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Properties

Label 16.8.66296687817...9481.3
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![216037, 238045, -495964, -742224, 41314, 846560, 351039, -478075, 2743, 110448, -38363, 3887, -13, 200, -40, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 40*x^14 + 200*x^13 - 13*x^12 + 3887*x^11 - 38363*x^10 + 110448*x^9 + 2743*x^8 - 478075*x^7 + 351039*x^6 + 846560*x^5 + 41314*x^4 - 742224*x^3 - 495964*x^2 + 238045*x + 216037)
 
gp: K = bnfinit(x^16 - 6*x^15 - 40*x^14 + 200*x^13 - 13*x^12 + 3887*x^11 - 38363*x^10 + 110448*x^9 + 2743*x^8 - 478075*x^7 + 351039*x^6 + 846560*x^5 + 41314*x^4 - 742224*x^3 - 495964*x^2 + 238045*x + 216037, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 40 x^{14} + 200 x^{13} - 13 x^{12} + 3887 x^{11} - 38363 x^{10} + 110448 x^{9} + 2743 x^{8} - 478075 x^{7} + 351039 x^{6} + 846560 x^{5} + 41314 x^{4} - 742224 x^{3} - 495964 x^{2} + 238045 x + 216037 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{13} a^{8} - \frac{3}{13} a^{7} - \frac{5}{13} a^{6} - \frac{6}{13} a^{5} + \frac{2}{13} a^{4} + \frac{2}{13} a^{3} - \frac{2}{13} a^{2} + \frac{3}{13} a - \frac{4}{13}$, $\frac{1}{13} a^{9} - \frac{1}{13} a^{7} + \frac{5}{13} a^{6} - \frac{3}{13} a^{5} - \frac{5}{13} a^{4} + \frac{4}{13} a^{3} - \frac{3}{13} a^{2} + \frac{5}{13} a + \frac{1}{13}$, $\frac{1}{13} a^{10} + \frac{2}{13} a^{7} + \frac{5}{13} a^{6} + \frac{2}{13} a^{5} + \frac{6}{13} a^{4} - \frac{1}{13} a^{3} + \frac{3}{13} a^{2} + \frac{4}{13} a - \frac{4}{13}$, $\frac{1}{13} a^{11} - \frac{2}{13} a^{7} - \frac{1}{13} a^{6} + \frac{5}{13} a^{5} - \frac{5}{13} a^{4} - \frac{1}{13} a^{3} - \frac{5}{13} a^{2} + \frac{3}{13} a - \frac{5}{13}$, $\frac{1}{13} a^{12} + \frac{6}{13} a^{7} - \frac{5}{13} a^{6} - \frac{4}{13} a^{5} + \frac{3}{13} a^{4} - \frac{1}{13} a^{3} - \frac{1}{13} a^{2} + \frac{1}{13} a + \frac{5}{13}$, $\frac{1}{13} a^{13} - \frac{2}{13}$, $\frac{1}{13} a^{14} - \frac{2}{13} a$, $\frac{1}{410509474236269484565800857992208236426901} a^{15} + \frac{13884755061631948250610333318044080779565}{410509474236269484565800857992208236426901} a^{14} + \frac{3948984988253100603105677500051799910625}{410509474236269484565800857992208236426901} a^{13} + \frac{3103405989933432493838342730116612676934}{410509474236269484565800857992208236426901} a^{12} + \frac{13772340237143903605487584124531502540285}{410509474236269484565800857992208236426901} a^{11} - \frac{5733279197929595708985568057967813238204}{410509474236269484565800857992208236426901} a^{10} + \frac{334415834046805640223531928233791383806}{31577651864328421889676989076323710494377} a^{9} - \frac{11459218963340485941036673985863622784742}{410509474236269484565800857992208236426901} a^{8} - \frac{93656431424205132596288855823615799865576}{410509474236269484565800857992208236426901} a^{7} + \frac{61365543674172238553779294212833843080960}{410509474236269484565800857992208236426901} a^{6} + \frac{10890246923722859262194205345924828666119}{410509474236269484565800857992208236426901} a^{5} - \frac{4708596887287233372443013253368045746009}{31577651864328421889676989076323710494377} a^{4} - \frac{81061057315685780066904648271551956347755}{410509474236269484565800857992208236426901} a^{3} + \frac{197085227696608680914933277450562580351879}{410509474236269484565800857992208236426901} a^{2} - \frac{171212870327037768644965037541805394444440}{410509474236269484565800857992208236426901} a + \frac{117756873072740998699841816084945673905705}{410509474236269484565800857992208236426901}$

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:  \( 5979911668.23 \) (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 ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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