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Properties

Label 14.2.73944117820...8125.1
Degree $14$
Signature $[2, 6]$
Discriminant $5^{12}\cdot 13^{13}$
Root discriminant $43.00$
Ramified primes $5, 13$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $\PGL(2,13)$ (as 14T39)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -14, 65, -39, -156, 481, 676, -195, -299, 104, 39, -26, 0, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 - 26*x^11 + 39*x^10 + 104*x^9 - 299*x^8 - 195*x^7 + 676*x^6 + 481*x^5 - 156*x^4 - 39*x^3 + 65*x^2 - 14*x + 1)
 
gp: K = bnfinit(x^14 - x^13 - 26*x^11 + 39*x^10 + 104*x^9 - 299*x^8 - 195*x^7 + 676*x^6 + 481*x^5 - 156*x^4 - 39*x^3 + 65*x^2 - 14*x + 1, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} - 26 x^{11} + 39 x^{10} + 104 x^{9} - 299 x^{8} - 195 x^{7} + 676 x^{6} + 481 x^{5} - 156 x^{4} - 39 x^{3} + 65 x^{2} - 14 x + 1 \)

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:  \(73944117820374267578125=5^{12}\cdot 13^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.00$
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:  $5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{4890883327079} a^{13} - \frac{1175579435387}{4890883327079} a^{12} + \frac{389155571353}{4890883327079} a^{11} + \frac{379861025860}{4890883327079} a^{10} + \frac{1124133692106}{4890883327079} a^{9} - \frac{320295405574}{4890883327079} a^{8} + \frac{438329105597}{4890883327079} a^{7} - \frac{1653726830296}{4890883327079} a^{6} + \frac{669747837682}{4890883327079} a^{5} + \frac{2029055500852}{4890883327079} a^{4} + \frac{1569755324636}{4890883327079} a^{3} - \frac{907284001388}{4890883327079} a^{2} + \frac{612957453835}{4890883327079} a + \frac{560289444737}{4890883327079}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  $7$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 172163.541515 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\PGL(2,13)$ (as 14T39):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2184
The 15 conjugacy class representatives for $\PGL(2,13)$
Character table for $\PGL(2,13)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 28 sibling: data not computed
Degree 42 sibling: 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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.13.0.1}{13} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.13.12.1$x^{13} - 5$$13$$1$$12$$C_{13}:C_4$$[\ ]_{13}^{4}$
13Data not computed