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Properties

Label 14.2.150584603848261632.1
Degree $14$
Signature $[2, 6]$
Discriminant $2^{14}\cdot 3^{13}\cdot 7^{8}$
Root discriminant $16.86$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
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![1, 28, 10, 20, 20, 14, 3, 12, -12, 2, -1, -4, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 4*x^12 - 4*x^11 - x^10 + 2*x^9 - 12*x^8 + 12*x^7 + 3*x^6 + 14*x^5 + 20*x^4 + 20*x^3 + 10*x^2 + 28*x + 1)
 
gp: K = bnfinit(x^14 - 2*x^13 + 4*x^12 - 4*x^11 - x^10 + 2*x^9 - 12*x^8 + 12*x^7 + 3*x^6 + 14*x^5 + 20*x^4 + 20*x^3 + 10*x^2 + 28*x + 1, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} + 4 x^{12} - 4 x^{11} - x^{10} + 2 x^{9} - 12 x^{8} + 12 x^{7} + 3 x^{6} + 14 x^{5} + 20 x^{4} + 20 x^{3} + 10 x^{2} + 28 x + 1 \)

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:  \(150584603848261632=2^{14}\cdot 3^{13}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.86$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{7} a^{11} - \frac{2}{7} a^{9} - \frac{2}{7} a^{8} + \frac{2}{7} a^{6} + \frac{1}{7} a^{5} + \frac{2}{7} a^{4} - \frac{1}{7} a^{3} - \frac{1}{7} a^{2} + \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{12} - \frac{2}{7} a^{10} - \frac{2}{7} a^{9} + \frac{2}{7} a^{7} + \frac{1}{7} a^{6} + \frac{2}{7} a^{5} - \frac{1}{7} a^{4} - \frac{1}{7} a^{3} + \frac{1}{7} a^{2} + \frac{1}{7} a$, $\frac{1}{35222404} a^{13} - \frac{710137}{35222404} a^{12} - \frac{1114885}{35222404} a^{11} + \frac{2117317}{5031772} a^{10} + \frac{8466067}{17611202} a^{9} - \frac{623717}{8805601} a^{8} - \frac{2232337}{8805601} a^{7} - \frac{1648988}{8805601} a^{6} + \frac{199987}{35222404} a^{5} - \frac{866725}{5031772} a^{4} - \frac{2406311}{35222404} a^{3} + \frac{2270115}{5031772} a^{2} + \frac{2397891}{35222404} a + \frac{1551381}{5031772}$

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:  \( 2031.94626444 \)
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

\(\Q(\sqrt{3}) \), 7.1.112021056.1

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 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.14.13.1$x^{14} - 3$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$