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Properties

Label 14.2.312145214074303896.1
Degree $14$
Signature $[2, 6]$
Discriminant $2^{3}\cdot 3^{3}\cdot 11^{6}\cdot 13^{8}$
Root discriminant $17.77$
Ramified primes $2, 3, 11, 13$
Class number $1$
Class group Trivial
Galois group 14T48

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6, -16, 9, 24, -21, -23, 36, -3, -34, 23, 5, -21, 15, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 6*x^13 + 15*x^12 - 21*x^11 + 5*x^10 + 23*x^9 - 34*x^8 - 3*x^7 + 36*x^6 - 23*x^5 - 21*x^4 + 24*x^3 + 9*x^2 - 16*x - 6)
 
gp: K = bnfinit(x^14 - 6*x^13 + 15*x^12 - 21*x^11 + 5*x^10 + 23*x^9 - 34*x^8 - 3*x^7 + 36*x^6 - 23*x^5 - 21*x^4 + 24*x^3 + 9*x^2 - 16*x - 6, 1)
 

Normalized defining polynomial

\( x^{14} - 6 x^{13} + 15 x^{12} - 21 x^{11} + 5 x^{10} + 23 x^{9} - 34 x^{8} - 3 x^{7} + 36 x^{6} - 23 x^{5} - 21 x^{4} + 24 x^{3} + 9 x^{2} - 16 x - 6 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(312145214074303896=2^{3}\cdot 3^{3}\cdot 11^{6}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.77$
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:  $2, 3, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{43459001} a^{13} - \frac{9395385}{43459001} a^{12} + \frac{18897747}{43459001} a^{11} - \frac{7479640}{43459001} a^{10} - \frac{21213455}{43459001} a^{9} - \frac{4795658}{43459001} a^{8} - \frac{7461423}{43459001} a^{7} + \frac{8172233}{43459001} a^{6} + \frac{7164480}{43459001} a^{5} - \frac{10174058}{43459001} a^{4} + \frac{15162437}{43459001} a^{3} - \frac{18719640}{43459001} a^{2} + \frac{13545580}{43459001} a + \frac{2161576}{43459001}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4213.63772224 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T48:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5376
The 40 conjugacy class representatives for [2^7]F_42(7)=2wrF_42(7)
Character table for [2^7]F_42(7)=2wrF_42(7) is not computed

Intermediate fields

7.1.38014691.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 14 sibling: data not computed
Degree 28 siblings: data not computed
Degree 32 sibling: data not computed
Degree 42 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 ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.12.0.1$x^{12} - 26 x^{10} + 275 x^{8} - 1500 x^{6} + 4375 x^{4} - 6250 x^{2} + 7221$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.12.8.1$x^{12} - 39 x^{9} - 338 x^{6} + 10985 x^{3} + 228488$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$