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Properties

Label 16.4.35449289197...0625.1
Degree $16$
Signature $[4, 6]$
Discriminant $5^{14}\cdot 41^{5}\cdot 2239^{2}$
Root discriminant $34.23$
Ramified primes $5, 41, 2239$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1782

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-495, 5220, -17560, 21665, -11770, 5230, -1520, -1560, 1246, -718, 538, -206, 90, -41, 8, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 8*x^14 - 41*x^13 + 90*x^12 - 206*x^11 + 538*x^10 - 718*x^9 + 1246*x^8 - 1560*x^7 - 1520*x^6 + 5230*x^5 - 11770*x^4 + 21665*x^3 - 17560*x^2 + 5220*x - 495)
 
gp: K = bnfinit(x^16 - 3*x^15 + 8*x^14 - 41*x^13 + 90*x^12 - 206*x^11 + 538*x^10 - 718*x^9 + 1246*x^8 - 1560*x^7 - 1520*x^6 + 5230*x^5 - 11770*x^4 + 21665*x^3 - 17560*x^2 + 5220*x - 495, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 8 x^{14} - 41 x^{13} + 90 x^{12} - 206 x^{11} + 538 x^{10} - 718 x^{9} + 1246 x^{8} - 1560 x^{7} - 1520 x^{6} + 5230 x^{5} - 11770 x^{4} + 21665 x^{3} - 17560 x^{2} + 5220 x - 495 \)

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:  \(3544928919759039306640625=5^{14}\cdot 41^{5}\cdot 2239^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.23$
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:  $5, 41, 2239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{3} a^{5} + \frac{2}{9} a^{4} - \frac{1}{9} a^{3} + \frac{2}{9} a^{2}$, $\frac{1}{135} a^{11} + \frac{7}{135} a^{10} - \frac{7}{135} a^{9} + \frac{4}{135} a^{8} - \frac{2}{27} a^{7} - \frac{1}{9} a^{6} + \frac{4}{27} a^{5} - \frac{4}{27} a^{4} - \frac{7}{27} a^{3} + \frac{13}{27} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{405} a^{12} - \frac{1}{405} a^{11} - \frac{2}{45} a^{10} - \frac{2}{27} a^{9} + \frac{1}{135} a^{8} - \frac{5}{81} a^{7} - \frac{8}{81} a^{6} - \frac{2}{9} a^{5} - \frac{20}{81} a^{4} - \frac{10}{27} a^{3} + \frac{31}{81} a^{2} - \frac{2}{9} a + \frac{4}{9}$, $\frac{1}{1215} a^{13} - \frac{1}{1215} a^{11} - \frac{19}{405} a^{10} - \frac{17}{135} a^{9} + \frac{37}{243} a^{8} - \frac{22}{243} a^{7} + \frac{1}{243} a^{6} - \frac{47}{243} a^{5} - \frac{95}{243} a^{4} - \frac{98}{243} a^{3} + \frac{112}{243} a^{2} + \frac{11}{27} a - \frac{5}{27}$, $\frac{1}{18225} a^{14} + \frac{4}{18225} a^{13} - \frac{2}{3645} a^{12} - \frac{1}{729} a^{11} + \frac{5}{243} a^{10} + \frac{869}{18225} a^{9} - \frac{56}{2025} a^{8} - \frac{113}{1215} a^{7} - \frac{376}{3645} a^{6} - \frac{359}{729} a^{5} + \frac{1052}{3645} a^{4} + \frac{349}{729} a^{3} + \frac{214}{3645} a^{2} + \frac{43}{135} a - \frac{56}{405}$, $\frac{1}{933653427525} a^{15} - \frac{741688}{133379061075} a^{14} - \frac{60812}{957593259} a^{13} - \frac{37374262}{186730685505} a^{12} - \frac{513040124}{186730685505} a^{11} + \frac{2294417228}{71819494425} a^{10} - \frac{151039767874}{933653427525} a^{9} - \frac{6501690737}{62243561835} a^{8} + \frac{871870418}{14363898885} a^{7} + \frac{247136165}{2872779777} a^{6} - \frac{36547501073}{186730685505} a^{5} + \frac{3154927073}{37346137101} a^{4} - \frac{27274997572}{62243561835} a^{3} - \frac{16766241374}{186730685505} a^{2} - \frac{9278230301}{20747853945} a + \frac{119169194}{4149570789}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 2287279.97947 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1782:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16384
The 130 conjugacy class representatives for t16n1782 are not computed
Character table for t16n1782 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.5125.1, 8.6.294043671875.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
2239Data not computed