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Properties

Label 16.4.31087867512...0961.5
Degree $16$
Signature $[4, 6]$
Discriminant $31^{10}\cdot 41^{14}$
Root discriminant $220.44$
Ramified primes $31, 41$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T817

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![268927201, 0, -423979746, 0, -437946455, 0, -61766136, 0, -2874498, 0, -1234, 0, 3848, 0, 112, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 112*x^14 + 3848*x^12 - 1234*x^10 - 2874498*x^8 - 61766136*x^6 - 437946455*x^4 - 423979746*x^2 + 268927201)
 
gp: K = bnfinit(x^16 + 112*x^14 + 3848*x^12 - 1234*x^10 - 2874498*x^8 - 61766136*x^6 - 437946455*x^4 - 423979746*x^2 + 268927201, 1)
 

Normalized defining polynomial

\( x^{16} + 112 x^{14} + 3848 x^{12} - 1234 x^{10} - 2874498 x^{8} - 61766136 x^{6} - 437946455 x^{4} - 423979746 x^{2} + 268927201 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(31087867512274251337747808514764310961=31^{10}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $220.44$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{67208} a^{12} + \frac{603}{8401} a^{10} - \frac{1}{8} a^{9} + \frac{5863}{67208} a^{8} - \frac{9821}{67208} a^{6} - \frac{1}{4} a^{5} + \frac{9587}{67208} a^{4} - \frac{3}{8} a^{3} - \frac{359}{2168} a^{2} - \frac{1}{8} a - \frac{149}{2168}$, $\frac{1}{1545784} a^{13} + \frac{26409}{386446} a^{11} - \frac{1}{8} a^{10} - \frac{78147}{1545784} a^{9} + \frac{259011}{1545784} a^{7} - \frac{1}{4} a^{6} + \frac{9587}{1545784} a^{5} + \frac{1}{8} a^{4} - \frac{20413}{49864} a^{3} - \frac{1}{8} a^{2} + \frac{2561}{49864} a - \frac{1}{2}$, $\frac{1}{2390959199945341166776} a^{14} - \frac{4633509427085571}{2390959199945341166776} a^{12} - \frac{1}{8} a^{11} - \frac{182344376375053660267}{2390959199945341166776} a^{10} - \frac{1}{8} a^{9} - \frac{5120910628382851272}{298869899993167645847} a^{8} - \frac{1}{4} a^{7} + \frac{208344648222020273475}{1195479599972670583388} a^{6} - \frac{1}{8} a^{5} - \frac{76506732239209876489}{597739799986335291694} a^{4} - \frac{1}{2} a^{3} + \frac{1851525920697999145}{9640964515908633737} a^{2} + \frac{3}{8} a + \frac{1295788397060773}{145799085306746824}$, $\frac{1}{54992061598742846835848} a^{15} - \frac{4633509427085571}{54992061598742846835848} a^{13} + \frac{1610875023583952214815}{54992061598742846835848} a^{11} - \frac{1}{8} a^{10} + \frac{3246601614897781294141}{54992061598742846835848} a^{9} - \frac{1}{8} a^{8} + \frac{4990263048112702607027}{27496030799371423417924} a^{7} - \frac{1}{4} a^{6} - \frac{1682037816082026990403}{6874007699842855854481} a^{5} - \frac{1}{8} a^{4} - \frac{149084189404862780369}{1773937470927188607608} a^{3} + \frac{287365896296923003}{838344740513794238} a - \frac{1}{8}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $9$
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:  \( 989393453954 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T817:

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 t16n817
Character table for t16n817 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.179859661768855001.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
31Data not computed
41Data not computed