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Properties

Label 14.14.1267782337...6128.1
Degree $14$
Signature $[14, 0]$
Discriminant $2^{14}\cdot 3^{7}\cdot 29^{12}$
Root discriminant $62.10$
Ramified primes $2, 3, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{14}$ (as 14T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-59, 1584, 8842, 5098, -15970, -9184, 11777, 4572, -4033, -924, 645, 74, -44, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 - 44*x^12 + 74*x^11 + 645*x^10 - 924*x^9 - 4033*x^8 + 4572*x^7 + 11777*x^6 - 9184*x^5 - 15970*x^4 + 5098*x^3 + 8842*x^2 + 1584*x - 59)
 
gp: K = bnfinit(x^14 - 2*x^13 - 44*x^12 + 74*x^11 + 645*x^10 - 924*x^9 - 4033*x^8 + 4572*x^7 + 11777*x^6 - 9184*x^5 - 15970*x^4 + 5098*x^3 + 8842*x^2 + 1584*x - 59, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} - 44 x^{12} + 74 x^{11} + 645 x^{10} - 924 x^{9} - 4033 x^{8} + 4572 x^{7} + 11777 x^{6} - 9184 x^{5} - 15970 x^{4} + 5098 x^{3} + 8842 x^{2} + 1584 x - 59 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12677823379379991227056128=2^{14}\cdot 3^{7}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.10$
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:  $2, 3, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(348=2^{2}\cdot 3\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{348}(1,·)$, $\chi_{348}(227,·)$, $\chi_{348}(169,·)$, $\chi_{348}(107,·)$, $\chi_{348}(239,·)$, $\chi_{348}(335,·)$, $\chi_{348}(49,·)$, $\chi_{348}(83,·)$, $\chi_{348}(277,·)$, $\chi_{348}(23,·)$, $\chi_{348}(25,·)$, $\chi_{348}(59,·)$, $\chi_{348}(313,·)$, $\chi_{348}(181,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{157471} a^{12} + \frac{75994}{157471} a^{11} + \frac{71884}{157471} a^{10} - \frac{39665}{157471} a^{9} + \frac{33536}{157471} a^{8} - \frac{39928}{157471} a^{7} + \frac{69399}{157471} a^{6} + \frac{60077}{157471} a^{5} - \frac{70896}{157471} a^{4} - \frac{4600}{9263} a^{3} - \frac{30724}{157471} a^{2} - \frac{15349}{157471} a + \frac{320}{2669}$, $\frac{1}{64485547816421} a^{13} + \frac{47747477}{64485547816421} a^{12} - \frac{1892147195967}{3793267518613} a^{11} + \frac{17479471879805}{64485547816421} a^{10} + \frac{30586694222046}{64485547816421} a^{9} - \frac{16626124351619}{64485547816421} a^{8} + \frac{12092521407334}{64485547816421} a^{7} + \frac{5087131237462}{64485547816421} a^{6} + \frac{28614617159203}{64485547816421} a^{5} + \frac{2884388454678}{64485547816421} a^{4} - \frac{30396892280229}{64485547816421} a^{3} + \frac{12037366113518}{64485547816421} a^{2} - \frac{13423815326656}{64485547816421} a - \frac{421582130325}{1092975386719}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $13$
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:  \( 64981687.802080065 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{14}$ (as 14T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 14
The 14 conjugacy class representatives for $C_{14}$
Character table for $C_{14}$

Intermediate fields

\(\Q(\sqrt{3}) \), 7.7.594823321.1

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

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 ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{14}$

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$2.14.14.15$x^{14} + 2 x^{13} + x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
$3$3.14.7.1$x^{14} - 54 x^{8} - 243 x^{4} - 729 x^{2} - 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.14.12.1$x^{14} + 2407 x^{7} + 1839267$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$