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Properties

Label 16.0.51952739971...6889.1
Degree $16$
Signature $[0, 8]$
Discriminant $7^{8}\cdot 37^{14}$
Root discriminant $62.33$
Ramified primes $7, 37$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![963, -438, 2977, -5499, 43350, 15954, 49532, -1644, 20353, -5127, 5317, -1524, 800, -162, 54, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 54*x^14 - 162*x^13 + 800*x^12 - 1524*x^11 + 5317*x^10 - 5127*x^9 + 20353*x^8 - 1644*x^7 + 49532*x^6 + 15954*x^5 + 43350*x^4 - 5499*x^3 + 2977*x^2 - 438*x + 963)
 
gp: K = bnfinit(x^16 - 6*x^15 + 54*x^14 - 162*x^13 + 800*x^12 - 1524*x^11 + 5317*x^10 - 5127*x^9 + 20353*x^8 - 1644*x^7 + 49532*x^6 + 15954*x^5 + 43350*x^4 - 5499*x^3 + 2977*x^2 - 438*x + 963, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 54 x^{14} - 162 x^{13} + 800 x^{12} - 1524 x^{11} + 5317 x^{10} - 5127 x^{9} + 20353 x^{8} - 1644 x^{7} + 49532 x^{6} + 15954 x^{5} + 43350 x^{4} - 5499 x^{3} + 2977 x^{2} - 438 x + 963 \)

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:  \(51952739971213319841055446889=7^{8}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.33$
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:  $7, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{21} a^{12} - \frac{1}{7} a^{11} - \frac{2}{21} a^{10} - \frac{1}{21} a^{9} - \frac{1}{21} a^{8} + \frac{8}{21} a^{6} - \frac{1}{7} a^{5} - \frac{8}{21} a^{4} - \frac{1}{7} a^{3} + \frac{1}{3} a^{2} + \frac{10}{21} a + \frac{3}{7}$, $\frac{1}{21} a^{13} + \frac{1}{7} a^{11} + \frac{1}{7} a^{9} - \frac{1}{7} a^{8} + \frac{1}{21} a^{7} - \frac{1}{7} a^{5} - \frac{2}{7} a^{4} - \frac{2}{21} a^{3} + \frac{1}{7} a^{2} + \frac{4}{21} a + \frac{2}{7}$, $\frac{1}{153279} a^{14} + \frac{200}{17031} a^{13} + \frac{1835}{153279} a^{12} + \frac{5554}{51093} a^{11} - \frac{295}{2433} a^{10} - \frac{767}{51093} a^{9} - \frac{2123}{21897} a^{8} + \frac{299}{51093} a^{7} + \frac{10874}{51093} a^{6} - \frac{7288}{51093} a^{5} - \frac{33067}{153279} a^{4} + \frac{355}{7299} a^{3} - \frac{8558}{21897} a^{2} + \frac{2512}{51093} a + \frac{4061}{17031}$, $\frac{1}{210239796200776550847320787} a^{15} - \frac{34592184135001620196}{70079932066925516949106929} a^{14} - \frac{4642141060598912525123002}{210239796200776550847320787} a^{13} + \frac{1080157404152012592772178}{70079932066925516949106929} a^{12} - \frac{3581533412319528068349566}{23359977355641838983035643} a^{11} - \frac{3107752960333357247013383}{70079932066925516949106929} a^{10} - \frac{22548959709832031113881533}{210239796200776550847320787} a^{9} - \frac{10897763269308501772285784}{70079932066925516949106929} a^{8} + \frac{4671514654305540183621194}{70079932066925516949106929} a^{7} - \frac{1144628919323216843861986}{10011418866703645278443847} a^{6} - \frac{58645973109043051848975538}{210239796200776550847320787} a^{5} - \frac{15391005702036495561497302}{70079932066925516949106929} a^{4} - \frac{24766387284288012937323464}{210239796200776550847320787} a^{3} + \frac{5727066353643095349132886}{23359977355641838983035643} a^{2} - \frac{2222653765538061609552485}{7786659118547279661011881} a - \frac{309123996170424336109200}{1112379874078182808715983}$

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:  $7$
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:  \( 87423196.4181 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

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 $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{37}) \), \(\Q(\sqrt{-259}) \), \(\Q(\sqrt{-7}) \), 4.0.50653.1, 4.4.2481997.1, \(\Q(\sqrt{-7}, \sqrt{37})\), 8.0.4651661979517.1 x2, 8.4.227931436996333.1 x2, 8.0.6160309108009.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$37$37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$