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Properties

Label 14.6.15695382201...1504.1
Degree $14$
Signature $[6, 4]$
Discriminant $2^{8}\cdot 3^{17}\cdot 7^{15}$
Root discriminant $45.38$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group 14T46

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![112, -784, 1792, -560, -4480, 8848, -7560, 3024, -252, -140, -28, 28, 7, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 7*x^12 + 28*x^11 - 28*x^10 - 140*x^9 - 252*x^8 + 3024*x^7 - 7560*x^6 + 8848*x^5 - 4480*x^4 - 560*x^3 + 1792*x^2 - 784*x + 112)
 
gp: K = bnfinit(x^14 - 7*x^13 + 7*x^12 + 28*x^11 - 28*x^10 - 140*x^9 - 252*x^8 + 3024*x^7 - 7560*x^6 + 8848*x^5 - 4480*x^4 - 560*x^3 + 1792*x^2 - 784*x + 112, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} + 7 x^{12} + 28 x^{11} - 28 x^{10} - 140 x^{9} - 252 x^{8} + 3024 x^{7} - 7560 x^{6} + 8848 x^{5} - 4480 x^{4} - 560 x^{3} + 1792 x^{2} - 784 x + 112 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(156953822015120676021504=2^{8}\cdot 3^{17}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.38$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{6} - \frac{1}{6} a^{3} - \frac{1}{3}$, $\frac{1}{12} a^{7} - \frac{1}{12} a^{6} - \frac{1}{4} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{24} a^{8} - \frac{1}{24} a^{7} - \frac{1}{24} a^{6} - \frac{1}{6} a^{5} + \frac{5}{12} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{48} a^{9} - \frac{1}{48} a^{8} - \frac{1}{48} a^{7} - \frac{1}{12} a^{6} + \frac{5}{24} a^{5} + \frac{1}{3} a^{4} - \frac{1}{6} a^{3} + \frac{1}{6} a^{2} + \frac{5}{12} a$, $\frac{1}{144} a^{10} + \frac{1}{144} a^{9} - \frac{1}{48} a^{8} - \frac{1}{24} a^{7} - \frac{1}{24} a^{6} + \frac{1}{12} a^{5} - \frac{1}{2} a^{3} - \frac{1}{12} a^{2} - \frac{7}{18} a + \frac{1}{9}$, $\frac{1}{288} a^{11} - \frac{1}{288} a^{10} + \frac{1}{288} a^{9} - \frac{1}{48} a^{7} - \frac{1}{48} a^{6} + \frac{1}{24} a^{5} + \frac{7}{24} a^{4} + \frac{1}{8} a^{3} + \frac{7}{18} a^{2} - \frac{17}{36} a + \frac{11}{36}$, $\frac{1}{288} a^{12} + \frac{1}{288} a^{9} - \frac{1}{48} a^{8} - \frac{1}{24} a^{7} + \frac{1}{48} a^{6} - \frac{1}{6} a^{5} - \frac{1}{12} a^{4} + \frac{1}{72} a^{3} - \frac{1}{12} a^{2} - \frac{1}{6} a + \frac{11}{36}$, $\frac{1}{864} a^{13} + \frac{1}{864} a^{12} + \frac{1}{864} a^{10} + \frac{7}{864} a^{9} - \frac{1}{48} a^{8} + \frac{1}{48} a^{7} - \frac{1}{16} a^{6} - \frac{1}{6} a^{5} + \frac{25}{216} a^{4} - \frac{101}{216} a^{3} - \frac{1}{12} a^{2} + \frac{35}{108} a + \frac{29}{108}$

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:  $9$
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:  \( 13981319.4928 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T46:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5040
The 15 conjugacy class representatives for 2[1/2]S(7)
Character table for 2[1/2]S(7)

Intermediate fields

\(\Q(\sqrt{21}) \), 7.3.28817416656.2

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

Sibling fields

Degree 7 sibling: 7.3.28817416656.2
Degree 21 sibling: Deg 21
Degree 30 sibling: data not computed
Degree 35 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 R R ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.9.1$x^{6} + 3 x^{4} + 15$$6$$1$$9$$C_6$$[2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
7Data not computed