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Properties

Label 14.0.35613405386...1963.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 7^{18}$
Root discriminant $21.14$
Ramified primes $3, 7$
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![1, 7, 56, -63, 0, -49, 49, 1, 14, -7, -7, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^10 - 7*x^9 + 14*x^8 + x^7 + 49*x^6 - 49*x^5 - 63*x^3 + 56*x^2 + 7*x + 1)
 
gp: K = bnfinit(x^14 - 7*x^10 - 7*x^9 + 14*x^8 + x^7 + 49*x^6 - 49*x^5 - 63*x^3 + 56*x^2 + 7*x + 1, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{10} - 7 x^{9} + 14 x^{8} + x^{7} + 49 x^{6} - 49 x^{5} - 63 x^{3} + 56 x^{2} + 7 x + 1 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-3561340538630151963=-\,3^{7}\cdot 7^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.14$
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:  $3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{4}{9} a^{7} - \frac{4}{9} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{2}{9} a^{2} - \frac{4}{9} a - \frac{1}{9}$, $\frac{1}{765} a^{12} - \frac{23}{765} a^{11} - \frac{23}{765} a^{10} + \frac{10}{153} a^{9} - \frac{106}{765} a^{8} + \frac{161}{765} a^{7} - \frac{38}{85} a^{6} - \frac{23}{85} a^{4} - \frac{133}{765} a^{3} + \frac{23}{765} a^{2} - \frac{106}{765} a - \frac{38}{85}$, $\frac{1}{31365} a^{13} + \frac{14}{31365} a^{12} + \frac{401}{31365} a^{11} - \frac{862}{10455} a^{10} - \frac{2591}{31365} a^{9} + \frac{1849}{31365} a^{8} + \frac{1531}{6273} a^{7} - \frac{4388}{10455} a^{6} - \frac{3469}{10455} a^{5} + \frac{11078}{31365} a^{4} + \frac{8107}{31365} a^{3} + \frac{2597}{6273} a^{2} + \frac{12311}{31365} a - \frac{301}{3485}$

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:  $6$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{637}{3485} a^{13} + \frac{637}{10455} a^{12} + \frac{294}{3485} a^{11} + \frac{77}{697} a^{10} - \frac{12131}{10455} a^{9} - \frac{5488}{3485} a^{8} + \frac{5886}{3485} a^{7} - \frac{343}{2091} a^{6} + \frac{30576}{3485} a^{5} - \frac{21021}{3485} a^{4} + \frac{11123}{10455} a^{3} - \frac{2121}{205} a^{2} + \frac{23366}{3485} a + \frac{1748}{2091} \) (order $6$)
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:  \( 19677.9819798 \)
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

\(\Q(\sqrt{-3}) \), 7.1.40353607.1

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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$3.2.1.2$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$7.7.9.4$x^{7} + 35 x^{3} + 7$$7$$1$$9$$D_{7}$$[3/2]_{2}$
7.7.9.4$x^{7} + 35 x^{3} + 7$$7$$1$$9$$D_{7}$$[3/2]_{2}$