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Properties

Label 16.0.71576937566...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{6}\cdot 149^{4}$
Root discriminant $41.30$
Ramified primes $5, 29, 149$
Class number $56$ (GRH)
Class group $[2, 28]$ (GRH)
Galois group 16T1025

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1680101, 310669, -39434, 344642, 288522, -52280, -85302, -5738, 11672, 5202, -1220, -1170, 240, 104, -24, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 24*x^14 + 104*x^13 + 240*x^12 - 1170*x^11 - 1220*x^10 + 5202*x^9 + 11672*x^8 - 5738*x^7 - 85302*x^6 - 52280*x^5 + 288522*x^4 + 344642*x^3 - 39434*x^2 + 310669*x + 1680101)
 
gp: K = bnfinit(x^16 - 4*x^15 - 24*x^14 + 104*x^13 + 240*x^12 - 1170*x^11 - 1220*x^10 + 5202*x^9 + 11672*x^8 - 5738*x^7 - 85302*x^6 - 52280*x^5 + 288522*x^4 + 344642*x^3 - 39434*x^2 + 310669*x + 1680101, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 24 x^{14} + 104 x^{13} + 240 x^{12} - 1170 x^{11} - 1220 x^{10} + 5202 x^{9} + 11672 x^{8} - 5738 x^{7} - 85302 x^{6} - 52280 x^{5} + 288522 x^{4} + 344642 x^{3} - 39434 x^{2} + 310669 x + 1680101 \)

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:  \(71576937566385674072265625=5^{12}\cdot 29^{6}\cdot 149^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.30$
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:  $5, 29, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{9590569} a^{14} + \frac{92644}{9590569} a^{13} + \frac{3121624}{9590569} a^{12} - \frac{3555664}{9590569} a^{11} + \frac{3427332}{9590569} a^{10} - \frac{2309983}{9590569} a^{9} + \frac{527557}{9590569} a^{8} + \frac{3810986}{9590569} a^{7} - \frac{2881793}{9590569} a^{6} - \frac{972326}{9590569} a^{5} + \frac{2206008}{9590569} a^{4} + \frac{1125526}{9590569} a^{3} - \frac{2943000}{9590569} a^{2} + \frac{4674955}{9590569} a - \frac{3731144}{9590569}$, $\frac{1}{26275282718424729698045030538099985794961} a^{15} + \frac{9283690135110284631960620271823}{2021175593724979207541925426007691214997} a^{14} + \frac{10047671972555781264229710508923073831576}{26275282718424729698045030538099985794961} a^{13} - \frac{5947675668291201801563363293722853878553}{26275282718424729698045030538099985794961} a^{12} - \frac{2380303411076234297519280272536798563380}{26275282718424729698045030538099985794961} a^{11} - \frac{12017601024473626616469446344494739510131}{26275282718424729698045030538099985794961} a^{10} - \frac{6042855705407772457897369670686497028812}{26275282718424729698045030538099985794961} a^{9} - \frac{2989659984913555646434406097691426769616}{26275282718424729698045030538099985794961} a^{8} - \frac{11884628987100451274993399293336523051863}{26275282718424729698045030538099985794961} a^{7} + \frac{11957595313478050388665975953606261927164}{26275282718424729698045030538099985794961} a^{6} + \frac{10739580933784070497926772844278617815211}{26275282718424729698045030538099985794961} a^{5} + \frac{9081851927937231346193073866715752990992}{26275282718424729698045030538099985794961} a^{4} + \frac{9479210284609426278829081491107551591999}{26275282718424729698045030538099985794961} a^{3} - \frac{3288514133006287439259309627305601782347}{26275282718424729698045030538099985794961} a^{2} - \frac{522589010393877709780559509483080746809}{2021175593724979207541925426007691214997} a - \frac{1940641676491945953550981265029058215753}{26275282718424729698045030538099985794961}$

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

Class group and class number

$C_{2}\times C_{28}$, which has order $56$ (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:  \( 29217.4391773 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1025:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 44 conjugacy class representatives for t16n1025
Character table for t16n1025 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.0.8460315453125.1, 8.4.56780640625.1, 8.4.78318125.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
5Data not computed
$29$29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
149Data not computed