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Properties

Label 14.2.793...432.1
Degree $14$
Signature $[2, 6]$
Discriminant $7.940\times 10^{19}$
Root discriminant $26.39$
Ramified primes $2, 13$
Class number $1$
Class group trivial
Galois group $D_{14}$ (as 14T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 26*x^10 - 26*x^8 - 104*x^7 + 130*x^4 + 169*x^2 - 2*x + 4)
 
gp: K = bnfinit(x^14 - 2*x^13 + 26*x^10 - 26*x^8 - 104*x^7 + 130*x^4 + 169*x^2 - 2*x + 4, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -2, 169, 0, 130, 0, 0, -104, -26, 0, 26, 0, 0, -2, 1]);
 

\( x^{14} - 2 x^{13} + 26 x^{10} - 26 x^{8} - 104 x^{7} + 130 x^{4} + 169 x^{2} - 2 x + 4 \)

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

Invariants

Degree:  $14$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(79396891942519570432\)\(\medspace = 2^{18}\cdot 13^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $26.39$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 13$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{10} a^{8} + \frac{1}{10} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{10} a^{2} - \frac{1}{10} a - \frac{1}{5}$, $\frac{1}{10} a^{9} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{3}{10} a^{3} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{10} a^{10} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{3}{10} a^{4} + \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{10} a^{11} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{10} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{10} a^{12} + \frac{1}{10} a^{7} - \frac{1}{10} a^{6} - \frac{2}{5} a^{4} + \frac{2}{5} a^{2} + \frac{1}{10} a + \frac{2}{5}$, $\frac{1}{2345770} a^{13} + \frac{19547}{1172885} a^{12} - \frac{42354}{1172885} a^{11} + \frac{7059}{1172885} a^{10} - \frac{2327}{2345770} a^{9} + \frac{19742}{1172885} a^{8} - \frac{84799}{2345770} a^{7} + \frac{81987}{167555} a^{6} + \frac{3091}{167555} a^{5} + \frac{72092}{167555} a^{4} + \frac{692031}{2345770} a^{3} + \frac{235552}{1172885} a^{2} + \frac{469076}{1172885} a + \frac{12}{1172885}$

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 55327.6750141 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{6}\cdot 55327.6750141 \cdot 1}{2\sqrt{79396891942519570432}}\approx 0.764099429573$

Galois group

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

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

Intermediate fields

\(\Q(\sqrt{13}) \), 7.1.2471326208.1

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

Sibling fields

Galois closure: Deg 28
Degree 14 sibling: Deg 14

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ R ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
13Data not computed