41} a^{11} + \frac{5}{41} a^{10} - \frac{6}{41} a^{9} - \frac{1}{41} a^{8} + \frac{18}{41} a^{7} + \frac{10}{41} a^{6} - \frac{18}{41} a^{5} - \frac{12}{41} a^{4} + \frac{5}{41} a^{3} + \frac{11}{41} a^{2} - \frac{13}{41} a - \frac{14}{41}$, $\frac{1}{4710373940043518100082} a^{15} + \frac{1869721331284131619}{2355186970021759050041} a^{14} + \frac{163990009057699168483}{4710373940043518100082} a^{13} - \frac{444113398950642991232}{2355186970021759050041} a^{12} - \frac{102954553358853140879}{2355186970021759050041} a^{11} - \frac{1115886339267979533555}{4710373940043518100082} a^{10} - \frac{137482144040398464173}{2355186970021759050041} a^{9} - \frac{433555929727210192169}{2355186970021759050041} a^{8} - \frac{553265725375471210300}{2355186970021759050041} a^{7} + \frac{391390514356938048553}{2355186970021759050041} a^{6} - \frac{378404920213220597033}{2355186970021759050041} a^{5} - \frac{219050501787654739487}{2355186970021759050041} a^{4} + \frac{21407303797223514012}{57443584634677050001} a^{3} - \frac{62290518320735980309}{2355186970021759050041} a^{2} + \frac{822040865815961836169}{2355186970021759050041} a - \frac{1172077064097083581951}{2355186970021759050041}$

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:  \( 194408629.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_2^2.C_2$ (as 16T317):

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 $C_2^4:C_2^2.C_2$
Character table for $C_2^4:C_2^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.4.8000.1, \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{40})^+\)

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41Data not computed