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Properties

Label 16.8.71576937566...5625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{12}\cdot 29^{6}\cdot 149^{4}$
Root discriminant $41.30$
Ramified primes $5, 29, 149$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1025

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![18301, 120019, 208986, -36648, -121338, 122230, -75142, 7302, 4472, -5798, 3270, -330, -90, 104, -24, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 24*x^14 + 104*x^13 - 90*x^12 - 330*x^11 + 3270*x^10 - 5798*x^9 + 4472*x^8 + 7302*x^7 - 75142*x^6 + 122230*x^5 - 121338*x^4 - 36648*x^3 + 208986*x^2 + 120019*x + 18301)
 
gp: K = bnfinit(x^16 - 4*x^15 - 24*x^14 + 104*x^13 - 90*x^12 - 330*x^11 + 3270*x^10 - 5798*x^9 + 4472*x^8 + 7302*x^7 - 75142*x^6 + 122230*x^5 - 121338*x^4 - 36648*x^3 + 208986*x^2 + 120019*x + 18301, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 24 x^{14} + 104 x^{13} - 90 x^{12} - 330 x^{11} + 3270 x^{10} - 5798 x^{9} + 4472 x^{8} + 7302 x^{7} - 75142 x^{6} + 122230 x^{5} - 121338 x^{4} - 36648 x^{3} + 208986 x^{2} + 120019 x + 18301 \)

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:  \(71576937566385674072265625=5^{12}\cdot 29^{6}\cdot 149^{4}\)
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{9590569} a^{14} + \frac{92644}{9590569} a^{13} - \frac{2775976}{9590569} a^{12} - \frac{3451297}{9590569} a^{11} + \frac{3465662}{9590569} a^{10} + \frac{92201}{9590569} a^{9} - \frac{4635223}{9590569} a^{8} + \frac{939106}{9590569} a^{7} + \frac{1204533}{9590569} a^{6} - \frac{1980476}{9590569} a^{5} + \frac{803212}{9590569} a^{4} + \frac{2689214}{9590569} a^{3} - \frac{669704}{9590569} a^{2} - \frac{872562}{9590569} a + \frac{3297301}{9590569}$, $\frac{1}{8674480826740464168926052697753342421} a^{15} - \frac{437398167300730399271976450939}{8674480826740464168926052697753342421} a^{14} - \frac{1476955174853714295075752569762568190}{8674480826740464168926052697753342421} a^{13} - \frac{2711280044789850584647425607236890931}{8674480826740464168926052697753342421} a^{12} + \frac{838525495522586633910419732785728018}{8674480826740464168926052697753342421} a^{11} - \frac{569049226348380853484072234337560615}{8674480826740464168926052697753342421} a^{10} - \frac{3080299220712052025720550915420148534}{8674480826740464168926052697753342421} a^{9} - \frac{1043871870902819365454848275975656295}{8674480826740464168926052697753342421} a^{8} + \frac{1721674988347782061965014058283807073}{8674480826740464168926052697753342421} a^{7} + \frac{3203196171785690315910455382247710750}{8674480826740464168926052697753342421} a^{6} - \frac{1394528090049599105444565506208391452}{8674480826740464168926052697753342421} a^{5} + \frac{1820421195944690079329906346608340051}{8674480826740464168926052697753342421} a^{4} - \frac{3114323991780893634497386394800835290}{8674480826740464168926052697753342421} a^{3} + \frac{4122714198269356917159029390572644922}{8674480826740464168926052697753342421} a^{2} - \frac{1894582424259745174721891993574900167}{8674480826740464168926052697753342421} a + \frac{2649273120333178204032669535802022819}{8674480826740464168926052697753342421}$

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:  $11$
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:  \( 11627974.5088 \) (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, 8.4.56780640625.1, 8.8.8460315453125.1, 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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