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Properties

Label 14.2.84196429082...9136.1
Degree $14$
Signature $[2, 6]$
Discriminant $2^{18}\cdot 13^{22}$
Root discriminant $137.25$
Ramified primes $2, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $\PSL(2,13)$ (as 14T30)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![35152, 70304, 0, 0, 17576, 0, 0, 0, 845, -338, 0, 0, 13, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 6*x^13 + 13*x^12 - 338*x^9 + 845*x^8 + 17576*x^4 + 70304*x + 35152)
 
gp: K = bnfinit(x^14 - 6*x^13 + 13*x^12 - 338*x^9 + 845*x^8 + 17576*x^4 + 70304*x + 35152, 1)
 

Normalized defining polynomial

\( x^{14} - 6 x^{13} + 13 x^{12} - 338 x^{9} + 845 x^{8} + 17576 x^{4} + 70304 x + 35152 \)

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:  \(841964290822597536686373339136=2^{18}\cdot 13^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $137.25$
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:  $2, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{13} a^{6} - \frac{6}{13} a^{5}$, $\frac{1}{39} a^{7} - \frac{10}{39} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{78} a^{8} - \frac{1}{78} a^{6} + \frac{4}{13} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a$, $\frac{1}{156} a^{9} - \frac{1}{156} a^{7} - \frac{1}{26} a^{6} - \frac{1}{78} a^{5} + \frac{1}{6} a^{4} + \frac{5}{12} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{16224} a^{10} - \frac{1}{2704} a^{9} + \frac{1}{1248} a^{8} - \frac{1}{156} a^{7} - \frac{7}{312} a^{6} - \frac{253}{624} a^{5} + \frac{25}{96} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a + \frac{11}{24}$, $\frac{1}{16224} a^{11} - \frac{23}{16224} a^{9} - \frac{1}{624} a^{8} - \frac{1}{104} a^{7} - \frac{1}{624} a^{6} + \frac{35}{416} a^{5} + \frac{11}{48} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{5}{24} a + \frac{1}{12}$, $\frac{1}{32448} a^{12} - \frac{1}{32448} a^{10} + \frac{25}{16224} a^{9} + \frac{5}{1248} a^{8} + \frac{5}{416} a^{7} + \frac{5}{192} a^{6} + \frac{161}{1248} a^{5} - \frac{13}{96} a^{4} - \frac{1}{6} a^{3} - \frac{5}{48} a^{2} + \frac{3}{8} a + \frac{3}{8}$, $\frac{1}{20182656} a^{13} - \frac{203}{6727552} a^{11} + \frac{1}{258752} a^{10} + \frac{2255}{3363776} a^{9} + \frac{239}{258752} a^{8} + \frac{2475}{517504} a^{7} - \frac{625}{19904} a^{6} - \frac{15941}{776256} a^{5} - \frac{3475}{9952} a^{4} + \frac{14915}{29856} a^{3} - \frac{1283}{14928} a^{2} - \frac{6143}{14928} a + \frac{915}{2488}$

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:  $7$
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:  \( 18415956755.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\PSL(2,13)$ (as 14T30):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1092
The 9 conjugacy class representatives for $\PSL(2,13)$
Character table for $\PSL(2,13)$

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 ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.13.0.1}{13} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.13.0.1}{13} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R ${\href{/LocalNumberField/17.13.0.1}{13} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.13.0.1}{13} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.13.0.1}{13} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.4.6.7$x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
2.4.6.7$x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.13.22.2$x^{13} + 39 x^{10} + 13$$13$$1$$22$$C_{13}:C_6$$[11/6]_{6}$