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Properties

Label 16.8.57681033264...3953.3
Degree $16$
Signature $[8, 4]$
Discriminant $17^{15}\cdot 67^{4}$
Root discriminant $40.74$
Ramified primes $17, 67$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times OD_{16}).D_4$ (as 16T591)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3637, -30742, -73336, -29014, 28054, 12403, 1759, 2723, -630, -198, -104, -56, -4, -44, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 9*x^14 - 44*x^13 - 4*x^12 - 56*x^11 - 104*x^10 - 198*x^9 - 630*x^8 + 2723*x^7 + 1759*x^6 + 12403*x^5 + 28054*x^4 - 29014*x^3 - 73336*x^2 - 30742*x - 3637)
 
gp: K = bnfinit(x^16 - 3*x^15 + 9*x^14 - 44*x^13 - 4*x^12 - 56*x^11 - 104*x^10 - 198*x^9 - 630*x^8 + 2723*x^7 + 1759*x^6 + 12403*x^5 + 28054*x^4 - 29014*x^3 - 73336*x^2 - 30742*x - 3637, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 9 x^{14} - 44 x^{13} - 4 x^{12} - 56 x^{11} - 104 x^{10} - 198 x^{9} - 630 x^{8} + 2723 x^{7} + 1759 x^{6} + 12403 x^{5} + 28054 x^{4} - 29014 x^{3} - 73336 x^{2} - 30742 x - 3637 \)

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:  \(57681033264163530732453953=17^{15}\cdot 67^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.74$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{46525784879675192907490701134140643} a^{15} - \frac{1627406499776280628803650637451438}{3578906529205784069806977010318511} a^{14} + \frac{6028051498823800071666018106852603}{46525784879675192907490701134140643} a^{13} - \frac{12577191574373226898670575685165201}{46525784879675192907490701134140643} a^{12} - \frac{15760044238443419957888881606524645}{46525784879675192907490701134140643} a^{11} - \frac{1537379843746778748737431479126203}{3578906529205784069806977010318511} a^{10} - \frac{331837457469359060474905075048340}{3578906529205784069806977010318511} a^{9} - \frac{9236954321175956949629592839283140}{46525784879675192907490701134140643} a^{8} + \frac{3383658212033885228174218656527616}{46525784879675192907490701134140643} a^{7} - \frac{1639325522595166631188933140239248}{3578906529205784069806977010318511} a^{6} + \frac{228700916147059570523183049576183}{460651335442328642648422783506343} a^{5} + \frac{1441507251253683995859688105301663}{3578906529205784069806977010318511} a^{4} + \frac{162547406671019944833345479007175}{3578906529205784069806977010318511} a^{3} - \frac{17634905389859565927142098264352}{124734007720308828170216356927991} a^{2} + \frac{10375388277176708611217522308423092}{46525784879675192907490701134140643} a + \frac{8185005540223955617821403709394077}{46525784879675192907490701134140643}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 2922057.59311 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).D_4$ (as 16T591):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $(C_2\times OD_{16}).D_4$
Character table for $(C_2\times OD_{16}).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
17Data not computed
67Data not computed