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Properties

Label 16.0.68372792983...489.15
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 $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7311616, 0, 11927344, 0, 5593912, 0, 76947, 0, 30602, 0, 14523, 0, 793, 0, -26, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 26*x^14 + 793*x^12 + 14523*x^10 + 30602*x^8 + 76947*x^6 + 5593912*x^4 + 11927344*x^2 + 7311616)
 
gp: K = bnfinit(x^16 - 26*x^14 + 793*x^12 + 14523*x^10 + 30602*x^8 + 76947*x^6 + 5593912*x^4 + 11927344*x^2 + 7311616, 1)
 

Normalized defining polynomial

\( x^{16} - 26 x^{14} + 793 x^{12} + 14523 x^{10} + 30602 x^{8} + 76947 x^{6} + 5593912 x^{4} + 11927344 x^{2} + 7311616 \)

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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{9} + \frac{3}{8} a^{3}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{4} a^{7} + \frac{3}{16} a^{4} + \frac{5}{16} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{9} + \frac{1}{8} a^{7} - \frac{5}{32} a^{5} + \frac{13}{32} a^{3} - \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{17152} a^{12} + \frac{407}{17152} a^{10} - \frac{1}{16} a^{9} + \frac{223}{4288} a^{8} - \frac{1253}{17152} a^{6} - \frac{3387}{17152} a^{4} + \frac{5}{16} a^{3} + \frac{1719}{4288} a^{2} + \frac{121}{268}$, $\frac{1}{445952} a^{13} - \frac{381}{34304} a^{11} - \frac{189}{8576} a^{9} + \frac{67355}{445952} a^{7} - \frac{6775}{34304} a^{5} - \frac{981}{8576} a^{3} - \frac{415}{6968} a$, $\frac{1}{588135043445745243136} a^{14} - \frac{1}{891904} a^{13} + \frac{1272295683698557}{45241157188134249472} a^{12} + \frac{381}{68608} a^{11} + \frac{647401492352922097}{22620578594067124736} a^{10} - \frac{883}{17152} a^{9} - \frac{28968952769886973933}{588135043445745243136} a^{8} - \frac{67355}{891904} a^{7} - \frac{8998428487830326297}{45241157188134249472} a^{6} + \frac{6775}{68608} a^{5} + \frac{457009581295779735}{22620578594067124736} a^{4} - \frac{2235}{17152} a^{3} - \frac{14323437732048332529}{73516880430718155392} a^{2} - \frac{6553}{13936} a + \frac{11232749946270161}{27188195425561448}$, $\frac{1}{7645755564794688160768} a^{15} + \frac{306440248069149}{294067521722872621568} a^{13} - \frac{1}{34304} a^{12} - \frac{5335616726212560051}{588135043445745243136} a^{11} + \frac{665}{34304} a^{10} + \frac{367727311084114117359}{7645755564794688160768} a^{9} + \frac{313}{8576} a^{8} - \frac{52154820141301300949}{294067521722872621568} a^{7} + \frac{1253}{34304} a^{6} + \frac{131095106075550330991}{588135043445745243136} a^{5} + \frac{6603}{34304} a^{4} + \frac{462045726300681545389}{1911438891198672040192} a^{3} + \frac{2033}{8576} a^{2} + \frac{254224576197372889}{706893081064597648} a + \frac{147}{536}$

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:  \( 5393311.71147 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{-1139}) \), \(\Q(\sqrt{-67}) \), 4.0.22054457.2, 4.4.4913.1, \(\Q(\sqrt{17}, \sqrt{-67})\), 8.0.8268784250602433.1 x2, 8.4.1842010303097.1 x2, 8.0.486399073564849.4

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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}$