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Properties

Label 16.8.57681033264...3953.1
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![-70141, 138577, -131072, 63033, 22291, -61449, 49921, -21262, 900, 4070, -2123, 411, 30, -23, 8, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 8*x^14 - 23*x^13 + 30*x^12 + 411*x^11 - 2123*x^10 + 4070*x^9 + 900*x^8 - 21262*x^7 + 49921*x^6 - 61449*x^5 + 22291*x^4 + 63033*x^3 - 131072*x^2 + 138577*x - 70141)
 
gp: K = bnfinit(x^16 - 5*x^15 + 8*x^14 - 23*x^13 + 30*x^12 + 411*x^11 - 2123*x^10 + 4070*x^9 + 900*x^8 - 21262*x^7 + 49921*x^6 - 61449*x^5 + 22291*x^4 + 63033*x^3 - 131072*x^2 + 138577*x - 70141, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 8 x^{14} - 23 x^{13} + 30 x^{12} + 411 x^{11} - 2123 x^{10} + 4070 x^{9} + 900 x^{8} - 21262 x^{7} + 49921 x^{6} - 61449 x^{5} + 22291 x^{4} + 63033 x^{3} - 131072 x^{2} + 138577 x - 70141 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{10458309593730227510051966430654409} a^{15} - \frac{3537646812107010103553764218659861}{10458309593730227510051966430654409} a^{14} - \frac{2626786609770511981637839752730090}{10458309593730227510051966430654409} a^{13} - \frac{3410239728823558062690127611692508}{10458309593730227510051966430654409} a^{12} + \frac{703873057320199110310149310040576}{10458309593730227510051966430654409} a^{11} - \frac{4534315989232486052088330606967946}{10458309593730227510051966430654409} a^{10} + \frac{3398266716086615064793272644227009}{10458309593730227510051966430654409} a^{9} + \frac{1260614657998396618049409698849559}{10458309593730227510051966430654409} a^{8} + \frac{982428992688424766370877129906445}{10458309593730227510051966430654409} a^{7} - \frac{1962615354926083586646020002911666}{10458309593730227510051966430654409} a^{6} - \frac{1368877776711190602332290064230897}{10458309593730227510051966430654409} a^{5} - \frac{4309403201267013273146404355309812}{10458309593730227510051966430654409} a^{4} - \frac{4540053214706332871762257302827213}{10458309593730227510051966430654409} a^{3} + \frac{3671353209022348230701132777769674}{10458309593730227510051966430654409} a^{2} + \frac{1497562887392297702935910050532497}{10458309593730227510051966430654409} a - \frac{342298715082426274694409632818209}{10458309593730227510051966430654409}$

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:  \( 3256507.75452 \) (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