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Properties

Label 16.0.21767823360...000.10
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{12}\cdot 5^{12}$
Root discriminant $21.56$
Ramified primes $2, 3, 5$
Class number $4$
Class group $[2, 2]$
Galois group $C_4 \times D_4$ (as 16T19)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1021, -282, 570, 1220, -178, 726, 840, -510, 1119, -562, 552, -198, 138, -32, 18, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 18*x^14 - 32*x^13 + 138*x^12 - 198*x^11 + 552*x^10 - 562*x^9 + 1119*x^8 - 510*x^7 + 840*x^6 + 726*x^5 - 178*x^4 + 1220*x^3 + 570*x^2 - 282*x + 1021)
 
gp: K = bnfinit(x^16 - 2*x^15 + 18*x^14 - 32*x^13 + 138*x^12 - 198*x^11 + 552*x^10 - 562*x^9 + 1119*x^8 - 510*x^7 + 840*x^6 + 726*x^5 - 178*x^4 + 1220*x^3 + 570*x^2 - 282*x + 1021, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 18 x^{14} - 32 x^{13} + 138 x^{12} - 198 x^{11} + 552 x^{10} - 562 x^{9} + 1119 x^{8} - 510 x^{7} + 840 x^{6} + 726 x^{5} - 178 x^{4} + 1220 x^{3} + 570 x^{2} - 282 x + 1021 \)

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:  \(2176782336000000000000=2^{24}\cdot 3^{12}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.56$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2318} a^{13} + \frac{68}{1159} a^{12} + \frac{445}{2318} a^{11} + \frac{178}{1159} a^{10} + \frac{441}{2318} a^{9} + \frac{63}{2318} a^{8} - \frac{569}{2318} a^{7} + \frac{27}{1159} a^{6} - \frac{1095}{2318} a^{5} - \frac{357}{2318} a^{4} + \frac{1145}{2318} a^{3} - \frac{417}{1159} a^{2} + \frac{158}{1159} a + \frac{229}{2318}$, $\frac{1}{2318} a^{14} + \frac{493}{2318} a^{12} + \frac{52}{1159} a^{11} - \frac{12}{61} a^{10} + \frac{355}{2318} a^{9} + \frac{135}{2318} a^{8} + \frac{472}{1159} a^{7} - \frac{163}{1159} a^{6} + \frac{211}{2318} a^{5} + \frac{1019}{2318} a^{4} + \frac{535}{1159} a^{3} - \frac{1001}{2318} a^{2} - \frac{1023}{2318} a - \frac{505}{1159}$, $\frac{1}{2366678} a^{15} - \frac{119}{1183339} a^{14} + \frac{31}{2366678} a^{13} - \frac{204401}{2366678} a^{12} - \frac{164077}{1183339} a^{11} + \frac{482187}{2366678} a^{10} + \frac{5991}{62281} a^{9} + \frac{185428}{1183339} a^{8} + \frac{83844}{1183339} a^{7} + \frac{604535}{2366678} a^{6} - \frac{51553}{1183339} a^{5} + \frac{528492}{1183339} a^{4} - \frac{837965}{2366678} a^{3} - \frac{383489}{2366678} a^{2} + \frac{1022513}{2366678} a - \frac{1}{2318}$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  \( \frac{1}{122} a^{15} + \frac{15}{122} a^{13} + \frac{45}{61} a^{11} + \frac{11}{122} a^{10} + \frac{275}{122} a^{9} + \frac{55}{61} a^{8} + \frac{225}{61} a^{7} + \frac{385}{122} a^{6} + \frac{439}{122} a^{5} + \frac{275}{61} a^{4} + \frac{445}{122} a^{3} + \frac{275}{122} a^{2} + \frac{160}{61} a + \frac{47}{61} \) (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:  \( 13014.3141955 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times D_4$ (as 16T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_4 \times D_4$
Character table for $C_4 \times D_4$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{20})^+\), 4.0.18000.1, 4.0.10800.2, \(\Q(\sqrt{-3}, \sqrt{5})\), 4.0.432.1, 8.0.116640000.1, 8.0.2916000000.5, 8.0.324000000.2

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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.12.15$x^{8} + 2 x^{7} + 2 x^{4} + 12$$4$$2$$12$$C_2^2:C_4$$[2, 2]^{4}$
2.8.12.15$x^{8} + 2 x^{7} + 2 x^{4} + 12$$4$$2$$12$$C_2^2:C_4$$[2, 2]^{4}$
3Data not computed
5Data not computed