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Properties

Label 16.8.66296687817...9481.2
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![-1021019, 2329344, -811071, -2537101, 3719006, -2463602, 1031729, -326727, 78779, -6762, -5645, 2767, -424, 21, -1, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - x^14 + 21*x^13 - 424*x^12 + 2767*x^11 - 5645*x^10 - 6762*x^9 + 78779*x^8 - 326727*x^7 + 1031729*x^6 - 2463602*x^5 + 3719006*x^4 - 2537101*x^3 - 811071*x^2 + 2329344*x - 1021019)
 
gp: K = bnfinit(x^16 - 4*x^15 - x^14 + 21*x^13 - 424*x^12 + 2767*x^11 - 5645*x^10 - 6762*x^9 + 78779*x^8 - 326727*x^7 + 1031729*x^6 - 2463602*x^5 + 3719006*x^4 - 2537101*x^3 - 811071*x^2 + 2329344*x - 1021019, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - x^{14} + 21 x^{13} - 424 x^{12} + 2767 x^{11} - 5645 x^{10} - 6762 x^{9} + 78779 x^{8} - 326727 x^{7} + 1031729 x^{6} - 2463602 x^{5} + 3719006 x^{4} - 2537101 x^{3} - 811071 x^{2} + 2329344 x - 1021019 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{17} a^{8} - \frac{2}{17} a^{7} + \frac{6}{17} a^{6} - \frac{3}{17} a^{5} + \frac{2}{17} a^{4} + \frac{3}{17} a^{3} + \frac{6}{17} a^{2} + \frac{2}{17} a + \frac{1}{17}$, $\frac{1}{17} a^{9} + \frac{2}{17} a^{7} - \frac{8}{17} a^{6} - \frac{4}{17} a^{5} + \frac{7}{17} a^{4} - \frac{5}{17} a^{3} - \frac{3}{17} a^{2} + \frac{5}{17} a + \frac{2}{17}$, $\frac{1}{17} a^{10} - \frac{4}{17} a^{7} + \frac{1}{17} a^{6} - \frac{4}{17} a^{5} + \frac{8}{17} a^{4} + \frac{8}{17} a^{3} - \frac{7}{17} a^{2} - \frac{2}{17} a - \frac{2}{17}$, $\frac{1}{17} a^{11} - \frac{7}{17} a^{7} + \frac{3}{17} a^{6} - \frac{4}{17} a^{5} - \frac{1}{17} a^{4} + \frac{5}{17} a^{3} + \frac{5}{17} a^{2} + \frac{6}{17} a + \frac{4}{17}$, $\frac{1}{34} a^{12} - \frac{1}{34} a^{10} - \frac{1}{34} a^{8} + \frac{6}{17} a^{7} + \frac{7}{17} a^{6} - \frac{15}{34} a^{5} - \frac{4}{17} a^{4} + \frac{15}{34} a^{3} - \frac{1}{17} a^{2} + \frac{1}{34} a - \frac{9}{34}$, $\frac{1}{34} a^{13} - \frac{1}{34} a^{11} - \frac{1}{34} a^{9} + \frac{2}{17} a^{7} + \frac{15}{34} a^{6} - \frac{3}{17} a^{5} - \frac{9}{34} a^{4} - \frac{2}{17} a^{3} - \frac{3}{34} a^{2} + \frac{1}{34} a - \frac{6}{17}$, $\frac{1}{12682} a^{14} + \frac{95}{12682} a^{13} + \frac{29}{12682} a^{12} + \frac{7}{12682} a^{11} + \frac{311}{12682} a^{10} + \frac{9}{12682} a^{9} - \frac{81}{6341} a^{8} + \frac{3233}{12682} a^{7} + \frac{5225}{12682} a^{6} - \frac{569}{12682} a^{5} + \frac{1345}{12682} a^{4} - \frac{5809}{12682} a^{3} + \frac{2878}{6341} a^{2} + \frac{319}{746} a - \frac{501}{6341}$, $\frac{1}{298082254075815377818801892414} a^{15} + \frac{4893201662470901107513902}{149041127037907688909400946207} a^{14} + \frac{1756078948751988272509100721}{149041127037907688909400946207} a^{13} + \frac{344669751492524820097946659}{149041127037907688909400946207} a^{12} + \frac{435554171594663003311786781}{149041127037907688909400946207} a^{11} - \frac{1523368726578203502654973298}{149041127037907688909400946207} a^{10} - \frac{1718085012416983177982189529}{298082254075815377818801892414} a^{9} + \frac{2769321030573388225906088367}{298082254075815377818801892414} a^{8} + \frac{4377496282629703705368123439}{149041127037907688909400946207} a^{7} - \frac{1082924794353625606717807892}{8767125119876922877023585071} a^{6} - \frac{31947720872626976427886327160}{149041127037907688909400946207} a^{5} + \frac{30290553934184048985040231548}{149041127037907688909400946207} a^{4} - \frac{32889726404888546581429757617}{298082254075815377818801892414} a^{3} + \frac{8282890128721839807228543099}{298082254075815377818801892414} a^{2} + \frac{56674866829101619216567608711}{298082254075815377818801892414} a - \frac{48999209129455937982049826520}{149041127037907688909400946207}$

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:  \( 6003656187.52 \) (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

\(\Q(\sqrt{221}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{13}, \sqrt{17})\), 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 ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$17.8.7.2$x^{8} - 153$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.4$x^{8} - 12393$$8$$1$$7$$C_8$$[\ ]_{8}$