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Properties

Label 16.0.17537439366...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 19^{4}\cdot 103^{4}\cdot 2213^{2}$
Root discriminant $58.24$
Ramified primes $5, 19, 103, 2213$
Class number $1490$ (GRH)
Class group $[1490]$ (GRH)
Galois group 16T1046

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![520451, 9972, 410887, 34975, 205353, 18493, 67729, 4072, 15227, 441, 2533, 13, 319, -1, 26, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 26*x^14 - x^13 + 319*x^12 + 13*x^11 + 2533*x^10 + 441*x^9 + 15227*x^8 + 4072*x^7 + 67729*x^6 + 18493*x^5 + 205353*x^4 + 34975*x^3 + 410887*x^2 + 9972*x + 520451)
 
gp: K = bnfinit(x^16 + 26*x^14 - x^13 + 319*x^12 + 13*x^11 + 2533*x^10 + 441*x^9 + 15227*x^8 + 4072*x^7 + 67729*x^6 + 18493*x^5 + 205353*x^4 + 34975*x^3 + 410887*x^2 + 9972*x + 520451, 1)
 

Normalized defining polynomial

\( x^{16} + 26 x^{14} - x^{13} + 319 x^{12} + 13 x^{11} + 2533 x^{10} + 441 x^{9} + 15227 x^{8} + 4072 x^{7} + 67729 x^{6} + 18493 x^{5} + 205353 x^{4} + 34975 x^{3} + 410887 x^{2} + 9972 x + 520451 \)

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:  \(17537439366439787736572265625=5^{12}\cdot 19^{4}\cdot 103^{4}\cdot 2213^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.24$
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:  $5, 19, 103, 2213$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7730186719} a^{13} - \frac{1937621248}{7730186719} a^{12} + \frac{3703440718}{7730186719} a^{11} - \frac{2039112614}{7730186719} a^{10} + \frac{2408008904}{7730186719} a^{9} + \frac{2587728061}{7730186719} a^{8} + \frac{438777706}{7730186719} a^{7} + \frac{2959872281}{7730186719} a^{6} + \frac{1319366653}{7730186719} a^{5} - \frac{2064237495}{7730186719} a^{4} + \frac{1884351839}{7730186719} a^{3} + \frac{1963631013}{7730186719} a^{2} - \frac{1808852836}{7730186719} a + \frac{1302034794}{7730186719}$, $\frac{1}{7730186719} a^{14} + \frac{1229341129}{7730186719} a^{12} - \frac{2615583605}{7730186719} a^{11} + \frac{2625603320}{7730186719} a^{10} - \frac{491176276}{7730186719} a^{9} - \frac{3278471887}{7730186719} a^{8} + \frac{3527913596}{7730186719} a^{7} - \frac{309728187}{7730186719} a^{6} + \frac{785978505}{7730186719} a^{5} - \frac{3784451586}{7730186719} a^{4} + \frac{152952969}{7730186719} a^{3} - \frac{2012774531}{7730186719} a^{2} + \frac{2509529535}{7730186719} a - \frac{584330134}{7730186719}$, $\frac{1}{606116210450071} a^{15} + \frac{20965}{606116210450071} a^{14} - \frac{29603}{606116210450071} a^{13} + \frac{43150156435205}{606116210450071} a^{12} - \frac{114471258458934}{606116210450071} a^{11} - \frac{273865898423516}{606116210450071} a^{10} + \frac{114581950770202}{606116210450071} a^{9} + \frac{298351765984047}{606116210450071} a^{8} - \frac{285167580272381}{606116210450071} a^{7} + \frac{111657555400079}{606116210450071} a^{6} - \frac{298629832769933}{606116210450071} a^{5} - \frac{72769425614691}{606116210450071} a^{4} - \frac{158738623422578}{606116210450071} a^{3} - \frac{284141628725519}{606116210450071} a^{2} + \frac{226392207479957}{606116210450071} a - \frac{230995562942046}{606116210450071}$

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

Class group and class number

$C_{1490}$, which has order $1490$ (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:  \( 10491.0986114 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1046:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 768
The 40 conjugacy class representatives for t16n1046
Character table for t16n1046 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.1957.1, 8.8.2393655625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$103$103.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
103.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
103.8.4.1$x^{8} + 106090 x^{4} - 1092727 x^{2} + 2813772025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
2213Data not computed