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Properties

Label 16.0.32978890425...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 5^{12}\cdot 29^{2}\cdot 89^{4}$
Root discriminant $22.13$
Ramified primes $2, 5, 29, 89$
Class number $2$
Class group $[2]$
Galois group 16T1086

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -22, 167, -533, 867, -810, 563, -429, 260, -31, -47, 5, 17, -12, 7, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 7*x^14 - 12*x^13 + 17*x^12 + 5*x^11 - 47*x^10 - 31*x^9 + 260*x^8 - 429*x^7 + 563*x^6 - 810*x^5 + 867*x^4 - 533*x^3 + 167*x^2 - 22*x + 1)
 
gp: K = bnfinit(x^16 - 3*x^15 + 7*x^14 - 12*x^13 + 17*x^12 + 5*x^11 - 47*x^10 - 31*x^9 + 260*x^8 - 429*x^7 + 563*x^6 - 810*x^5 + 867*x^4 - 533*x^3 + 167*x^2 - 22*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 7 x^{14} - 12 x^{13} + 17 x^{12} + 5 x^{11} - 47 x^{10} - 31 x^{9} + 260 x^{8} - 429 x^{7} + 563 x^{6} - 810 x^{5} + 867 x^{4} - 533 x^{3} + 167 x^{2} - 22 x + 1 \)

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:  \(3297889042562500000000=2^{8}\cdot 5^{12}\cdot 29^{2}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.13$
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:  $2, 5, 29, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11} a^{14} + \frac{1}{11} a^{13} - \frac{5}{11} a^{12} - \frac{4}{11} a^{11} + \frac{4}{11} a^{10} - \frac{3}{11} a^{9} - \frac{2}{11} a^{8} - \frac{2}{11} a^{7} - \frac{2}{11} a^{6} + \frac{2}{11} a^{5} - \frac{2}{11} a^{4} - \frac{3}{11} a^{3} - \frac{4}{11} a^{2} + \frac{5}{11} a - \frac{2}{11}$, $\frac{1}{1658403526571} a^{15} - \frac{55381488564}{1658403526571} a^{14} - \frac{648114470290}{1658403526571} a^{13} + \frac{542697109656}{1658403526571} a^{12} + \frac{304880731594}{1658403526571} a^{11} - \frac{124260585293}{1658403526571} a^{10} + \frac{359745507789}{1658403526571} a^{9} - \frac{50247605297}{1658403526571} a^{8} - \frac{127992835024}{1658403526571} a^{7} - \frac{167047613772}{1658403526571} a^{6} + \frac{428426665651}{1658403526571} a^{5} + \frac{42887684283}{150763956961} a^{4} - \frac{665421642748}{1658403526571} a^{3} - \frac{3263585931}{150763956961} a^{2} - \frac{531781931707}{1658403526571} a + \frac{668418055385}{1658403526571}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{1257343679519}{1658403526571} a^{15} - \frac{4249463244043}{1658403526571} a^{14} + \frac{9859549401969}{1658403526571} a^{13} - \frac{17400801311904}{1658403526571} a^{12} + \frac{24919142218004}{1658403526571} a^{11} + \frac{1884693310228}{1658403526571} a^{10} - \frac{6018792098959}{150763956961} a^{9} - \frac{20327406094357}{1658403526571} a^{8} + \frac{359535063237065}{1658403526571} a^{7} - \frac{643991964815314}{1658403526571} a^{6} + \frac{816862514906740}{1658403526571} a^{5} - \frac{1164550745023922}{1658403526571} a^{4} + \frac{1321832594091254}{1658403526571} a^{3} - \frac{842459576130563}{1658403526571} a^{2} + \frac{241713767942523}{1658403526571} a - \frac{17284148606309}{1658403526571} \) (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:  \( 28633.1838965 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1086:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 97 conjugacy class representatives for t16n1086 are not computed
Character table for t16n1086 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.0.11125.1, 4.4.2225.1, 8.0.123765625.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.8.5$x^{8} + 2 x^{7} + 8 x^{2} + 16$$2$$4$$8$$C_8:C_2$$[2, 2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
$89$89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.8.4.1$x^{8} + 427734 x^{4} - 704969 x^{2} + 45739093689$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$