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Properties

Label 14.14.6206865031...0000.1
Degree $14$
Signature $[14, 0]$
Discriminant $2^{12}\cdot 3^{6}\cdot 5^{10}\cdot 7^{14}\cdot 11^{12}$
Root discriminant $500.60$
Ramified primes $2, 3, 5, 7, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T23

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![38710804, 117450102, 86806489, -21354564, -31565996, 970662, 4304454, -12606, -279510, 0, 9317, 0, -154, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 154*x^12 + 9317*x^10 - 279510*x^8 - 12606*x^7 + 4304454*x^6 + 970662*x^5 - 31565996*x^4 - 21354564*x^3 + 86806489*x^2 + 117450102*x + 38710804)
 
gp: K = bnfinit(x^14 - 154*x^12 + 9317*x^10 - 279510*x^8 - 12606*x^7 + 4304454*x^6 + 970662*x^5 - 31565996*x^4 - 21354564*x^3 + 86806489*x^2 + 117450102*x + 38710804, 1)
 

Normalized defining polynomial

\( x^{14} - 154 x^{12} + 9317 x^{10} - 279510 x^{8} - 12606 x^{7} + 4304454 x^{6} + 970662 x^{5} - 31565996 x^{4} - 21354564 x^{3} + 86806489 x^{2} + 117450102 x + 38710804 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(62068650315722446859495441640000000000=2^{12}\cdot 3^{6}\cdot 5^{10}\cdot 7^{14}\cdot 11^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $500.60$
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:  $2, 3, 5, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{902} a^{7} - \frac{7}{82} a^{5} - \frac{5}{41} a^{3} - \frac{27}{82} a - \frac{20}{41}$, $\frac{1}{902} a^{8} - \frac{7}{82} a^{6} - \frac{5}{41} a^{4} - \frac{27}{82} a^{2} - \frac{20}{41} a$, $\frac{1}{902} a^{9} + \frac{25}{82} a^{5} + \frac{23}{82} a^{3} - \frac{20}{41} a^{2} - \frac{29}{82} a + \frac{18}{41}$, $\frac{1}{9922} a^{10} - \frac{35}{82} a^{6} + \frac{17}{82} a^{4} + \frac{21}{451} a^{3} - \frac{25}{82} a^{2} - \frac{17}{41} a$, $\frac{1}{883058} a^{11} - \frac{5}{441529} a^{10} - \frac{1}{7298} a^{9} + \frac{1}{7298} a^{8} + \frac{39}{80278} a^{7} + \frac{273}{7298} a^{6} + \frac{1134}{3649} a^{5} - \frac{12794}{40139} a^{4} + \frac{13276}{40139} a^{3} - \frac{1937}{7298} a^{2} + \frac{3121}{7298} a + \frac{219}{3649}$, $\frac{1}{9713638} a^{12} - \frac{6}{441529} a^{10} - \frac{9}{80278} a^{9} - \frac{35}{80278} a^{8} + \frac{1}{80278} a^{7} + \frac{511}{7298} a^{6} - \frac{199376}{441529} a^{5} + \frac{2045}{7298} a^{4} + \frac{27169}{80278} a^{3} + \frac{1806}{3649} a^{2} - \frac{212}{3649} a + \frac{1704}{3649}$, $\frac{1}{9713638} a^{13} - \frac{1}{21538} a^{10} + \frac{1}{7298} a^{9} + \frac{2}{3649} a^{8} + \frac{37}{80278} a^{7} + \frac{202255}{883058} a^{6} + \frac{3095}{7298} a^{5} + \frac{4023}{80278} a^{4} + \frac{16486}{40139} a^{3} - \frac{3643}{7298} a^{2} - \frac{218}{3649} a - \frac{843}{3649}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 409687527821000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T23:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 588
The 16 conjugacy class representatives for [1/6_+.F_42(7)^2]2_2
Character table for [1/6_+.F_42(7)^2]2_2

Intermediate fields

\(\Q(\sqrt{5}) \)

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

Sibling fields

Degree 14 siblings: data not computed
Degree 28 siblings: 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 R R R ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.12.6.1$x^{12} - 243 x^{2} + 1458$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
7Data not computed
$11$11.7.6.1$x^{7} - 11$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
11.7.6.1$x^{7} - 11$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$