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Properties

Label 16.8.41796373745...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 3^{6}\cdot 5^{12}\cdot 241^{3}$
Root discriminant $39.93$
Ramified primes $2, 3, 5, 241$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2.D_4^2.C_2$ (as 16T660)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3559, 14002, -20281, 9078, 8552, -12576, 5387, -808, 571, -294, -201, 120, 21, -2, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 10*x^14 - 2*x^13 + 21*x^12 + 120*x^11 - 201*x^10 - 294*x^9 + 571*x^8 - 808*x^7 + 5387*x^6 - 12576*x^5 + 8552*x^4 + 9078*x^3 - 20281*x^2 + 14002*x - 3559)
 
gp: K = bnfinit(x^16 - 10*x^14 - 2*x^13 + 21*x^12 + 120*x^11 - 201*x^10 - 294*x^9 + 571*x^8 - 808*x^7 + 5387*x^6 - 12576*x^5 + 8552*x^4 + 9078*x^3 - 20281*x^2 + 14002*x - 3559, 1)
 

Normalized defining polynomial

\( x^{16} - 10 x^{14} - 2 x^{13} + 21 x^{12} + 120 x^{11} - 201 x^{10} - 294 x^{9} + 571 x^{8} - 808 x^{7} + 5387 x^{6} - 12576 x^{5} + 8552 x^{4} + 9078 x^{3} - 20281 x^{2} + 14002 x - 3559 \)

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:  \(41796373745664000000000000=2^{24}\cdot 3^{6}\cdot 5^{12}\cdot 241^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.93$
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:  $2, 3, 5, 241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11} a^{14} - \frac{2}{11} a^{11} - \frac{1}{11} a^{10} + \frac{1}{11} a^{9} - \frac{2}{11} a^{8} + \frac{2}{11} a^{7} + \frac{1}{11} a^{6} + \frac{4}{11} a^{5} - \frac{4}{11} a^{4} + \frac{4}{11} a^{3} - \frac{2}{11} a^{2} - \frac{1}{11} a + \frac{5}{11}$, $\frac{1}{7242422470047506599561} a^{15} - \frac{25980690019721923242}{7242422470047506599561} a^{14} - \frac{139474765148977719438}{658402042731591509051} a^{13} - \frac{1880249646081996503635}{7242422470047506599561} a^{12} + \frac{1715358195009351942385}{7242422470047506599561} a^{11} + \frac{3541482653895929599988}{7242422470047506599561} a^{10} - \frac{6357729254480923828}{658402042731591509051} a^{9} - \frac{2020429944341044802711}{7242422470047506599561} a^{8} - \frac{1005728351995263298166}{7242422470047506599561} a^{7} - \frac{2293560610641532985626}{7242422470047506599561} a^{6} - \frac{3496909442258114812200}{7242422470047506599561} a^{5} - \frac{1312837843491820471368}{7242422470047506599561} a^{4} + \frac{1247115050758029811650}{7242422470047506599561} a^{3} - \frac{2372604669528334265964}{7242422470047506599561} a^{2} - \frac{3212533812965819177074}{7242422470047506599561} a - \frac{2369428730306589609978}{7242422470047506599561}$

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:  \( 2753186.32443 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2.D_4^2.C_2$ (as 16T660):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 25 conjugacy class representatives for $C_2.D_4^2.C_2$
Character table for $C_2.D_4^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.5552640000.1

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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
5Data not computed
241Data not computed