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Properties

Label 16.0.18721417284...5625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{10}\cdot 61^{8}$
Root discriminant $21.36$
Ramified primes $5, 61$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2.D_4$ (as 16T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![304, -872, 1860, -2574, 3045, -2822, 2314, -1668, 1146, -710, 410, -178, 62, -20, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 10*x^14 - 20*x^13 + 62*x^12 - 178*x^11 + 410*x^10 - 710*x^9 + 1146*x^8 - 1668*x^7 + 2314*x^6 - 2822*x^5 + 3045*x^4 - 2574*x^3 + 1860*x^2 - 872*x + 304)
 
gp: K = bnfinit(x^16 - 4*x^15 + 10*x^14 - 20*x^13 + 62*x^12 - 178*x^11 + 410*x^10 - 710*x^9 + 1146*x^8 - 1668*x^7 + 2314*x^6 - 2822*x^5 + 3045*x^4 - 2574*x^3 + 1860*x^2 - 872*x + 304, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 10 x^{14} - 20 x^{13} + 62 x^{12} - 178 x^{11} + 410 x^{10} - 710 x^{9} + 1146 x^{8} - 1668 x^{7} + 2314 x^{6} - 2822 x^{5} + 3045 x^{4} - 2574 x^{3} + 1860 x^{2} - 872 x + 304 \)

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:  \(1872141728489072265625=5^{10}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.36$
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:  $5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{12} a^{11} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{12} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{120} a^{12} + \frac{1}{60} a^{11} + \frac{1}{20} a^{10} - \frac{1}{8} a^{9} + \frac{1}{60} a^{8} - \frac{1}{10} a^{7} + \frac{1}{8} a^{6} - \frac{1}{20} a^{5} + \frac{13}{60} a^{4} + \frac{31}{120} a^{3} + \frac{3}{20} a^{2} + \frac{1}{10} a - \frac{7}{15}$, $\frac{1}{120} a^{13} + \frac{1}{60} a^{11} + \frac{1}{40} a^{10} + \frac{1}{60} a^{9} + \frac{7}{60} a^{8} - \frac{7}{40} a^{7} + \frac{1}{5} a^{6} - \frac{11}{60} a^{5} + \frac{3}{40} a^{4} + \frac{23}{60} a^{3} + \frac{1}{20} a^{2} + \frac{1}{3} a - \frac{1}{15}$, $\frac{1}{1800} a^{14} + \frac{1}{450} a^{13} + \frac{1}{1800} a^{12} + \frac{49}{1800} a^{11} + \frac{47}{450} a^{10} - \frac{83}{1800} a^{9} - \frac{29}{600} a^{8} - \frac{7}{450} a^{7} + \frac{439}{1800} a^{6} + \frac{367}{1800} a^{5} + \frac{47}{225} a^{4} + \frac{739}{1800} a^{3} + \frac{61}{300} a^{2} + \frac{29}{90} a + \frac{43}{225}$, $\frac{1}{3859021800} a^{15} - \frac{43901}{482377725} a^{14} + \frac{9173813}{3859021800} a^{13} + \frac{15336047}{3859021800} a^{12} + \frac{841491}{42878020} a^{11} - \frac{98357183}{1286340600} a^{10} + \frac{293382979}{3859021800} a^{9} - \frac{12883628}{482377725} a^{8} + \frac{21267531}{85756040} a^{7} - \frac{155339311}{3859021800} a^{6} - \frac{262223429}{1929510900} a^{5} + \frac{297683177}{3859021800} a^{4} + \frac{183448567}{964755450} a^{3} + \frac{467598757}{964755450} a^{2} - \frac{31602679}{160792575} a + \frac{121151479}{482377725}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 22677.4816599 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.D_4$ (as 16T33):

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 $C_2^2.D_4$
Character table for $C_2^2.D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{305}) \), 4.0.1525.1 x2, 4.0.18605.1 x2, \(\Q(\sqrt{5}, \sqrt{61})\), 8.4.43268253125.1 x2, 8.0.8653650625.1 x2, 8.0.8653650625.2

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 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$61$61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
61.4.2.1$x^{4} + 183 x^{2} + 14884$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$