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Properties

Label 16.0.11019960576...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{16}\cdot 5^{8}$
Root discriminant $13.42$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $C_2\times S_4$ (as 16T61)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 12, -12, 3, 30, -32, 26, 6, -6, 24, -30, 29, -20, 12, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 12*x^14 - 20*x^13 + 29*x^12 - 30*x^11 + 24*x^10 - 6*x^9 + 6*x^8 + 26*x^7 - 32*x^6 + 30*x^5 + 3*x^4 - 12*x^3 + 12*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^16 - 4*x^15 + 12*x^14 - 20*x^13 + 29*x^12 - 30*x^11 + 24*x^10 - 6*x^9 + 6*x^8 + 26*x^7 - 32*x^6 + 30*x^5 + 3*x^4 - 12*x^3 + 12*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 12 x^{14} - 20 x^{13} + 29 x^{12} - 30 x^{11} + 24 x^{10} - 6 x^{9} + 6 x^{8} + 26 x^{7} - 32 x^{6} + 30 x^{5} + 3 x^{4} - 12 x^{3} + 12 x^{2} - 4 x + 1 \)

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:  \(1101996057600000000=2^{16}\cdot 3^{16}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.42$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{12} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{2}$, $\frac{1}{18} a^{13} - \frac{1}{18} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{7} - \frac{2}{9} a^{6} - \frac{2}{9} a^{5} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2} - \frac{5}{18} a + \frac{1}{18}$, $\frac{1}{18} a^{14} + \frac{1}{18} a^{12} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{3} a^{7} - \frac{4}{9} a^{6} - \frac{1}{9} a^{5} + \frac{4}{9} a^{4} - \frac{2}{9} a^{3} - \frac{1}{2} a^{2} + \frac{1}{9} a + \frac{1}{18}$, $\frac{1}{27954} a^{15} - \frac{212}{13977} a^{14} - \frac{503}{27954} a^{13} + \frac{16}{13977} a^{12} + \frac{283}{13977} a^{11} + \frac{706}{13977} a^{10} + \frac{556}{4659} a^{9} + \frac{1393}{13977} a^{8} - \frac{5788}{13977} a^{7} + \frac{176}{4659} a^{6} + \frac{6515}{13977} a^{5} - \frac{679}{1553} a^{4} + \frac{2131}{27954} a^{3} - \frac{1805}{13977} a^{2} - \frac{1069}{27954} a + \frac{5515}{13977}$

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:  \( \frac{2819}{9318} a^{15} - \frac{4382}{4659} a^{14} + \frac{26333}{9318} a^{13} - \frac{35585}{9318} a^{12} + \frac{29042}{4659} a^{11} - \frac{25580}{4659} a^{10} + \frac{22903}{4659} a^{9} - \frac{3776}{4659} a^{8} + \frac{18082}{4659} a^{7} + \frac{14714}{1553} a^{6} - \frac{1033}{1553} a^{5} + \frac{31580}{4659} a^{4} + \frac{22027}{9318} a^{3} - \frac{667}{4659} a^{2} + \frac{27263}{9318} a + \frac{319}{3106} \) (order $6$)
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:  \( 984.52774662 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_4$ (as 16T61):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48
The 10 conjugacy class representatives for $C_2\times S_4$
Character table for $C_2\times S_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), 4.2.10800.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 8.0.1049760000.4, 8.4.116640000.2, 8.0.1049760000.5

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

Sibling fields

Degree 6 siblings: 6.2.233280.1, 6.0.699840.1
Degree 8 siblings: 8.4.116640000.2, 8.0.1049760000.5
Degree 12 siblings: 12.4.34012224000000.1, 12.2.16325867520000.1, 12.0.489776025600.1, 12.0.306110016000000.2, 12.0.306110016000000.6, 12.0.306110016000000.3
Degree 24 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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.8.11$x^{8} + 20 x^{2} + 4$$4$$2$$8$$S_4$$[4/3, 4/3]_{3}^{2}$
2.8.8.11$x^{8} + 20 x^{2} + 4$$4$$2$$8$$S_4$$[4/3, 4/3]_{3}^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.12.14.6$x^{12} + 3 x^{11} + 3 x^{10} - 6 x^{9} + 3 x^{8} + 9 x^{7} + 9 x^{4} + 9 x^{3} + 9$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$