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Properties

Label 16.8.96313927189...0625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 29^{8}\cdot 149^{4}$
Root discriminant $42.07$
Ramified primes $5, 29, 149$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1439

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1745, -5115, 2634, -15050, -14843, 12072, 5945, -1840, -1966, 326, 748, -231, -213, 117, -3, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 - 3*x^14 + 117*x^13 - 213*x^12 - 231*x^11 + 748*x^10 + 326*x^9 - 1966*x^8 - 1840*x^7 + 5945*x^6 + 12072*x^5 - 14843*x^4 - 15050*x^3 + 2634*x^2 - 5115*x + 1745)
 
gp: K = bnfinit(x^16 - 7*x^15 - 3*x^14 + 117*x^13 - 213*x^12 - 231*x^11 + 748*x^10 + 326*x^9 - 1966*x^8 - 1840*x^7 + 5945*x^6 + 12072*x^5 - 14843*x^4 - 15050*x^3 + 2634*x^2 - 5115*x + 1745, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} - 3 x^{14} + 117 x^{13} - 213 x^{12} - 231 x^{11} + 748 x^{10} + 326 x^{9} - 1966 x^{8} - 1840 x^{7} + 5945 x^{6} + 12072 x^{5} - 14843 x^{4} - 15050 x^{3} + 2634 x^{2} - 5115 x + 1745 \)

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:  \(96313927189328563031640625=5^{8}\cdot 29^{8}\cdot 149^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.07$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{8} a^{13} - \frac{1}{4} a^{11} + \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{1}{8}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{1}{8} a^{12} + \frac{1}{8} a^{11} - \frac{7}{16} a^{10} - \frac{3}{16} a^{9} - \frac{7}{16} a^{8} + \frac{7}{16} a^{7} - \frac{5}{16} a^{6} + \frac{3}{16} a^{5} - \frac{1}{2} a^{4} - \frac{3}{16} a^{3} - \frac{5}{16} a^{2} + \frac{3}{16} a - \frac{7}{16}$, $\frac{1}{21173538945896770955826577952} a^{15} + \frac{201960427820870912024029097}{10586769472948385477913288976} a^{14} - \frac{440738481926448071383293761}{21173538945896770955826577952} a^{13} - \frac{11374105781539249868376545}{5293384736474192738956644488} a^{12} - \frac{523120057627606620416246943}{3024791277985252993689511136} a^{11} + \frac{4749796190730615938906291}{378098909748156624211188892} a^{10} - \frac{1922339069847629094701627555}{5293384736474192738956644488} a^{9} + \frac{2434122079568353759792789853}{10586769472948385477913288976} a^{8} - \frac{244809652414962653663261497}{5293384736474192738956644488} a^{7} + \frac{1541516320284899540407752027}{5293384736474192738956644488} a^{6} + \frac{894887937293712713267838771}{3024791277985252993689511136} a^{5} + \frac{6579049945836491768632159141}{21173538945896770955826577952} a^{4} + \frac{2851403908543554609100178593}{10586769472948385477913288976} a^{3} - \frac{698347841325149695269670527}{2646692368237096369478322244} a^{2} + \frac{318850989813172784740122873}{10586769472948385477913288976} a + \frac{8885344941671359342824909831}{21173538945896770955826577952}$

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:  \( 14920839.3484 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1439:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 74 conjugacy class representatives for t16n1439 are not computed
Character table for t16n1439 is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{5}, \sqrt{29})\), 8.4.65865543125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$149$149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.4.2.1$x^{4} + 745 x^{2} + 199809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
149.4.2.1$x^{4} + 745 x^{2} + 199809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$