↑

Properties

Label 14.0.62764785704439251.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,251^{7}$
Root discriminant $15.84$
Ramified prime $251$
Class number $1$
Class group Trivial
Galois group $D_{7}$ (as 14T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -6, 81, 42, -198, 148, 154, -326, 190, 22, -74, 28, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 + 2*x^12 + 28*x^11 - 74*x^10 + 22*x^9 + 190*x^8 - 326*x^7 + 154*x^6 + 148*x^5 - 198*x^4 + 42*x^3 + 81*x^2 - 6*x + 4)
 
gp: K = bnfinit(x^14 - 4*x^13 + 2*x^12 + 28*x^11 - 74*x^10 + 22*x^9 + 190*x^8 - 326*x^7 + 154*x^6 + 148*x^5 - 198*x^4 + 42*x^3 + 81*x^2 - 6*x + 4, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} + 2 x^{12} + 28 x^{11} - 74 x^{10} + 22 x^{9} + 190 x^{8} - 326 x^{7} + 154 x^{6} + 148 x^{5} - 198 x^{4} + 42 x^{3} + 81 x^{2} - 6 x + 4 \)

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:  \(-62764785704439251=-\,251^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.84$
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:  $251$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{4} - \frac{3}{8} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{32} a^{11} - \frac{1}{8} a^{9} + \frac{1}{32} a^{8} + \frac{1}{16} a^{7} + \frac{1}{8} a^{6} + \frac{7}{32} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{7}{32} a^{2} + \frac{3}{16} a + \frac{3}{8}$, $\frac{1}{2656} a^{12} - \frac{3}{2656} a^{11} - \frac{7}{664} a^{10} - \frac{251}{2656} a^{9} - \frac{233}{2656} a^{8} + \frac{295}{1328} a^{7} - \frac{565}{2656} a^{6} - \frac{221}{2656} a^{5} + \frac{145}{664} a^{4} - \frac{941}{2656} a^{3} - \frac{199}{2656} a^{2} - \frac{323}{1328} a - \frac{209}{664}$, $\frac{1}{459488} a^{13} - \frac{75}{459488} a^{12} + \frac{1375}{114872} a^{11} + \frac{11061}{459488} a^{10} + \frac{13855}{459488} a^{9} + \frac{3371}{229744} a^{8} - \frac{1877}{459488} a^{7} - \frac{109605}{459488} a^{6} + \frac{25703}{114872} a^{5} + \frac{71507}{459488} a^{4} + \frac{192385}{459488} a^{3} - \frac{76159}{229744} a^{2} + \frac{38643}{114872} a - \frac{5423}{14359}$

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:  \( -1 \) (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:  \( 1334.44273317 \)
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

\(\Q(\sqrt{-251}) \), 7.1.15813251.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.15813251.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
251Data not computed