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Properties

Label 14.2.13543368199...6816.1
Degree $14$
Signature $[2, 6]$
Discriminant $2^{12}\cdot 41\cdot 73^{8}$
Root discriminant $27.41$
Ramified primes $2, 41, 73$
Class number $1$
Class group Trivial
Galois group 14T44

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-29, 262, -485, 204, 329, -496, 315, -120, 89, -78, 37, -12, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 3*x^12 - 12*x^11 + 37*x^10 - 78*x^9 + 89*x^8 - 120*x^7 + 315*x^6 - 496*x^5 + 329*x^4 + 204*x^3 - 485*x^2 + 262*x - 29)
 
gp: K = bnfinit(x^14 + 3*x^12 - 12*x^11 + 37*x^10 - 78*x^9 + 89*x^8 - 120*x^7 + 315*x^6 - 496*x^5 + 329*x^4 + 204*x^3 - 485*x^2 + 262*x - 29, 1)
 

Normalized defining polynomial

\( x^{14} + 3 x^{12} - 12 x^{11} + 37 x^{10} - 78 x^{9} + 89 x^{8} - 120 x^{7} + 315 x^{6} - 496 x^{5} + 329 x^{4} + 204 x^{3} - 485 x^{2} + 262 x - 29 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(135433681992324386816=2^{12}\cdot 41\cdot 73^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.41$
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:  $2, 41, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{26} a^{11} - \frac{1}{13} a^{10} - \frac{1}{26} a^{9} - \frac{1}{13} a^{8} - \frac{1}{13} a^{6} - \frac{4}{13} a^{5} - \frac{5}{13} a^{4} - \frac{1}{26} a^{3} - \frac{6}{13} a^{2} - \frac{1}{26} a - \frac{1}{13}$, $\frac{1}{52} a^{12} + \frac{2}{13} a^{10} - \frac{1}{13} a^{9} + \frac{9}{52} a^{8} - \frac{1}{26} a^{7} + \frac{7}{26} a^{6} - \frac{21}{52} a^{4} - \frac{7}{26} a^{3} + \frac{7}{26} a^{2} - \frac{1}{13} a - \frac{17}{52}$, $\frac{1}{653520133916} a^{13} - \frac{5906295001}{653520133916} a^{12} + \frac{5862141039}{326760066958} a^{11} - \frac{1911090613}{163380033479} a^{10} + \frac{29013913785}{653520133916} a^{9} + \frac{132175017293}{653520133916} a^{8} + \frac{38200750973}{163380033479} a^{7} - \frac{31534066569}{326760066958} a^{6} - \frac{209385311017}{653520133916} a^{5} + \frac{149814728987}{653520133916} a^{4} + \frac{54537346053}{326760066958} a^{3} + \frac{143415351947}{326760066958} a^{2} + \frac{268368225059}{653520133916} a + \frac{25097632517}{653520133916}$

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:  $7$
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:  \( 147633.299547 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T44:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2688
The 32 conjugacy class representatives for [2^7]F_21(7)=2wrF_21(7)
Character table for [2^7]F_21(7)=2wrF_21(7) is not computed

Intermediate fields

7.7.1817487424.1

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

Sibling fields

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 ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
$41$41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.3.0.1$x^{3} - x + 13$$1$$3$$0$$C_3$$[\ ]^{3}$
41.3.0.1$x^{3} - x + 13$$1$$3$$0$$C_3$$[\ ]^{3}$
41.6.0.1$x^{6} - x + 7$$1$$6$$0$$C_6$$[\ ]^{6}$
73Data not computed