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Properties

Label 16.0.10460890548...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 3^{10}\cdot 5^{8}\cdot 11^{6}$
Root discriminant $15.44$
Ramified primes $2, 3, 5, 11$
Class number $1$
Class group Trivial
Galois group 16T1220

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19, 37, -55, -295, 858, -875, 230, 264, -125, -192, 227, -80, -3, 2, 8, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 8*x^14 + 2*x^13 - 3*x^12 - 80*x^11 + 227*x^10 - 192*x^9 - 125*x^8 + 264*x^7 + 230*x^6 - 875*x^5 + 858*x^4 - 295*x^3 - 55*x^2 + 37*x + 19)
 
gp: K = bnfinit(x^16 - 5*x^15 + 8*x^14 + 2*x^13 - 3*x^12 - 80*x^11 + 227*x^10 - 192*x^9 - 125*x^8 + 264*x^7 + 230*x^6 - 875*x^5 + 858*x^4 - 295*x^3 - 55*x^2 + 37*x + 19, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 8 x^{14} + 2 x^{13} - 3 x^{12} - 80 x^{11} + 227 x^{10} - 192 x^{9} - 125 x^{8} + 264 x^{7} + 230 x^{6} - 875 x^{5} + 858 x^{4} - 295 x^{3} - 55 x^{2} + 37 x + 19 \)

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:  \(10460890548900000000=2^{8}\cdot 3^{10}\cdot 5^{8}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.44$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{12} a^{13} + \frac{5}{12} a^{11} + \frac{1}{6} a^{10} + \frac{5}{12} a^{8} + \frac{1}{6} a^{7} - \frac{1}{12} a^{6} + \frac{1}{4} a^{5} - \frac{1}{3} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a + \frac{1}{4}$, $\frac{1}{144} a^{14} + \frac{5}{144} a^{13} + \frac{7}{48} a^{12} - \frac{7}{48} a^{11} + \frac{1}{8} a^{10} + \frac{61}{144} a^{9} - \frac{23}{48} a^{8} + \frac{29}{144} a^{7} + \frac{5}{24} a^{6} - \frac{29}{144} a^{5} + \frac{5}{24} a^{4} + \frac{11}{24} a^{3} + \frac{1}{6} a^{2} - \frac{1}{144} a + \frac{55}{144}$, $\frac{1}{14056812864} a^{15} + \frac{3705823}{7028406432} a^{14} + \frac{3926841}{780934048} a^{13} - \frac{61852141}{585700536} a^{12} - \frac{1880285497}{4685604288} a^{11} - \frac{5696010641}{14056812864} a^{10} + \frac{38614381}{292850268} a^{9} + \frac{378794057}{878550804} a^{8} + \frac{86414225}{4685604288} a^{7} - \frac{1709632991}{14056812864} a^{6} - \frac{1015537469}{4685604288} a^{5} - \frac{18008057}{97616756} a^{4} - \frac{673616153}{2342802144} a^{3} + \frac{6909728087}{14056812864} a^{2} + \frac{2466684647}{7028406432} a + \frac{111061447}{246610752}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{215237}{6181536} a^{15} + \frac{54019}{772692} a^{14} + \frac{152875}{1545384} a^{13} - \frac{1028897}{3090768} a^{12} - \frac{3718591}{6181536} a^{11} + \frac{13058257}{6181536} a^{10} - \frac{440269}{3090768} a^{9} - \frac{17331907}{3090768} a^{8} + \frac{8756349}{2060512} a^{7} + \frac{40110799}{6181536} a^{6} - \frac{66038963}{6181536} a^{5} - \frac{3564091}{1545384} a^{4} + \frac{12483127}{1030256} a^{3} - \frac{15644705}{2060512} a^{2} - \frac{293053}{1545384} a + \frac{191165}{325344} \) (order $6$)
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:  \( 2267.97988665 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1220:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 55 conjugacy class representatives for t16n1220 are not computed
Character table for t16n1220 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), 4.2.2475.1, 4.2.275.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 8.0.6125625.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.3$x^{8} + 2 x^{7} + 2 x^{6} + 16$$2$$4$$8$$C_2^3: C_4$$[2, 2, 2]^{4}$
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$