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Properties

Label 14.2.41961914141...0221.1
Degree $14$
Signature $[2, 6]$
Discriminant $3^{7}\cdot 7^{7}\cdot 13^{12}$
Root discriminant $41.30$
Ramified primes $3, 7, 13$
Class number $1$
Class group Trivial
Galois group $D_{14}$ (as 14T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3647, 273, -423, -6656, -6373, -1950, 1108, 1742, 760, 0, -115, -26, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 3*x^12 - 26*x^11 - 115*x^10 + 760*x^8 + 1742*x^7 + 1108*x^6 - 1950*x^5 - 6373*x^4 - 6656*x^3 - 423*x^2 + 273*x - 3647)
 
gp: K = bnfinit(x^14 - 3*x^12 - 26*x^11 - 115*x^10 + 760*x^8 + 1742*x^7 + 1108*x^6 - 1950*x^5 - 6373*x^4 - 6656*x^3 - 423*x^2 + 273*x - 3647, 1)
 

Normalized defining polynomial

\( x^{14} - 3 x^{12} - 26 x^{11} - 115 x^{10} + 760 x^{8} + 1742 x^{7} + 1108 x^{6} - 1950 x^{5} - 6373 x^{4} - 6656 x^{3} - 423 x^{2} + 273 x - 3647 \)

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:  \(41961914141343110590221=3^{7}\cdot 7^{7}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.30$
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:  $3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{105} a^{10} - \frac{41}{105} a^{9} - \frac{47}{105} a^{8} + \frac{11}{35} a^{7} + \frac{47}{105} a^{6} + \frac{17}{105} a^{5} + \frac{10}{21} a^{4} - \frac{16}{35} a^{3} - \frac{26}{105} a^{2} - \frac{1}{3} a - \frac{2}{15}$, $\frac{1}{315} a^{11} + \frac{19}{105} a^{9} - \frac{4}{315} a^{8} - \frac{2}{9} a^{7} - \frac{52}{105} a^{6} + \frac{13}{35} a^{5} + \frac{16}{45} a^{4} + \frac{106}{315} a^{3} + \frac{6}{35} a^{2} + \frac{1}{15} a - \frac{7}{45}$, $\frac{1}{2205} a^{12} - \frac{1}{735} a^{11} + \frac{1}{245} a^{10} - \frac{1042}{2205} a^{9} - \frac{13}{45} a^{8} + \frac{22}{49} a^{7} + \frac{178}{735} a^{6} - \frac{211}{441} a^{5} - \frac{148}{441} a^{4} + \frac{52}{147} a^{3} - \frac{17}{245} a^{2} + \frac{44}{315} a - \frac{17}{35}$, $\frac{1}{231756750335562795} a^{13} + \frac{52188751849519}{231756750335562795} a^{12} - \frac{329304120987647}{231756750335562795} a^{11} - \frac{95007086819236}{231756750335562795} a^{10} - \frac{4833467738695574}{231756750335562795} a^{9} + \frac{6015459723360698}{46351350067112559} a^{8} + \frac{45644718316964828}{231756750335562795} a^{7} - \frac{34081626893013206}{231756750335562795} a^{6} - \frac{112815304198402039}{231756750335562795} a^{5} - \frac{11450275234296380}{46351350067112559} a^{4} - \frac{97914743259592262}{231756750335562795} a^{3} - \frac{112728986823904156}{231756750335562795} a^{2} - \frac{14013048575951}{6621621438158937} a + \frac{7638154826789978}{33108107190794685}$

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:  \( 3029371.24082 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{14}$ (as 14T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 28
The 10 conjugacy class representatives for $D_{14}$
Character table for $D_{14}$

Intermediate fields

\(\Q(\sqrt{21}) \), 7.1.1655595487.1

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 14 sibling: 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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$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}$
7.2.1.2$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}$
13Data not computed