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Properties

Label 16.0.39899568153...3809.1
Degree $16$
Signature $[0, 8]$
Discriminant $37^{6}\cdot 41^{15}$
Root discriminant $125.91$
Ramified primes $37, 41$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group 16T841

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8278943, -27400483, 38788769, -31135328, 19361822, -9301087, 2771348, -347359, 22141, -24954, 4741, 1132, -256, 49, -15, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 15*x^14 + 49*x^13 - 256*x^12 + 1132*x^11 + 4741*x^10 - 24954*x^9 + 22141*x^8 - 347359*x^7 + 2771348*x^6 - 9301087*x^5 + 19361822*x^4 - 31135328*x^3 + 38788769*x^2 - 27400483*x + 8278943)
 
gp: K = bnfinit(x^16 - 3*x^15 - 15*x^14 + 49*x^13 - 256*x^12 + 1132*x^11 + 4741*x^10 - 24954*x^9 + 22141*x^8 - 347359*x^7 + 2771348*x^6 - 9301087*x^5 + 19361822*x^4 - 31135328*x^3 + 38788769*x^2 - 27400483*x + 8278943, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 15 x^{14} + 49 x^{13} - 256 x^{12} + 1132 x^{11} + 4741 x^{10} - 24954 x^{9} + 22141 x^{8} - 347359 x^{7} + 2771348 x^{6} - 9301087 x^{5} + 19361822 x^{4} - 31135328 x^{3} + 38788769 x^{2} - 27400483 x + 8278943 \)

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:  \(3989956815365189429491572242763809=37^{6}\cdot 41^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $125.91$
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:  $37, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{165909809913308184536373147252381650587968311952668761} a^{15} + \frac{50138394460902643558837444707416263256867602935555271}{165909809913308184536373147252381650587968311952668761} a^{14} + \frac{48713631168004900130915683285368030205743826945445529}{165909809913308184536373147252381650587968311952668761} a^{13} - \frac{3270125382853277107168989316091359010473497569399072}{165909809913308184536373147252381650587968311952668761} a^{12} - \frac{5468886382049925743523564508876483990728634344228817}{165909809913308184536373147252381650587968311952668761} a^{11} - \frac{4982474966187079047386230177091687246925099743010959}{165909809913308184536373147252381650587968311952668761} a^{10} + \frac{31053933574713866023499267428665885899343306622144666}{165909809913308184536373147252381650587968311952668761} a^{9} + \frac{63923551530225313194225580973408347842860647931045601}{165909809913308184536373147252381650587968311952668761} a^{8} + \frac{62306184084757066883602084507638076267781434572514673}{165909809913308184536373147252381650587968311952668761} a^{7} - \frac{4103047200627196050441642889801427149298925456048898}{165909809913308184536373147252381650587968311952668761} a^{6} + \frac{36039254839220857437410213191431509411236509937737154}{165909809913308184536373147252381650587968311952668761} a^{5} + \frac{75966199140686245672170558304763257249339694584391758}{165909809913308184536373147252381650587968311952668761} a^{4} - \frac{61847611799676182080277782748408174482770901592752971}{165909809913308184536373147252381650587968311952668761} a^{3} - \frac{40708419124234902863204030016293371437468015621674396}{165909809913308184536373147252381650587968311952668761} a^{2} + \frac{69488881615889251975159843786525780966383482161669085}{165909809913308184536373147252381650587968311952668761} a - \frac{44829201172769695255182328827430024584148198516292809}{165909809913308184536373147252381650587968311952668761}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( 1165850711.88 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T841:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n841
Character table for t16n841 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.0.194754273881.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ $16$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ R R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
37Data not computed
41Data not computed