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Properties

Label 16.0.67184640000...0000.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{14}$
Root discriminant $20.03$
Ramified primes $2, 3, 5$
Class number $4$
Class group $[4]$
Galois group $C_2\times C_2^3.C_4$ (as 16T99)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 6, 16, -12, 14, 24, 88, 438, 634, 366, 78, -7, -12, -6, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 6*x^14 - 12*x^13 - 7*x^12 + 78*x^11 + 366*x^10 + 634*x^9 + 438*x^8 + 88*x^7 + 24*x^6 + 14*x^5 - 12*x^4 + 16*x^3 + 6*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 - 6*x^14 - 12*x^13 - 7*x^12 + 78*x^11 + 366*x^10 + 634*x^9 + 438*x^8 + 88*x^7 + 24*x^6 + 14*x^5 - 12*x^4 + 16*x^3 + 6*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 6 x^{14} - 12 x^{13} - 7 x^{12} + 78 x^{11} + 366 x^{10} + 634 x^{9} + 438 x^{8} + 88 x^{7} + 24 x^{6} + 14 x^{5} - 12 x^{4} + 16 x^{3} + 6 x^{2} - 6 x + 1 \)

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:  \(671846400000000000000=2^{24}\cdot 3^{8}\cdot 5^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.03$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} - \frac{3}{11} a^{12} + \frac{5}{11} a^{11} + \frac{4}{11} a^{10} + \frac{4}{11} a^{9} + \frac{1}{11} a^{8} + \frac{5}{11} a^{7} + \frac{3}{11} a^{6} + \frac{3}{11} a^{5} + \frac{4}{11} a^{4} + \frac{3}{11} a^{3} + \frac{2}{11} a^{2} + \frac{3}{11} a - \frac{1}{11}$, $\frac{1}{11} a^{14} - \frac{4}{11} a^{12} - \frac{3}{11} a^{11} + \frac{5}{11} a^{10} + \frac{2}{11} a^{9} - \frac{3}{11} a^{8} - \frac{4}{11} a^{7} + \frac{1}{11} a^{6} + \frac{2}{11} a^{5} + \frac{4}{11} a^{4} - \frac{2}{11} a^{2} - \frac{3}{11} a - \frac{3}{11}$, $\frac{1}{5394713734751} a^{15} + \frac{242248861686}{5394713734751} a^{14} + \frac{169827045687}{5394713734751} a^{13} + \frac{51149243492}{5394713734751} a^{12} + \frac{945422843357}{5394713734751} a^{11} + \frac{645618266739}{5394713734751} a^{10} - \frac{1904939312142}{5394713734751} a^{9} - \frac{777815507202}{5394713734751} a^{8} + \frac{2573566081760}{5394713734751} a^{7} - \frac{1537712639964}{5394713734751} a^{6} - \frac{620428129175}{5394713734751} a^{5} - \frac{2241386412532}{5394713734751} a^{4} + \frac{141929321407}{5394713734751} a^{3} - \frac{382285055575}{5394713734751} a^{2} - \frac{67387660690}{283932301829} a + \frac{1834197649917}{5394713734751}$

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( -\frac{2817760306252}{5394713734751} a^{15} + \frac{5608078530828}{5394713734751} a^{14} + \frac{15887017750822}{5394713734751} a^{13} + \frac{37051382045166}{5394713734751} a^{12} + \frac{23427298474896}{5394713734751} a^{11} - \frac{208536202242066}{5394713734751} a^{10} - \frac{1033499390657384}{5394713734751} a^{9} - \frac{1875849239841426}{5394713734751} a^{8} - \frac{1575284008320894}{5394713734751} a^{7} - \frac{695567907651292}{5394713734751} a^{6} - \frac{305385126444918}{5394713734751} a^{5} - \frac{141354443457587}{5394713734751} a^{4} - \frac{19875564779350}{5394713734751} a^{3} - \frac{41415050139238}{5394713734751} a^{2} - \frac{1489321964022}{283932301829} a + \frac{8614084387397}{5394713734751} \) (order $10$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 9684.50346137 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^3.C_4$ (as 16T99):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2\times C_2^3.C_4$
Character table for $C_2\times C_2^3.C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{15}) \), 4.0.18000.1, \(\Q(\zeta_{5})\), \(\Q(\sqrt{3}, \sqrt{5})\), 8.0.20000000.1, 8.0.1620000000.1, 8.0.324000000.3

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 R R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
3Data not computed
5Data not computed