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Properties

Label 16.0.18357405541...9424.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{22}\cdot 17^{8}\cdot 89^{4}$
Root discriminant $32.85$
Ramified primes $2, 17, 89$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T392)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![104, -136, 152, -220, 964, -1906, 1779, -896, -171, 508, -98, 76, -18, -18, 7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 7*x^14 - 18*x^13 - 18*x^12 + 76*x^11 - 98*x^10 + 508*x^9 - 171*x^8 - 896*x^7 + 1779*x^6 - 1906*x^5 + 964*x^4 - 220*x^3 + 152*x^2 - 136*x + 104)
 
gp: K = bnfinit(x^16 + 7*x^14 - 18*x^13 - 18*x^12 + 76*x^11 - 98*x^10 + 508*x^9 - 171*x^8 - 896*x^7 + 1779*x^6 - 1906*x^5 + 964*x^4 - 220*x^3 + 152*x^2 - 136*x + 104, 1)
 

Normalized defining polynomial

\( x^{16} + 7 x^{14} - 18 x^{13} - 18 x^{12} + 76 x^{11} - 98 x^{10} + 508 x^{9} - 171 x^{8} - 896 x^{7} + 1779 x^{6} - 1906 x^{5} + 964 x^{4} - 220 x^{3} + 152 x^{2} - 136 x + 104 \)

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:  \(1835740554155063901159424=2^{22}\cdot 17^{8}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.85$
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:  $2, 17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7}$, $\frac{1}{24} a^{12} - \frac{1}{12} a^{11} + \frac{1}{24} a^{10} - \frac{1}{12} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{5}{24} a^{6} - \frac{1}{12} a^{5} - \frac{1}{4} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{48} a^{13} - \frac{1}{48} a^{12} - \frac{1}{48} a^{11} - \frac{1}{48} a^{10} + \frac{1}{48} a^{9} - \frac{1}{16} a^{8} + \frac{1}{48} a^{7} + \frac{5}{48} a^{6} + \frac{5}{24} a^{5} - \frac{1}{6} a^{4} - \frac{1}{12} a^{3} + \frac{1}{6} a^{2} + \frac{1}{6}$, $\frac{1}{1632} a^{14} - \frac{1}{204} a^{13} - \frac{5}{272} a^{12} - \frac{21}{272} a^{11} + \frac{1}{204} a^{10} + \frac{79}{816} a^{9} - \frac{31}{816} a^{8} + \frac{173}{816} a^{7} - \frac{145}{1632} a^{6} - \frac{57}{272} a^{5} - \frac{89}{408} a^{4} + \frac{33}{136} a^{3} + \frac{5}{51} a^{2} - \frac{53}{204} a + \frac{89}{204}$, $\frac{1}{894618670163424} a^{15} - \frac{190418169869}{894618670163424} a^{14} - \frac{598174695047}{447309335081712} a^{13} + \frac{469612529869}{37275777923476} a^{12} + \frac{40143845242363}{447309335081712} a^{11} - \frac{24060175964921}{447309335081712} a^{10} - \frac{11921530728877}{223654667540856} a^{9} + \frac{303817210043}{5885649145812} a^{8} - \frac{5523166812137}{47085193166496} a^{7} + \frac{2096996351645}{298206223387808} a^{6} + \frac{93719449479145}{447309335081712} a^{5} + \frac{12323081680375}{111827333770428} a^{4} + \frac{19078091523253}{223654667540856} a^{3} + \frac{50475868839043}{111827333770428} a^{2} - \frac{7309552711099}{18637888961738} a - \frac{26135090940173}{111827333770428}$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  \( 918961.713844 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T392):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.205768.2, 4.0.25721.1, 4.0.2312.1, 8.0.42340469824.6, 8.0.169361879296.7, 8.0.169361879296.14

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$89$89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$