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Properties

Label 16.0.584325558976905216.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{12}$
Root discriminant $12.89$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $(C_2^2\times C_4):C_2$ (as 16T54)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -32, 112, -256, 500, -824, 1208, -1528, 1654, -1504, 1136, -704, 350, -136, 40, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 40*x^14 - 136*x^13 + 350*x^12 - 704*x^11 + 1136*x^10 - 1504*x^9 + 1654*x^8 - 1528*x^7 + 1208*x^6 - 824*x^5 + 500*x^4 - 256*x^3 + 112*x^2 - 32*x + 4)
 
gp: K = bnfinit(x^16 - 8*x^15 + 40*x^14 - 136*x^13 + 350*x^12 - 704*x^11 + 1136*x^10 - 1504*x^9 + 1654*x^8 - 1528*x^7 + 1208*x^6 - 824*x^5 + 500*x^4 - 256*x^3 + 112*x^2 - 32*x + 4, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 40 x^{14} - 136 x^{13} + 350 x^{12} - 704 x^{11} + 1136 x^{10} - 1504 x^{9} + 1654 x^{8} - 1528 x^{7} + 1208 x^{6} - 824 x^{5} + 500 x^{4} - 256 x^{3} + 112 x^{2} - 32 x + 4 \)

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:  \(584325558976905216=2^{40}\cdot 3^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.89$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{142} a^{14} + \frac{3}{71} a^{13} - \frac{19}{142} a^{12} + \frac{9}{71} a^{11} - \frac{9}{71} a^{10} + \frac{10}{71} a^{9} + \frac{7}{71} a^{8} - \frac{25}{71} a^{7} - \frac{27}{71} a^{6} + \frac{19}{71} a^{5} - \frac{26}{71} a^{4} - \frac{14}{71} a^{3} + \frac{9}{71} a^{2} + \frac{12}{71} a + \frac{2}{71}$, $\frac{1}{4786678} a^{15} + \frac{2747}{2393339} a^{14} + \frac{665235}{4786678} a^{13} + \frac{437831}{4786678} a^{12} + \frac{252940}{2393339} a^{11} - \frac{911217}{4786678} a^{10} - \frac{526958}{2393339} a^{9} + \frac{155967}{2393339} a^{8} + \frac{15654}{184103} a^{7} - \frac{887338}{2393339} a^{6} - \frac{901327}{2393339} a^{5} + \frac{638440}{2393339} a^{4} - \frac{353723}{2393339} a^{3} - \frac{568012}{2393339} a^{2} + \frac{94258}{2393339} a + \frac{1106932}{2393339}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{42626}{184103} a^{15} - \frac{334048}{184103} a^{14} + \frac{1655042}{184103} a^{13} - \frac{5552664}{184103} a^{12} + \frac{14115058}{184103} a^{11} - \frac{27935298}{184103} a^{10} + \frac{44189346}{184103} a^{9} - \frac{113914699}{368206} a^{8} + \frac{60384304}{184103} a^{7} - \frac{53101948}{184103} a^{6} + \frac{39523088}{184103} a^{5} - \frac{25205790}{184103} a^{4} + \frac{14480404}{184103} a^{3} - \frac{6812520}{184103} a^{2} + \frac{2786828}{184103} a - \frac{349774}{184103} \) (order $12$)
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:  \( 1216.56129898 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2\times C_4):C_2$ (as 16T54):

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 $(C_2^2\times C_4):C_2$
Character table for $(C_2^2\times C_4):C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), 4.0.1728.1 x2, \(\Q(\zeta_{12})\), 4.2.6912.1 x2, 8.0.47775744.1, 8.0.47775744.2, 8.0.5308416.2

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 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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
3Data not computed