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Properties

Label 14.2.203...592.1
Degree $14$
Signature $[2, 6]$
Discriminant $2.033\times 10^{22}$
Root discriminant $39.21$
Ramified primes $2, 13$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $\PGL(2,13)$ (as 14T39)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 13*x^12 - 52*x^11 + 104*x^10 - 364*x^9 + 806*x^8 - 1300*x^7 + 2990*x^6 - 4212*x^5 + 4680*x^4 - 7020*x^3 + 5265*x^2 - 1802*x + 445)
 
gp: K = bnfinit(x^14 - 2*x^13 + 13*x^12 - 52*x^11 + 104*x^10 - 364*x^9 + 806*x^8 - 1300*x^7 + 2990*x^6 - 4212*x^5 + 4680*x^4 - 7020*x^3 + 5265*x^2 - 1802*x + 445, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![445, -1802, 5265, -7020, 4680, -4212, 2990, -1300, 806, -364, 104, -52, 13, -2, 1]);
 

\( x^{14} - 2 x^{13} + 13 x^{12} - 52 x^{11} + 104 x^{10} - 364 x^{9} + 806 x^{8} - 1300 x^{7} + 2990 x^{6} - 4212 x^{5} + 4680 x^{4} - 7020 x^{3} + 5265 x^{2} - 1802 x + 445 \)

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

Invariants

Degree:  $14$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(20325604337285010030592\)\(\medspace = 2^{26}\cdot 13^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $39.21$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 13$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{3}{8} a + \frac{3}{8}$, $\frac{1}{40} a^{8} - \frac{3}{40} a^{6} + \frac{1}{20} a^{5} + \frac{1}{5} a^{4} - \frac{1}{20} a^{3} - \frac{19}{40} a^{2} + \frac{1}{10} a - \frac{3}{8}$, $\frac{1}{40} a^{9} + \frac{1}{20} a^{7} - \frac{3}{40} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{9}{40} a^{3} - \frac{2}{5} a^{2} - \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{80} a^{10} - \frac{1}{80} a^{9} - \frac{1}{16} a^{7} - \frac{3}{80} a^{6} - \frac{1}{20} a^{5} - \frac{13}{80} a^{4} + \frac{17}{80} a^{3} - \frac{9}{20} a^{2} - \frac{23}{80} a + \frac{7}{16}$, $\frac{1}{160} a^{11} + \frac{1}{160} a^{9} - \frac{1}{160} a^{8} + \frac{3}{80} a^{7} - \frac{3}{32} a^{6} + \frac{7}{160} a^{5} + \frac{1}{5} a^{4} - \frac{5}{32} a^{3} - \frac{47}{160} a^{2} - \frac{11}{80} a - \frac{13}{32}$, $\frac{1}{160} a^{12} - \frac{1}{160} a^{10} + \frac{1}{160} a^{9} - \frac{1}{80} a^{8} - \frac{1}{32} a^{7} - \frac{3}{160} a^{6} + \frac{3}{20} a^{5} + \frac{17}{160} a^{4} + \frac{3}{32} a^{3} + \frac{21}{80} a^{2} + \frac{29}{160} a + \frac{1}{16}$, $\frac{1}{160} a^{13} - \frac{1}{160} a^{10} + \frac{1}{160} a^{9} - \frac{1}{80} a^{8} - \frac{7}{160} a^{7} - \frac{17}{160} a^{6} - \frac{1}{4} a^{5} - \frac{3}{32} a^{4} + \frac{3}{32} a^{3} - \frac{11}{80} a^{2} - \frac{5}{16} a - \frac{11}{32}$

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

Class group and class number

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

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 1523520.8954 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{6}\cdot 1523520.8954 \cdot 1}{2\sqrt{20325604337285010030592}}\approx 1.3150306863$ (assuming GRH)

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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 ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.6.10.3$x^{6} + 2 x^{5} + 4 x + 2$$6$$1$$10$$S_4$$[8/3, 8/3]_{3}^{2}$
2.8.16.8$x^{8} + 8 x^{5} + 12$$4$$2$$16$$S_4$$[8/3, 8/3]_{3}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.13.13.5$x^{13} + 130 x + 13$$13$$1$$13$$F_{13}$$[13/12]_{12}$

Additional information

This field is unusual in that it has non-solvable Galois group and is ramified at only two small primes.