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Properties

Label 16.0.13916709339...0753.4
Degree $16$
Signature $[0, 8]$
Discriminant $17^{11}\cdot 67^{8}$
Root discriminant $57.41$
Ramified primes $17, 67$
Class number $20$ (GRH)
Class group $[2, 10]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T289)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2803, 10381, 22338, 30752, 38306, 32756, 16281, 7037, 5016, -546, 1005, -535, 256, 44, 13, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 13*x^14 + 44*x^13 + 256*x^12 - 535*x^11 + 1005*x^10 - 546*x^9 + 5016*x^8 + 7037*x^7 + 16281*x^6 + 32756*x^5 + 38306*x^4 + 30752*x^3 + 22338*x^2 + 10381*x + 2803)
 
gp: K = bnfinit(x^16 - 7*x^15 + 13*x^14 + 44*x^13 + 256*x^12 - 535*x^11 + 1005*x^10 - 546*x^9 + 5016*x^8 + 7037*x^7 + 16281*x^6 + 32756*x^5 + 38306*x^4 + 30752*x^3 + 22338*x^2 + 10381*x + 2803, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 13 x^{14} + 44 x^{13} + 256 x^{12} - 535 x^{11} + 1005 x^{10} - 546 x^{9} + 5016 x^{8} + 7037 x^{7} + 16281 x^{6} + 32756 x^{5} + 38306 x^{4} + 30752 x^{3} + 22338 x^{2} + 10381 x + 2803 \)

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:  \(13916709339102552311118140753=17^{11}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.41$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1943} a^{14} - \frac{345}{1943} a^{13} + \frac{225}{1943} a^{12} - \frac{843}{1943} a^{11} - \frac{284}{1943} a^{10} + \frac{321}{1943} a^{9} + \frac{5}{67} a^{8} - \frac{849}{1943} a^{7} - \frac{284}{1943} a^{6} - \frac{971}{1943} a^{5} - \frac{602}{1943} a^{4} - \frac{723}{1943} a^{3} + \frac{189}{1943} a^{2} + \frac{439}{1943} a - \frac{325}{1943}$, $\frac{1}{43477352170894285453647741727685779} a^{15} - \frac{803553561585002801179702705750}{43477352170894285453647741727685779} a^{14} + \frac{15863391689939796120107299910606294}{43477352170894285453647741727685779} a^{13} + \frac{4716259793548795970965425323175011}{43477352170894285453647741727685779} a^{12} - \frac{780012192110748989998069249414499}{2288281693204962392297249564615041} a^{11} - \frac{662820107545569720017658503467428}{1499219040375665015643025576816751} a^{10} - \frac{8100652552681527449928646433068879}{43477352170894285453647741727685779} a^{9} - \frac{7272672892896435637735709754452382}{43477352170894285453647741727685779} a^{8} + \frac{21548155661210840376356171933824752}{43477352170894285453647741727685779} a^{7} - \frac{11713465317740979754791295071106311}{43477352170894285453647741727685779} a^{6} - \frac{13689708124316272160326819148265088}{43477352170894285453647741727685779} a^{5} + \frac{10689800522960085554416902159080592}{43477352170894285453647741727685779} a^{4} + \frac{16319077646950613202796155554316318}{43477352170894285453647741727685779} a^{3} + \frac{15946460397812617828637373066894173}{43477352170894285453647741727685779} a^{2} + \frac{6123151920890458740181090779290337}{43477352170894285453647741727685779} a + \frac{20326062195488423941223852703341853}{43477352170894285453647741727685779}$

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

Class group and class number

$C_{2}\times C_{10}$, which has order $20$ (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:  \( 2439203.29407 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T289):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 44 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-67}) \), 4.0.76313.1, 8.0.99002457473.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
$67$67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$