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Properties

Label 14.0.35657645410...1663.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 149^{7}$
Root discriminant $21.14$
Ramified primes $3, 149$
Class number $2$
Class group $[2]$
Galois group $D_{7}$ (as 14T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![729, -729, 1674, -2493, 2859, -2247, 1252, -379, 55, -14, 26, -5, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 + 4*x^12 - 5*x^11 + 26*x^10 - 14*x^9 + 55*x^8 - 379*x^7 + 1252*x^6 - 2247*x^5 + 2859*x^4 - 2493*x^3 + 1674*x^2 - 729*x + 729)
 
gp: K = bnfinit(x^14 - 4*x^13 + 4*x^12 - 5*x^11 + 26*x^10 - 14*x^9 + 55*x^8 - 379*x^7 + 1252*x^6 - 2247*x^5 + 2859*x^4 - 2493*x^3 + 1674*x^2 - 729*x + 729, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} + 4 x^{12} - 5 x^{11} + 26 x^{10} - 14 x^{9} + 55 x^{8} - 379 x^{7} + 1252 x^{6} - 2247 x^{5} + 2859 x^{4} - 2493 x^{3} + 1674 x^{2} - 729 x + 729 \)

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:  \(-3565764541089561663=-\,3^{7}\cdot 149^{7}\)
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:  $3, 149$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{2}{9} a^{5} + \frac{4}{9} a^{4} - \frac{1}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{1}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{27} a^{10} + \frac{1}{27} a^{9} - \frac{1}{27} a^{8} + \frac{4}{27} a^{7} - \frac{1}{27} a^{6} - \frac{2}{27} a^{5} - \frac{11}{27} a^{4} - \frac{2}{9} a^{3} - \frac{4}{9} a^{2}$, $\frac{1}{135} a^{11} - \frac{2}{135} a^{10} + \frac{2}{135} a^{9} + \frac{1}{135} a^{8} + \frac{4}{27} a^{7} - \frac{1}{27} a^{6} + \frac{46}{135} a^{5} + \frac{1}{45} a^{4} + \frac{4}{9} a^{3} - \frac{1}{3} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{3911895} a^{12} + \frac{2293}{1303965} a^{11} - \frac{433}{60183} a^{10} - \frac{24607}{3911895} a^{9} + \frac{8371}{3911895} a^{8} - \frac{1952}{60183} a^{7} + \frac{167897}{1303965} a^{6} + \frac{1883084}{3911895} a^{5} + \frac{175537}{434655} a^{4} - \frac{4768}{20061} a^{3} + \frac{53953}{434655} a^{2} + \frac{30884}{144885} a + \frac{1836}{48295}$, $\frac{1}{5433622155} a^{13} + \frac{71}{5433622155} a^{12} + \frac{1587026}{1086724431} a^{11} + \frac{3033214}{5433622155} a^{10} + \frac{224240816}{5433622155} a^{9} - \frac{7247231}{5433622155} a^{8} - \frac{775360814}{5433622155} a^{7} - \frac{301496254}{5433622155} a^{6} - \frac{621862877}{5433622155} a^{5} - \frac{224358106}{1811207385} a^{4} + \frac{561421454}{1811207385} a^{3} + \frac{6431881}{46441215} a^{2} - \frac{19960484}{67081755} a - \frac{8358502}{22360585}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -1 \) (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:  \( 13570.4354807 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_7$ (as 14T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 14
The 5 conjugacy class representatives for $D_{7}$
Character table for $D_{7}$

Intermediate fields

\(\Q(\sqrt{-447}) \), 7.1.89314623.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.89314623.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$149$149.2.1.2$x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.1.2$x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.1.2$x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.1.2$x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.1.2$x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.1.2$x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.1.2$x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$