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Properties

Label 14.0.14759211473...3091.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{7}\cdot 13^{11}$
Root discriminant $19.85$
Ramified primes $7, 13$
Class number $2$
Class group $[2]$
Galois group $F_7$ (as 14T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -30, 113, -203, 182, -76, 73, -197, 248, -163, 64, -20, 9, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 + 9*x^12 - 20*x^11 + 64*x^10 - 163*x^9 + 248*x^8 - 197*x^7 + 73*x^6 - 76*x^5 + 182*x^4 - 203*x^3 + 113*x^2 - 30*x + 4)
 
gp: K = bnfinit(x^14 - 4*x^13 + 9*x^12 - 20*x^11 + 64*x^10 - 163*x^9 + 248*x^8 - 197*x^7 + 73*x^6 - 76*x^5 + 182*x^4 - 203*x^3 + 113*x^2 - 30*x + 4, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} + 9 x^{12} - 20 x^{11} + 64 x^{10} - 163 x^{9} + 248 x^{8} - 197 x^{7} + 73 x^{6} - 76 x^{5} + 182 x^{4} - 203 x^{3} + 113 x^{2} - 30 x + 4 \)

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:  \(-1475921147386413091=-\,7^{7}\cdot 13^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.85$
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:  $7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} + \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{8} - \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{2} a^{3} + \frac{1}{6} a$, $\frac{1}{327209214} a^{13} + \frac{3744792}{54534869} a^{12} + \frac{9002329}{109069738} a^{11} + \frac{13953151}{109069738} a^{10} + \frac{161842753}{327209214} a^{9} + \frac{28234525}{109069738} a^{8} - \frac{130631041}{327209214} a^{7} - \frac{55525975}{327209214} a^{6} - \frac{15990289}{54534869} a^{5} - \frac{4320010}{54534869} a^{4} - \frac{19470599}{109069738} a^{3} + \frac{80181827}{163604607} a^{2} + \frac{10801123}{54534869} a - \frac{4718677}{54534869}$

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:  \( 1466.80774384 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_7$ (as 14T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 7 conjugacy class representatives for $F_7$
Character table for $F_7$

Intermediate fields

\(\Q(\sqrt{-91}) \), 7.1.127353499.1

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 7 sibling: 7.1.127353499.1
Degree 21 sibling: Deg 21

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$