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Properties

Label 14.2.31003224101...0544.2
Degree $14$
Signature $[2, 6]$
Discriminant $2^{12}\cdot 3^{13}\cdot 7^{15}$
Root discriminant $40.41$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $F_7 \times C_2$ (as 14T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1097, -1645, 1666, -413, 707, 532, -1869, 1035, 315, -476, 203, -77, 28, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 28*x^12 - 77*x^11 + 203*x^10 - 476*x^9 + 315*x^8 + 1035*x^7 - 1869*x^6 + 532*x^5 + 707*x^4 - 413*x^3 + 1666*x^2 - 1645*x - 1097)
 
gp: K = bnfinit(x^14 - 7*x^13 + 28*x^12 - 77*x^11 + 203*x^10 - 476*x^9 + 315*x^8 + 1035*x^7 - 1869*x^6 + 532*x^5 + 707*x^4 - 413*x^3 + 1666*x^2 - 1645*x - 1097, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} + 28 x^{12} - 77 x^{11} + 203 x^{10} - 476 x^{9} + 315 x^{8} + 1035 x^{7} - 1869 x^{6} + 532 x^{5} + 707 x^{4} - 413 x^{3} + 1666 x^{2} - 1645 x - 1097 \)

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:  \(31003224101752232300544=2^{12}\cdot 3^{13}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.41$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{508} a^{10} - \frac{5}{508} a^{9} - \frac{37}{508} a^{8} - \frac{19}{127} a^{7} + \frac{123}{508} a^{6} - \frac{251}{508} a^{5} + \frac{23}{508} a^{4} + \frac{41}{254} a^{3} - \frac{227}{508} a^{2} + \frac{113}{508} a - \frac{11}{508}$, $\frac{1}{508} a^{11} - \frac{31}{254} a^{9} - \frac{7}{508} a^{8} - \frac{3}{508} a^{7} + \frac{55}{254} a^{6} - \frac{54}{127} a^{5} + \frac{197}{508} a^{4} + \frac{183}{508} a^{3} + \frac{62}{127} a^{2} - \frac{52}{127} a + \frac{199}{508}$, $\frac{1}{6492748} a^{12} - \frac{3}{3246374} a^{11} - \frac{2637}{6492748} a^{10} + \frac{3310}{1623187} a^{9} + \frac{11700}{1623187} a^{8} - \frac{133353}{3246374} a^{7} + \frac{392837}{6492748} a^{6} - \frac{94683}{3246374} a^{5} - \frac{179013}{1623187} a^{4} + \frac{352511}{1623187} a^{3} + \frac{1586383}{6492748} a^{2} - \frac{1137269}{3246374} a - \frac{306879}{6492748}$, $\frac{1}{7057617076} a^{13} + \frac{537}{7057617076} a^{12} - \frac{2076417}{7057617076} a^{11} - \frac{3923727}{7057617076} a^{10} - \frac{375555867}{3528808538} a^{9} - \frac{278570114}{1764404269} a^{8} - \frac{1447655967}{7057617076} a^{7} - \frac{1322643835}{7057617076} a^{6} - \frac{368013646}{1764404269} a^{5} - \frac{1750978479}{3528808538} a^{4} - \frac{3121890215}{7057617076} a^{3} + \frac{3316610987}{7057617076} a^{2} - \frac{2129872079}{7057617076} a + \frac{468248097}{7057617076}$

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:  $7$
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:  \( 1049488.352113655 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_7$ (as 14T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 84
The 14 conjugacy class representatives for $F_7 \times C_2$
Character table for $F_7 \times C_2$

Intermediate fields

\(\Q(\sqrt{21}) \), 7.1.38423222208.5

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 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.14.12.1$x^{14} - 2 x^{7} + 4$$7$$2$$12$$(C_7:C_3) \times C_2$$[\ ]_{7}^{6}$
$3$3.14.13.1$x^{14} - 3$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
$7$7.14.15.5$x^{14} - 21 x^{13} + 7 x^{12} + 14 x^{11} - 21 x^{9} + 7 x^{7} + 14 x^{6} - 21 x^{4} + 14 x^{3} + 7 x^{2} + 14$$14$$1$$15$$F_7$$[7/6]_{6}$