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Properties

Label 14.0.27033255809...3104.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{18}\cdot 13^{13}\cdot 23^{7}$
Root discriminant $126.55$
Ramified primes $2, 13, 23$
Class number $2$ (GRH)
Class group $[2]$ (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![4014, 16218, 7267, -29406, 11050, -1950, -130, 1872, -702, 78, 78, -78, 26, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 6*x^13 + 26*x^12 - 78*x^11 + 78*x^10 + 78*x^9 - 702*x^8 + 1872*x^7 - 130*x^6 - 1950*x^5 + 11050*x^4 - 29406*x^3 + 7267*x^2 + 16218*x + 4014)
 
gp: K = bnfinit(x^14 - 6*x^13 + 26*x^12 - 78*x^11 + 78*x^10 + 78*x^9 - 702*x^8 + 1872*x^7 - 130*x^6 - 1950*x^5 + 11050*x^4 - 29406*x^3 + 7267*x^2 + 16218*x + 4014, 1)
 

Normalized defining polynomial

\( x^{14} - 6 x^{13} + 26 x^{12} - 78 x^{11} + 78 x^{10} + 78 x^{9} - 702 x^{8} + 1872 x^{7} - 130 x^{6} - 1950 x^{5} + 11050 x^{4} - 29406 x^{3} + 7267 x^{2} + 16218 x + 4014 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-270332558098599894702382383104=-\,2^{18}\cdot 13^{13}\cdot 23^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $126.55$
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:  $2, 13, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{138} a^{12} + \frac{7}{69} a^{11} - \frac{1}{138} a^{10} + \frac{10}{69} a^{9} + \frac{1}{46} a^{8} - \frac{11}{69} a^{7} + \frac{7}{138} a^{6} - \frac{10}{69} a^{5} - \frac{19}{46} a^{4} + \frac{10}{23} a^{3} - \frac{13}{138} a^{2} + \frac{5}{23} a + \frac{6}{23}$, $\frac{1}{575334449302257816522} a^{13} + \frac{26705777981903614}{22128248050086839097} a^{12} - \frac{4029683466927838759}{44256496100173678194} a^{11} - \frac{1454312266002689981}{22128248050086839097} a^{10} - \frac{1526270799258370867}{14752165366724559398} a^{9} - \frac{2529229799099164442}{22128248050086839097} a^{8} - \frac{3726229707645703337}{44256496100173678194} a^{7} - \frac{6667132643240170948}{22128248050086839097} a^{6} - \frac{6638601552834853579}{14752165366724559398} a^{5} - \frac{1213533857266044618}{7376082683362279699} a^{4} - \frac{13333053802716398095}{44256496100173678194} a^{3} + \frac{1225778139234673942}{7376082683362279699} a^{2} + \frac{3282098483735651026}{7376082683362279699} a - \frac{35127405824960808997}{95889074883709636087}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $6$
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:  \( 3362555629.18 \) (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 R ${\href{/LocalNumberField/3.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.13.0.1}{13} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.13.13.1$x^{13} + 13 x + 13$$13$$1$$13$$F_{13}$$[13/12]_{12}$
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$