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Properties

Label 14.0.19330173275...4923.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 7^{14}\cdot 19^{4}$
Root discriminant $28.12$
Ramified primes $3, 7, 19$
Class number $1$
Class group Trivial
Galois group 14T37

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19, -105, -63, 595, 686, -294, -539, -106, 217, 42, -7, -14, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 14*x^11 - 7*x^10 + 42*x^9 + 217*x^8 - 106*x^7 - 539*x^6 - 294*x^5 + 686*x^4 + 595*x^3 - 63*x^2 - 105*x + 19)
 
gp: K = bnfinit(x^14 - 14*x^11 - 7*x^10 + 42*x^9 + 217*x^8 - 106*x^7 - 539*x^6 - 294*x^5 + 686*x^4 + 595*x^3 - 63*x^2 - 105*x + 19, 1)
 

Normalized defining polynomial

\( x^{14} - 14 x^{11} - 7 x^{10} + 42 x^{9} + 217 x^{8} - 106 x^{7} - 539 x^{6} - 294 x^{5} + 686 x^{4} + 595 x^{3} - 63 x^{2} - 105 x + 19 \)

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:  \(-193301732750862154923=-\,3^{7}\cdot 7^{14}\cdot 19^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.12$
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:  $3, 7, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{21} a^{11} - \frac{1}{7} a^{10} + \frac{2}{7} a^{9} - \frac{10}{21} a^{8} + \frac{8}{21} a^{7} - \frac{2}{21} a^{4} - \frac{8}{21} a^{3} + \frac{3}{7} a^{2} + \frac{2}{7} a + \frac{5}{21}$, $\frac{1}{63} a^{12} - \frac{1}{63} a^{11} + \frac{23}{63} a^{9} - \frac{4}{21} a^{8} - \frac{5}{63} a^{7} - \frac{2}{63} a^{5} + \frac{10}{21} a^{4} - \frac{1}{9} a^{3} + \frac{8}{21} a^{2} - \frac{25}{63} a + \frac{10}{63}$, $\frac{1}{90506498122404} a^{13} + \frac{548196416317}{90506498122404} a^{12} + \frac{1157398242601}{90506498122404} a^{11} + \frac{956659758611}{90506498122404} a^{10} + \frac{11189866730737}{22626624530601} a^{9} - \frac{416214219751}{6464749865886} a^{8} + \frac{5113227076325}{12929499731772} a^{7} - \frac{42480460470611}{90506498122404} a^{6} - \frac{2962948768511}{45253249061202} a^{5} - \frac{5230218786985}{22626624530601} a^{4} + \frac{15907062341537}{45253249061202} a^{3} + \frac{18196045764125}{90506498122404} a^{2} - \frac{2341874056787}{6464749865886} a - \frac{1740580547605}{12929499731772}$

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:  \( \frac{1881874645}{26440694748} a^{13} + \frac{273022169}{26440694748} a^{12} + \frac{79312975}{8813564916} a^{11} - \frac{26377038625}{26440694748} a^{10} - \frac{469589414}{734463743} a^{9} + \frac{36945357323}{13220347374} a^{8} + \frac{139562410889}{8813564916} a^{7} - \frac{131099138723}{26440694748} a^{6} - \frac{55259259951}{1468927486} a^{5} - \frac{182527497019}{6610173687} a^{4} + \frac{61365341891}{1468927486} a^{3} + \frac{1263596630681}{26440694748} a^{2} + \frac{88497508883}{13220347374} a - \frac{38424454049}{8813564916} \) (order $6$)
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:  \( 178819.961267 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T37:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1764
The 25 conjugacy class representatives for [1/2.F_42(7)^2]2
Character table for [1/2.F_42(7)^2]2 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \)

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

Sibling fields

Degree 21 siblings: data not computed
Degree 28 sibling: data not computed
Degree 42 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ R ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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

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$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$7.7.7.6$x^{7} + 28 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
7.7.7.4$x^{7} + 14 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
19.6.4.2$x^{6} - 19 x^{3} + 722$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$