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Properties

Label 14.2.536...933.1
Degree $14$
Signature $[2, 6]$
Discriminant $5.366\times 10^{20}$
Root discriminant $30.25$
Ramified primes $11, 13$
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 - 4*x^13 + 52*x^10 - 26*x^8 - 208*x^7 + 260*x^4 + 169*x^2 - 4*x + 16)
 
gp: K = bnfinit(x^14 - 4*x^13 + 52*x^10 - 26*x^8 - 208*x^7 + 260*x^4 + 169*x^2 - 4*x + 16, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -4, 169, 0, 260, 0, 0, -208, -26, 0, 52, 0, 0, -4, 1]);
 

\( x^{14} - 4 x^{13} + 52 x^{10} - 26 x^{8} - 208 x^{7} + 260 x^{4} + 169 x^{2} - 4 x + 16 \)

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:  \(536561726709678316933\)\(\medspace = 11^{6}\cdot 13^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $30.25$
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:  $11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{1}{16} a^{4} - \frac{5}{16} a^{3} + \frac{7}{16} a^{2}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{8} - \frac{1}{16} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{7}{16} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{9} - \frac{1}{16} a^{6} - \frac{1}{4} a^{5} - \frac{7}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{15232} a^{13} + \frac{313}{15232} a^{12} + \frac{213}{15232} a^{11} - \frac{71}{15232} a^{10} + \frac{393}{15232} a^{9} + \frac{821}{15232} a^{8} + \frac{335}{15232} a^{7} - \frac{91}{2176} a^{6} - \frac{15}{2176} a^{5} + \frac{5}{2176} a^{4} - \frac{69}{15232} a^{3} - \frac{929}{15232} a^{2} - \frac{39}{3808} a + \frac{3}{952}$

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:  \( 277701.766085 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{6}\cdot 277701.766085 \cdot 1}{2\sqrt{536561726709678316933}}\approx 1.47529204984$

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

\(\Q(\sqrt{13}) \), 7.1.6424482779.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ R R ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13Data not computed