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Properties

Label 16.4.14683396539...5625.1
Degree $16$
Signature $[4, 6]$
Discriminant $3^{4}\cdot 5^{12}\cdot 41^{4}\cdot 1621^{2}$
Root discriminant $28.05$
Ramified primes $3, 5, 41, 1621$
Class number $1$
Class group Trivial
Galois group 16T1559

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![55, -530, 540, -320, -519, 1601, -12, -1393, 160, 619, -42, -179, 0, 33, 3, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 3*x^14 + 33*x^13 - 179*x^11 - 42*x^10 + 619*x^9 + 160*x^8 - 1393*x^7 - 12*x^6 + 1601*x^5 - 519*x^4 - 320*x^3 + 540*x^2 - 530*x + 55)
 
gp: K = bnfinit(x^16 - 6*x^15 + 3*x^14 + 33*x^13 - 179*x^11 - 42*x^10 + 619*x^9 + 160*x^8 - 1393*x^7 - 12*x^6 + 1601*x^5 - 519*x^4 - 320*x^3 + 540*x^2 - 530*x + 55, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 3 x^{14} + 33 x^{13} - 179 x^{11} - 42 x^{10} + 619 x^{9} + 160 x^{8} - 1393 x^{7} - 12 x^{6} + 1601 x^{5} - 519 x^{4} - 320 x^{3} + 540 x^{2} - 530 x + 55 \)

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:  \(146833965391572509765625=3^{4}\cdot 5^{12}\cdot 41^{4}\cdot 1621^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.05$
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:  $3, 5, 41, 1621$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{29} a^{12} - \frac{13}{29} a^{11} + \frac{1}{29} a^{10} - \frac{1}{29} a^{9} + \frac{10}{29} a^{8} - \frac{4}{29} a^{7} + \frac{9}{29} a^{6} - \frac{13}{29} a^{5} - \frac{9}{29} a^{4} + \frac{8}{29} a^{3} - \frac{7}{29} a^{2} - \frac{9}{29} a - \frac{12}{29}$, $\frac{1}{29} a^{13} + \frac{6}{29} a^{11} + \frac{12}{29} a^{10} - \frac{3}{29} a^{9} + \frac{10}{29} a^{8} - \frac{14}{29} a^{7} - \frac{12}{29} a^{6} - \frac{4}{29} a^{5} + \frac{7}{29} a^{4} + \frac{10}{29} a^{3} - \frac{13}{29} a^{2} - \frac{13}{29} a - \frac{11}{29}$, $\frac{1}{2523} a^{14} + \frac{5}{2523} a^{13} + \frac{14}{2523} a^{12} - \frac{11}{841} a^{11} - \frac{979}{2523} a^{10} + \frac{596}{2523} a^{9} - \frac{10}{29} a^{8} + \frac{49}{841} a^{7} - \frac{311}{2523} a^{6} - \frac{1190}{2523} a^{5} + \frac{194}{841} a^{4} - \frac{63}{841} a^{3} - \frac{151}{841} a^{2} + \frac{490}{2523} a + \frac{632}{2523}$, $\frac{1}{1423262136424323} a^{15} - \frac{148657401637}{1423262136424323} a^{14} + \frac{13866294583682}{1423262136424323} a^{13} - \frac{1413780850327}{474420712141441} a^{12} + \frac{572213483911271}{1423262136424323} a^{11} - \frac{99899936329501}{1423262136424323} a^{10} + \frac{268348255496}{861017626391} a^{9} + \frac{181450165626411}{474420712141441} a^{8} - \frac{152304311227223}{1423262136424323} a^{7} - \frac{44310119743046}{109481702801871} a^{6} + \frac{76639536946735}{474420712141441} a^{5} + \frac{5363003564682}{474420712141441} a^{4} - \frac{98311162486118}{474420712141441} a^{3} - \frac{410618139448871}{1423262136424323} a^{2} - \frac{162093150684022}{1423262136424323} a - \frac{2161308907158}{16359334901429}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 398441.698819 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1559:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 94 conjugacy class representatives for t16n1559 are not computed
Character table for t16n1559 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.5125.1, 8.4.42576578125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.8.4.1$x^{8} + 57154 x^{4} - 68921 x^{2} + 816644929$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
1621Data not computed