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Properties

Label 16.0.68372792983...489.25
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 67^{8}$
Root discriminant $97.65$
Ramified primes $17, 67$
Class number $2560$ (GRH)
Class group $[2, 2, 2, 4, 80]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![784570571, 495299900, -141482603, -98922188, 35639923, 5426132, -8912064, 105532, 1305907, 39274, -91308, -4866, 3758, 208, -86, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 86*x^14 + 208*x^13 + 3758*x^12 - 4866*x^11 - 91308*x^10 + 39274*x^9 + 1305907*x^8 + 105532*x^7 - 8912064*x^6 + 5426132*x^5 + 35639923*x^4 - 98922188*x^3 - 141482603*x^2 + 495299900*x + 784570571)
 
gp: K = bnfinit(x^16 - 4*x^15 - 86*x^14 + 208*x^13 + 3758*x^12 - 4866*x^11 - 91308*x^10 + 39274*x^9 + 1305907*x^8 + 105532*x^7 - 8912064*x^6 + 5426132*x^5 + 35639923*x^4 - 98922188*x^3 - 141482603*x^2 + 495299900*x + 784570571, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 86 x^{14} + 208 x^{13} + 3758 x^{12} - 4866 x^{11} - 91308 x^{10} + 39274 x^{9} + 1305907 x^{8} + 105532 x^{7} - 8912064 x^{6} + 5426132 x^{5} + 35639923 x^{4} - 98922188 x^{3} - 141482603 x^{2} + 495299900 x + 784570571 \)

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:  \(68372792983010839504523425519489=17^{14}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.65$
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:  $17, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{34} a^{8} - \frac{1}{17} a^{7} + \frac{3}{17} a^{6} + \frac{7}{17} a^{5} - \frac{15}{34} a^{4} - \frac{7}{17} a^{3} - \frac{11}{34} a^{2} - \frac{15}{34} a + \frac{1}{34}$, $\frac{1}{34} a^{9} + \frac{1}{17} a^{7} - \frac{4}{17} a^{6} + \frac{13}{34} a^{5} - \frac{5}{17} a^{4} - \frac{5}{34} a^{3} - \frac{3}{34} a^{2} + \frac{5}{34} a + \frac{1}{17}$, $\frac{1}{34} a^{10} - \frac{2}{17} a^{7} + \frac{1}{34} a^{6} - \frac{2}{17} a^{5} - \frac{9}{34} a^{4} - \frac{9}{34} a^{3} - \frac{7}{34} a^{2} - \frac{1}{17} a - \frac{1}{17}$, $\frac{1}{34} a^{11} - \frac{7}{34} a^{7} - \frac{7}{17} a^{6} + \frac{13}{34} a^{5} - \frac{1}{34} a^{4} + \frac{5}{34} a^{3} - \frac{6}{17} a^{2} + \frac{3}{17} a + \frac{2}{17}$, $\frac{1}{19618} a^{12} - \frac{27}{9809} a^{11} + \frac{105}{19618} a^{10} + \frac{217}{19618} a^{9} - \frac{245}{19618} a^{8} - \frac{3840}{9809} a^{7} - \frac{380}{9809} a^{6} + \frac{645}{9809} a^{5} + \frac{4235}{9809} a^{4} + \frac{34}{577} a^{3} - \frac{961}{9809} a^{2} + \frac{6641}{19618} a + \frac{2112}{9809}$, $\frac{1}{19618} a^{13} + \frac{37}{9809} a^{11} + \frac{117}{19618} a^{10} - \frac{67}{19618} a^{9} - \frac{69}{9809} a^{8} - \frac{6387}{19618} a^{7} - \frac{3142}{9809} a^{6} + \frac{4851}{19618} a^{5} - \frac{2487}{19618} a^{4} - \frac{8161}{19618} a^{3} + \frac{9021}{19618} a^{2} + \frac{1395}{9809} a - \frac{3083}{9809}$, $\frac{1}{922046} a^{14} + \frac{13}{922046} a^{13} + \frac{8}{461023} a^{12} - \frac{13099}{922046} a^{11} + \frac{4019}{922046} a^{10} + \frac{7177}{922046} a^{9} + \frac{2149}{461023} a^{8} - \frac{120477}{922046} a^{7} + \frac{158162}{461023} a^{6} + \frac{74235}{461023} a^{5} - \frac{92776}{461023} a^{4} + \frac{396147}{922046} a^{3} + \frac{198073}{922046} a^{2} + \frac{340211}{922046} a - \frac{352269}{922046}$, $\frac{1}{8180002858973166669509554612902150035692114} a^{15} + \frac{569286418960879694268854343362899133}{8180002858973166669509554612902150035692114} a^{14} - \frac{5170203249024394520759846498741757311}{4090001429486583334754777306451075017846057} a^{13} + \frac{84657250209294309831699026021797380573}{4090001429486583334754777306451075017846057} a^{12} + \frac{62075802312800225816551936441515778377207}{8180002858973166669509554612902150035692114} a^{11} - \frac{5119603814399128000221167448405425552262}{4090001429486583334754777306451075017846057} a^{10} - \frac{7064625466190053324132351965807391810800}{4090001429486583334754777306451075017846057} a^{9} - \frac{36655033940060517712514669055315087608825}{8180002858973166669509554612902150035692114} a^{8} + \frac{1625648538182559603811358664366517247541512}{4090001429486583334754777306451075017846057} a^{7} - \frac{539945096074422934962705399446879797984579}{4090001429486583334754777306451075017846057} a^{6} + \frac{451094457686943320682341410855409051189175}{8180002858973166669509554612902150035692114} a^{5} - \frac{171025528236812333720281562574570265752776}{4090001429486583334754777306451075017846057} a^{4} + \frac{1647987866770659656575990678906488919211920}{4090001429486583334754777306451075017846057} a^{3} - \frac{3857647934144397133870096431101019623046363}{8180002858973166669509554612902150035692114} a^{2} + \frac{632481199844340354374212961261004244407316}{4090001429486583334754777306451075017846057} a - \frac{1116344500458283136489602874733355160190535}{8180002858973166669509554612902150035692114}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{80}$, which has order $2560$ (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:  $7$
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:  \( 726530.496253 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-1139}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-67}) \), \(\Q(\sqrt{17}, \sqrt{-67})\), 4.4.4913.1, 4.0.22054457.2, 8.0.486399073564849.4, 8.4.1842010303097.2 x2

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
$67$67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$