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Properties

Label 16.8.66296687817...9481.4
Degree $16$
Signature $[8, 4]$
Discriminant $13^{14}\cdot 17^{14}$
Root discriminant $112.55$
Ramified primes $13, 17$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1214071, 8356311, 18151962, 19008036, 11276590, 4107675, 846586, -84916, -214461, -111033, -13169, 7111, 1586, -164, -55, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 55*x^14 - 164*x^13 + 1586*x^12 + 7111*x^11 - 13169*x^10 - 111033*x^9 - 214461*x^8 - 84916*x^7 + 846586*x^6 + 4107675*x^5 + 11276590*x^4 + 19008036*x^3 + 18151962*x^2 + 8356311*x + 1214071)
 
gp: K = bnfinit(x^16 - 2*x^15 - 55*x^14 - 164*x^13 + 1586*x^12 + 7111*x^11 - 13169*x^10 - 111033*x^9 - 214461*x^8 - 84916*x^7 + 846586*x^6 + 4107675*x^5 + 11276590*x^4 + 19008036*x^3 + 18151962*x^2 + 8356311*x + 1214071, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 55 x^{14} - 164 x^{13} + 1586 x^{12} + 7111 x^{11} - 13169 x^{10} - 111033 x^{9} - 214461 x^{8} - 84916 x^{7} + 846586 x^{6} + 4107675 x^{5} + 11276590 x^{4} + 19008036 x^{3} + 18151962 x^{2} + 8356311 x + 1214071 \)

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:  \(662966878170355779548558581239481=13^{14}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $112.55$
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:  $13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{13} a^{8} - \frac{1}{13} a^{7} - \frac{2}{13} a^{6} - \frac{6}{13} a^{5} + \frac{5}{13} a^{4} + \frac{6}{13} a^{3} - \frac{2}{13} a^{2} + \frac{1}{13} a + \frac{1}{13}$, $\frac{1}{13} a^{9} - \frac{3}{13} a^{7} + \frac{5}{13} a^{6} - \frac{1}{13} a^{5} - \frac{2}{13} a^{4} + \frac{4}{13} a^{3} - \frac{1}{13} a^{2} + \frac{2}{13} a + \frac{1}{13}$, $\frac{1}{13} a^{10} + \frac{2}{13} a^{7} + \frac{6}{13} a^{6} + \frac{6}{13} a^{5} + \frac{6}{13} a^{4} + \frac{4}{13} a^{3} - \frac{4}{13} a^{2} + \frac{4}{13} a + \frac{3}{13}$, $\frac{1}{13} a^{11} - \frac{5}{13} a^{7} - \frac{3}{13} a^{6} + \frac{5}{13} a^{5} - \frac{6}{13} a^{4} - \frac{3}{13} a^{3} - \frac{5}{13} a^{2} + \frac{1}{13} a - \frac{2}{13}$, $\frac{1}{13} a^{12} + \frac{5}{13} a^{7} - \frac{5}{13} a^{6} + \frac{3}{13} a^{5} - \frac{4}{13} a^{4} - \frac{1}{13} a^{3} + \frac{4}{13} a^{2} + \frac{3}{13} a + \frac{5}{13}$, $\frac{1}{13} a^{13} - \frac{5}{13}$, $\frac{1}{689} a^{14} - \frac{9}{689} a^{13} + \frac{19}{689} a^{12} - \frac{25}{689} a^{11} + \frac{9}{689} a^{10} - \frac{11}{689} a^{9} - \frac{8}{689} a^{8} + \frac{149}{689} a^{7} - \frac{200}{689} a^{6} - \frac{98}{689} a^{5} + \frac{305}{689} a^{4} + \frac{15}{53} a^{3} + \frac{127}{689} a^{2} - \frac{6}{689} a + \frac{3}{13}$, $\frac{1}{1508141497573547214829021807518454869910187} a^{15} + \frac{807203969794435843633541651185165194726}{1508141497573547214829021807518454869910187} a^{14} + \frac{30936457371272640727789224962423666940963}{1508141497573547214829021807518454869910187} a^{13} + \frac{33584464218426589783205962112771045551457}{1508141497573547214829021807518454869910187} a^{12} + \frac{33471988582060108811259543927081108113146}{1508141497573547214829021807518454869910187} a^{11} + \frac{42217255582558168887073767089731654001696}{1508141497573547214829021807518454869910187} a^{10} - \frac{28855151340633101937835430975083726718789}{1508141497573547214829021807518454869910187} a^{9} - \frac{37218548569058822033860686089988508174008}{1508141497573547214829021807518454869910187} a^{8} - \frac{675991633646767070604729960060865307937312}{1508141497573547214829021807518454869910187} a^{7} - \frac{459313784579747860796553297442944549790678}{1508141497573547214829021807518454869910187} a^{6} - \frac{469915385630093397190733250954955886779935}{1508141497573547214829021807518454869910187} a^{5} + \frac{100416663382098680930761543293721694758052}{1508141497573547214829021807518454869910187} a^{4} - \frac{653000823935024791720797393500932886312375}{1508141497573547214829021807518454869910187} a^{3} + \frac{39206318142917220210013460660882894635525}{116010884428734401140693985193727297685399} a^{2} + \frac{260603521689399627797871077250440464098875}{1508141497573547214829021807518454869910187} a - \frac{12285235910187971937929763268904990649603}{28455499954217871977906071839970846602079}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 4584367839.59 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{221}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{13}, \sqrt{17})\), 8.8.116507435287321.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.8.7.1$x^{8} - 13$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
13.8.7.1$x^{8} - 13$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$17$17.8.7.4$x^{8} - 12393$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.4$x^{8} - 12393$$8$$1$$7$$C_8$$[\ ]_{8}$