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Properties

Label 16.0.24031838291...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 13^{14}$
Root discriminant $38.57$
Ramified primes $5, 13$
Class number $50$ (GRH)
Class group $[5, 10]$ (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![15616, -33984, 73424, -74104, 18416, 34498, -35249, 10008, 5707, -6688, 3521, -1408, 531, -166, 39, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 39*x^14 - 166*x^13 + 531*x^12 - 1408*x^11 + 3521*x^10 - 6688*x^9 + 5707*x^8 + 10008*x^7 - 35249*x^6 + 34498*x^5 + 18416*x^4 - 74104*x^3 + 73424*x^2 - 33984*x + 15616)
 
gp: K = bnfinit(x^16 - 6*x^15 + 39*x^14 - 166*x^13 + 531*x^12 - 1408*x^11 + 3521*x^10 - 6688*x^9 + 5707*x^8 + 10008*x^7 - 35249*x^6 + 34498*x^5 + 18416*x^4 - 74104*x^3 + 73424*x^2 - 33984*x + 15616, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 39 x^{14} - 166 x^{13} + 531 x^{12} - 1408 x^{11} + 3521 x^{10} - 6688 x^{9} + 5707 x^{8} + 10008 x^{7} - 35249 x^{6} + 34498 x^{5} + 18416 x^{4} - 74104 x^{3} + 73424 x^{2} - 33984 x + 15616 \)

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:  \(24031838291621636962890625=5^{14}\cdot 13^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.57$
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:  $5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{8} a^{5} - \frac{1}{16} a^{4}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{16} a^{7} - \frac{1}{8} a^{6} - \frac{1}{16} a^{5}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{1}{16} a^{7} + \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{3}{16} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{32} a^{14} - \frac{1}{32} a^{12} - \frac{1}{32} a^{10} - \frac{1}{16} a^{9} - \frac{3}{32} a^{8} - \frac{1}{16} a^{7} + \frac{7}{32} a^{6} + \frac{3}{16} a^{5} + \frac{7}{32} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{83076158761305858592591808} a^{15} + \frac{57434636983132575518125}{20769039690326464648147952} a^{14} - \frac{487293299978947809774433}{83076158761305858592591808} a^{13} + \frac{182216546961557638271507}{10384519845163232324073976} a^{12} + \frac{2047098572516990925981167}{83076158761305858592591808} a^{11} - \frac{520296241115048026847677}{41538079380652929296295904} a^{10} + \frac{2071514292799811186689769}{83076158761305858592591808} a^{9} + \frac{3527454785693537793923199}{41538079380652929296295904} a^{8} + \frac{10102602485270625191356539}{83076158761305858592591808} a^{7} - \frac{2998977930351995166150855}{41538079380652929296295904} a^{6} + \frac{5884909888309910938467851}{83076158761305858592591808} a^{5} - \frac{1427471797886070929122603}{20769039690326464648147952} a^{4} - \frac{1257907227017928116001813}{2596129961290808081018494} a^{3} - \frac{204162468074717016082415}{10384519845163232324073976} a^{2} + \frac{398677113019847273794864}{1298064980645404040509247} a - \frac{510345521537606648962582}{1298064980645404040509247}$

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

Class group and class number

$C_{5}\times C_{10}$, which has order $50$ (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:  \( 136143.590528 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{5}, \sqrt{13})\), 4.4.274625.1, 4.4.274625.2, 8.8.75418890625.1, 8.0.4902227890625.1 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
5Data not computed
13Data not computed