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Properties

Label 16.0.17615759052...3125.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 1652141^{3}$
Root discriminant $32.76$
Ramified primes $5, 1652141$
Class number $30$ (GRH)
Class group $[30]$ (GRH)
Galois group 16T1651

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25771, -47515, 49513, -24189, 4200, 2664, -165, -1610, 1367, -453, -24, 85, -19, -5, 11, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 11*x^14 - 5*x^13 - 19*x^12 + 85*x^11 - 24*x^10 - 453*x^9 + 1367*x^8 - 1610*x^7 - 165*x^6 + 2664*x^5 + 4200*x^4 - 24189*x^3 + 49513*x^2 - 47515*x + 25771)
 
gp: K = bnfinit(x^16 - 3*x^15 + 11*x^14 - 5*x^13 - 19*x^12 + 85*x^11 - 24*x^10 - 453*x^9 + 1367*x^8 - 1610*x^7 - 165*x^6 + 2664*x^5 + 4200*x^4 - 24189*x^3 + 49513*x^2 - 47515*x + 25771, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 11 x^{14} - 5 x^{13} - 19 x^{12} + 85 x^{11} - 24 x^{10} - 453 x^{9} + 1367 x^{8} - 1610 x^{7} - 165 x^{6} + 2664 x^{5} + 4200 x^{4} - 24189 x^{3} + 49513 x^{2} - 47515 x + 25771 \)

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:  \(1761575905283218445703125=5^{8}\cdot 1652141^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.76$
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:  $5, 1652141$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{395621827370532310674212481563} a^{15} + \frac{11678977517907197243065069771}{56517403910076044382030354509} a^{14} - \frac{97474124516083226720489605305}{395621827370532310674212481563} a^{13} - \frac{27221942384352416942689780525}{56517403910076044382030354509} a^{12} + \frac{17580724214506821088850086813}{395621827370532310674212481563} a^{11} - \frac{19344339992423653591524175221}{56517403910076044382030354509} a^{10} + \frac{153069847691640288581756002066}{395621827370532310674212481563} a^{9} - \frac{6103532400504818984454384232}{56517403910076044382030354509} a^{8} + \frac{69980037721737635774776614003}{395621827370532310674212481563} a^{7} - \frac{99989761435907723082145539853}{395621827370532310674212481563} a^{6} - \frac{9936700191730243129860929846}{56517403910076044382030354509} a^{5} - \frac{176671166239532275237059004560}{395621827370532310674212481563} a^{4} - \frac{23698253617295233629581287102}{395621827370532310674212481563} a^{3} - \frac{126635147998212620784991712385}{395621827370532310674212481563} a^{2} + \frac{20942434438086190139210530513}{56517403910076044382030354509} a - \frac{46738076951528740856714951342}{395621827370532310674212481563}$

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

Class group and class number

$C_{30}$, which has order $30$ (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:  \( 6960.86418224 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1651:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4608
The 44 conjugacy class representatives for t16n1651
Character table for t16n1651 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.8.1032588125.1

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
Arithmetically equvalently 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 $16$ $16$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
1652141Data not computed