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Properties

Label 16.4.26536817251...0000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{8}\cdot 5^{14}\cdot 19^{8}$
Root discriminant $25.21$
Ramified primes $2, 5, 19$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times Q_8).C_2^3$ (as 16T226)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-29, -96, 28, 443, 671, 469, 170, 109, 14, -37, -40, 7, -24, -4, 2, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 2*x^14 - 4*x^13 - 24*x^12 + 7*x^11 - 40*x^10 - 37*x^9 + 14*x^8 + 109*x^7 + 170*x^6 + 469*x^5 + 671*x^4 + 443*x^3 + 28*x^2 - 96*x - 29)
 
gp: K = bnfinit(x^16 - 3*x^15 + 2*x^14 - 4*x^13 - 24*x^12 + 7*x^11 - 40*x^10 - 37*x^9 + 14*x^8 + 109*x^7 + 170*x^6 + 469*x^5 + 671*x^4 + 443*x^3 + 28*x^2 - 96*x - 29, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 2 x^{14} - 4 x^{13} - 24 x^{12} + 7 x^{11} - 40 x^{10} - 37 x^{9} + 14 x^{8} + 109 x^{7} + 170 x^{6} + 469 x^{5} + 671 x^{4} + 443 x^{3} + 28 x^{2} - 96 x - 29 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(26536817251562500000000=2^{8}\cdot 5^{14}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.21$
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:  $2, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{869} a^{14} + \frac{29}{869} a^{13} - \frac{116}{869} a^{12} - \frac{159}{869} a^{11} - \frac{222}{869} a^{10} + \frac{190}{869} a^{9} + \frac{146}{869} a^{8} - \frac{318}{869} a^{7} - \frac{375}{869} a^{6} + \frac{76}{869} a^{5} + \frac{326}{869} a^{4} + \frac{56}{869} a^{3} + \frac{377}{869} a^{2} - \frac{12}{869} a - \frac{172}{869}$, $\frac{1}{177676415194751} a^{15} + \frac{70777517431}{177676415194751} a^{14} + \frac{537944343038}{4333571102311} a^{13} + \frac{69385813225208}{177676415194751} a^{12} + \frac{88812620662301}{177676415194751} a^{11} + \frac{24323153420133}{177676415194751} a^{10} - \frac{77017234463106}{177676415194751} a^{9} + \frac{57231525669533}{177676415194751} a^{8} + \frac{65433462774325}{177676415194751} a^{7} - \frac{24618119115215}{177676415194751} a^{6} + \frac{2119338626359}{177676415194751} a^{5} + \frac{68432027354810}{177676415194751} a^{4} + \frac{24090519110791}{177676415194751} a^{3} + \frac{87220870108299}{177676415194751} a^{2} + \frac{1996235640603}{177676415194751} a - \frac{63926161591454}{177676415194751}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 52328.5996464 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times Q_8).C_2^3$ (as 16T226):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $(C_2\times Q_8).C_2^3$
Character table for $(C_2\times Q_8).C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.45125.1, 4.2.475.1, 4.2.2375.1, 8.4.2036265625.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$