↑

Properties

Label 14.14.3503613497...0000.1
Degree $14$
Signature $[14, 0]$
Discriminant $2^{12}\cdot 3^{6}\cdot 5^{10}\cdot 7^{14}\cdot 11^{6}$
Root discriminant $179.13$
Ramified primes $2, 3, 5, 7, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T23

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![73616884, 159935622, 86806489, -29079204, -31565996, 1321782, 4304454, -17166, -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 - 17166*x^7 + 4304454*x^6 + 1321782*x^5 - 31565996*x^4 - 29079204*x^3 + 86806489*x^2 + 159935622*x + 73616884)
 
gp: K = bnfinit(x^14 - 154*x^12 + 9317*x^10 - 279510*x^8 - 17166*x^7 + 4304454*x^6 + 1321782*x^5 - 31565996*x^4 - 29079204*x^3 + 86806489*x^2 + 159935622*x + 73616884, 1)
 

Normalized defining polynomial

\( x^{14} - 154 x^{12} + 9317 x^{10} - 279510 x^{8} - 17166 x^{7} + 4304454 x^{6} + 1321782 x^{5} - 31565996 x^{4} - 29079204 x^{3} + 86806489 x^{2} + 159935622 x + 73616884 \)

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:  \(35036134976849482947240000000000=2^{12}\cdot 3^{6}\cdot 5^{10}\cdot 7^{14}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $179.13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{202} a^{7} - \frac{77}{202} a^{5} + \frac{39}{101} a^{3} - \frac{25}{202} a + \frac{1}{101}$, $\frac{1}{202} a^{8} - \frac{77}{202} a^{6} + \frac{39}{101} a^{4} - \frac{25}{202} a^{2} + \frac{1}{101} a$, $\frac{1}{202} a^{9} + \frac{7}{202} a^{5} - \frac{79}{202} a^{3} + \frac{1}{101} a^{2} + \frac{95}{202} a - \frac{24}{101}$, $\frac{1}{202} a^{10} + \frac{7}{202} a^{6} - \frac{79}{202} a^{4} + \frac{1}{101} a^{3} + \frac{95}{202} a^{2} - \frac{24}{101} a$, $\frac{1}{1157662} a^{11} + \frac{15}{52621} a^{10} - \frac{11}{105242} a^{9} - \frac{87}{52621} a^{8} - \frac{37}{105242} a^{7} - \frac{4205}{52621} a^{6} + \frac{15400}{52621} a^{5} - \frac{67884}{578831} a^{4} + \frac{7687}{52621} a^{3} - \frac{444}{52621} a^{2} + \frac{4627}{105242} a - \frac{297}{52621}$, $\frac{1}{1157662} a^{12} - \frac{6}{52621} a^{10} - \frac{191}{105242} a^{9} + \frac{73}{105242} a^{8} + \frac{153}{105242} a^{7} + \frac{26565}{105242} a^{6} - \frac{155675}{578831} a^{5} - \frac{6125}{105242} a^{4} + \frac{753}{105242} a^{3} + \frac{16239}{52621} a^{2} + \frac{24000}{52621} a + \frac{24028}{52621}$, $\frac{1}{1157662} a^{13} + \frac{61}{52621} a^{10} + \frac{92}{52621} a^{9} + \frac{109}{105242} a^{8} - \frac{201}{105242} a^{7} + \frac{194419}{1157662} a^{6} - \frac{16490}{52621} a^{5} + \frac{13377}{52621} a^{4} + \frac{13795}{105242} a^{3} - \frac{38477}{105242} a^{2} - \frac{3371}{105242} a + \frac{6644}{52621}$

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:  \( 278502504145 \) (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$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.7.6.1$x^{7} - 11$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$