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Properties

Label 16.4.31342639780...813.17
Degree $16$
Signature $[4, 6]$
Discriminant $13^{11}\cdot 53^{10}$
Root discriminant $69.74$
Ramified primes $13, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4:D_4.D_4$ (as 16T681)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![234563, 1056685, 2177663, 2609981, 1896839, 713049, -36282, -172465, -68979, -4977, 3949, 1653, 297, -78, -40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 40*x^14 - 78*x^13 + 297*x^12 + 1653*x^11 + 3949*x^10 - 4977*x^9 - 68979*x^8 - 172465*x^7 - 36282*x^6 + 713049*x^5 + 1896839*x^4 + 2609981*x^3 + 2177663*x^2 + 1056685*x + 234563)
 
gp: K = bnfinit(x^16 - 40*x^14 - 78*x^13 + 297*x^12 + 1653*x^11 + 3949*x^10 - 4977*x^9 - 68979*x^8 - 172465*x^7 - 36282*x^6 + 713049*x^5 + 1896839*x^4 + 2609981*x^3 + 2177663*x^2 + 1056685*x + 234563, 1)
 

Normalized defining polynomial

\( x^{16} - 40 x^{14} - 78 x^{13} + 297 x^{12} + 1653 x^{11} + 3949 x^{10} - 4977 x^{9} - 68979 x^{8} - 172465 x^{7} - 36282 x^{6} + 713049 x^{5} + 1896839 x^{4} + 2609981 x^{3} + 2177663 x^{2} + 1056685 x + 234563 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(313426397802392026311505288813=13^{11}\cdot 53^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.74$
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:  $13, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{72404570801199923512405485130171890469479} a^{15} + \frac{5116972270285625394644732239148027050678}{10343510114457131930343640732881698638497} a^{14} - \frac{2085087578238772958245976074556328578029}{24134856933733307837468495043390630156493} a^{13} - \frac{7974196702468098646533276954942635074489}{24134856933733307837468495043390630156493} a^{12} - \frac{9462036091818756335704896896928615807570}{24134856933733307837468495043390630156493} a^{11} - \frac{11551412631489998500304558054689388675900}{24134856933733307837468495043390630156493} a^{10} + \frac{2704650209997524480218891046700285795556}{72404570801199923512405485130171890469479} a^{9} - \frac{3343301265693799170321088553674850212775}{10343510114457131930343640732881698638497} a^{8} - \frac{2775525281139957749871372992729688741437}{72404570801199923512405485130171890469479} a^{7} - \frac{410016534085824247382478599605689245249}{24134856933733307837468495043390630156493} a^{6} - \frac{2726496237419873329034031872535829391275}{24134856933733307837468495043390630156493} a^{5} - \frac{7637158141934633534400681206761605118902}{24134856933733307837468495043390630156493} a^{4} + \frac{924805953582118492510636012283813325941}{10343510114457131930343640732881698638497} a^{3} + \frac{33985043147107665812234676385982419596067}{72404570801199923512405485130171890469479} a^{2} - \frac{7728710291121942416403270949277755702881}{24134856933733307837468495043390630156493} a - \frac{4093855779314210563487930521419677162690}{10343510114457131930343640732881698638497}$

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:  $9$
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:  \( 301743446.941 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:D_4.D_4$ (as 16T681):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 19 conjugacy class representatives for $C_4:D_4.D_4$
Character table for $C_4:D_4.D_4$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.8957.1, 8.4.2929680361933.5

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 $16$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$13$13.4.3.1$x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$53$53.4.3.4$x^{4} + 424$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.0.1$x^{4} - x + 18$$1$$4$$0$$C_4$$[\ ]^{4}$
53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$