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Properties

Label 16.8.33662328752...4641.2
Degree $16$
Signature $[8, 4]$
Discriminant $31^{6}\cdot 41^{14}$
Root discriminant $93.42$
Ramified primes $31, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T257)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1270343, 802816, 2713383, -4514914, 4072213, -2710906, 1418618, -647354, 236931, -63054, 11216, -1104, -358, 274, -44, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 44*x^14 + 274*x^13 - 358*x^12 - 1104*x^11 + 11216*x^10 - 63054*x^9 + 236931*x^8 - 647354*x^7 + 1418618*x^6 - 2710906*x^5 + 4072213*x^4 - 4514914*x^3 + 2713383*x^2 + 802816*x - 1270343)
 
gp: K = bnfinit(x^16 - 4*x^15 - 44*x^14 + 274*x^13 - 358*x^12 - 1104*x^11 + 11216*x^10 - 63054*x^9 + 236931*x^8 - 647354*x^7 + 1418618*x^6 - 2710906*x^5 + 4072213*x^4 - 4514914*x^3 + 2713383*x^2 + 802816*x - 1270343, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 44 x^{14} + 274 x^{13} - 358 x^{12} - 1104 x^{11} + 11216 x^{10} - 63054 x^{9} + 236931 x^{8} - 647354 x^{7} + 1418618 x^{6} - 2710906 x^{5} + 4072213 x^{4} - 4514914 x^{3} + 2713383 x^{2} + 802816 x - 1270343 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(33662328752972862921089838254641=31^{6}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.42$
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:  $31, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{50738} a^{14} - \frac{3971}{50738} a^{13} + \frac{2268}{25369} a^{12} + \frac{5547}{50738} a^{11} - \frac{4700}{25369} a^{10} - \frac{1064}{25369} a^{9} + \frac{6303}{25369} a^{8} - \frac{22239}{50738} a^{7} + \frac{881}{50738} a^{6} + \frac{21867}{50738} a^{5} + \frac{24877}{50738} a^{4} + \frac{7477}{50738} a^{3} + \frac{9779}{50738} a^{2} - \frac{12059}{25369} a - \frac{18499}{50738}$, $\frac{1}{5485864985439816553134891234130308431857906} a^{15} - \frac{29916315901142326867572801496206637}{119257934466082968546410679002832791996911} a^{14} + \frac{397700010430269077473060982480266261437971}{5485864985439816553134891234130308431857906} a^{13} - \frac{294288599194870973640833420843852904804242}{2742932492719908276567445617065154215928953} a^{12} - \frac{307287939297072460643460341112368958129648}{2742932492719908276567445617065154215928953} a^{11} + \frac{751481591160094532306631781347574899113631}{5485864985439816553134891234130308431857906} a^{10} + \frac{420890840034630237332454317775387172783334}{2742932492719908276567445617065154215928953} a^{9} + \frac{664837173438354054084270031965678538883541}{5485864985439816553134891234130308431857906} a^{8} - \frac{209935513854333029912367890319467466710041}{5485864985439816553134891234130308431857906} a^{7} + \frac{1081247898576754615286162958452844796542414}{2742932492719908276567445617065154215928953} a^{6} - \frac{1000116605235058200364583860275237747571231}{2742932492719908276567445617065154215928953} a^{5} + \frac{574151797093610034754974993095825425734389}{2742932492719908276567445617065154215928953} a^{4} + \frac{546146138571706698331943175697626698952572}{2742932492719908276567445617065154215928953} a^{3} + \frac{62564123810694977908430634970484788133505}{238515868932165937092821358005665583993822} a^{2} - \frac{731462304361452269269598331284219502608005}{2742932492719908276567445617065154215928953} a + \frac{728037714950410358274285449968835122109514}{2742932492719908276567445617065154215928953}$

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:  $11$
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:  \( 8507235789.71 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T257):

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

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.6.5801924573188871.1, 8.4.187158857199641.1, 8.6.147253231471.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ 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
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$