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Properties

Label 14.0.10440732699...1504.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{14}\cdot 3^{7}\cdot 7^{7}\cdot 29^{12}$
Root discriminant $164.29$
Ramified primes $2, 3, 7, 29$
Class number $1103456$ (GRH)
Class group $[2, 2, 2, 2, 68966]$ (GRH)
Galois group $C_{14}$ (as 14T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4979767381, -277132320, 1144024522, -58021334, 118383398, -5536192, 7248641, -310308, 287927, -10668, 7557, -214, 124, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 124*x^12 - 214*x^11 + 7557*x^10 - 10668*x^9 + 287927*x^8 - 310308*x^7 + 7248641*x^6 - 5536192*x^5 + 118383398*x^4 - 58021334*x^3 + 1144024522*x^2 - 277132320*x + 4979767381)
 
gp: K = bnfinit(x^14 - 2*x^13 + 124*x^12 - 214*x^11 + 7557*x^10 - 10668*x^9 + 287927*x^8 - 310308*x^7 + 7248641*x^6 - 5536192*x^5 + 118383398*x^4 - 58021334*x^3 + 1144024522*x^2 - 277132320*x + 4979767381, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} + 124 x^{12} - 214 x^{11} + 7557 x^{10} - 10668 x^{9} + 287927 x^{8} - 310308 x^{7} + 7248641 x^{6} - 5536192 x^{5} + 118383398 x^{4} - 58021334 x^{3} + 1144024522 x^{2} - 277132320 x + 4979767381 \)

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:  \(-10440732699324736115103484821504=-\,2^{14}\cdot 3^{7}\cdot 7^{7}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $164.29$
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:  $2, 3, 7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2436=2^{2}\cdot 3\cdot 7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{2436}(1,·)$, $\chi_{2436}(1763,·)$, $\chi_{2436}(1093,·)$, $\chi_{2436}(2017,·)$, $\chi_{2436}(169,·)$, $\chi_{2436}(587,·)$, $\chi_{2436}(335,·)$, $\chi_{2436}(1009,·)$, $\chi_{2436}(83,·)$, $\chi_{2436}(755,·)$, $\chi_{2436}(1847,·)$, $\chi_{2436}(923,·)$, $\chi_{2436}(2269,·)$, $\chi_{2436}(1765,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{17} a^{11} - \frac{8}{17} a^{10} - \frac{7}{17} a^{9} - \frac{1}{17} a^{8} + \frac{4}{17} a^{6} - \frac{6}{17} a^{5} - \frac{7}{17} a^{4} - \frac{7}{17} a^{3} - \frac{2}{17} a^{2} - \frac{1}{17} a$, $\frac{1}{17} a^{12} - \frac{3}{17} a^{10} - \frac{6}{17} a^{9} - \frac{8}{17} a^{8} + \frac{4}{17} a^{7} - \frac{8}{17} a^{6} - \frac{4}{17} a^{5} + \frac{5}{17} a^{4} - \frac{7}{17} a^{3} - \frac{8}{17} a$, $\frac{1}{168777884480148773063824545913456958361541} a^{13} + \frac{2494178770679041662003410990720177205137}{168777884480148773063824545913456958361541} a^{12} + \frac{3897371542625804791957900133428331871923}{168777884480148773063824545913456958361541} a^{11} + \frac{12315797087197090223874976108883716566659}{168777884480148773063824545913456958361541} a^{10} + \frac{10230710693900264324474856448650518246109}{168777884480148773063824545913456958361541} a^{9} + \frac{25776052232723199412737230698456885680485}{168777884480148773063824545913456958361541} a^{8} + \frac{4140670742064972067574502030859039288915}{9928110851773457239048502700791585785973} a^{7} + \frac{67630390875780366605812910256145782397715}{168777884480148773063824545913456958361541} a^{6} + \frac{46157587552193420844914505414963564839257}{168777884480148773063824545913456958361541} a^{5} - \frac{5870558189628209561500976244807776007081}{168777884480148773063824545913456958361541} a^{4} - \frac{35287851466925110738236285402278625944930}{168777884480148773063824545913456958361541} a^{3} - \frac{22000269587954699180469204620749521526196}{168777884480148773063824545913456958361541} a^{2} + \frac{2186810434196174455346846786303965832173}{9928110851773457239048502700791585785973} a - \frac{63320148276543767377270281730216019823}{9928110851773457239048502700791585785973}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{68966}$, which has order $1103456$ (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:  \( 6020.985100147561 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{14}$ (as 14T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 14
The 14 conjugacy class representatives for $C_{14}$
Character table for $C_{14}$

Intermediate fields

\(\Q(\sqrt{-21}) \), 7.7.594823321.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$2$2.14.14.15$x^{14} + 2 x^{13} + x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
$3$3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$7$7.14.7.2$x^{14} - 686 x^{8} + 117649 x^{2} - 3294172$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.14.12.1$x^{14} + 2407 x^{7} + 1839267$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$