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Properties

Label 14.0.24522397616...0000.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{14}\cdot 5^{7}\cdot 7^{24}$
Root discriminant $125.68$
Ramified primes $2, 5, 7$
Class number $13538$ (GRH)
Class group $[13538]$ (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![8465729, 4042276, 2839473, 789740, 509600, 141946, 57890, 6260, 3871, 896, 308, -42, -7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^12 - 42*x^11 + 308*x^10 + 896*x^9 + 3871*x^8 + 6260*x^7 + 57890*x^6 + 141946*x^5 + 509600*x^4 + 789740*x^3 + 2839473*x^2 + 4042276*x + 8465729)
 
gp: K = bnfinit(x^14 - 7*x^12 - 42*x^11 + 308*x^10 + 896*x^9 + 3871*x^8 + 6260*x^7 + 57890*x^6 + 141946*x^5 + 509600*x^4 + 789740*x^3 + 2839473*x^2 + 4042276*x + 8465729, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{12} - 42 x^{11} + 308 x^{10} + 896 x^{9} + 3871 x^{8} + 6260 x^{7} + 57890 x^{6} + 141946 x^{5} + 509600 x^{4} + 789740 x^{3} + 2839473 x^{2} + 4042276 x + 8465729 \)

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:  \(-245223976167125010433280000000=-\,2^{14}\cdot 5^{7}\cdot 7^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $125.68$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(980=2^{2}\cdot 5\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{980}(1,·)$, $\chi_{980}(99,·)$, $\chi_{980}(421,·)$, $\chi_{980}(519,·)$, $\chi_{980}(841,·)$, $\chi_{980}(939,·)$, $\chi_{980}(141,·)$, $\chi_{980}(239,·)$, $\chi_{980}(561,·)$, $\chi_{980}(659,·)$, $\chi_{980}(281,·)$, $\chi_{980}(379,·)$, $\chi_{980}(701,·)$, $\chi_{980}(799,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{39463} a^{12} - \frac{19548}{39463} a^{11} + \frac{11143}{39463} a^{10} - \frac{11394}{39463} a^{9} - \frac{9301}{39463} a^{8} - \frac{18301}{39463} a^{7} - \frac{30}{39463} a^{6} - \frac{717}{39463} a^{5} + \frac{10043}{39463} a^{4} + \frac{9907}{39463} a^{3} + \frac{311}{39463} a^{2} + \frac{4091}{39463} a + \frac{9167}{39463}$, $\frac{1}{222028717935240253644269531168347} a^{13} - \frac{2669292850442739937512379798}{222028717935240253644269531168347} a^{12} - \frac{7996573688779038219648721506316}{222028717935240253644269531168347} a^{11} - \frac{2573258991607566550971948384259}{11685721996591592297066817429913} a^{10} - \frac{23129810380436648605440688065802}{222028717935240253644269531168347} a^{9} + \frac{94817629350432100649963525484052}{222028717935240253644269531168347} a^{8} + \frac{14289785206100338697430777065527}{222028717935240253644269531168347} a^{7} - \frac{4516813129380524842146056019867}{222028717935240253644269531168347} a^{6} - \frac{39740391137492939684134372916890}{222028717935240253644269531168347} a^{5} + \frac{44136859977047325930055911498493}{222028717935240253644269531168347} a^{4} - \frac{94483881377489672459527030344409}{222028717935240253644269531168347} a^{3} + \frac{2388538133632943227686054018392}{7162216707588395278847404231237} a^{2} - \frac{75252139591614960377011705340606}{222028717935240253644269531168347} a + \frac{67231414094871997594327390805240}{222028717935240253644269531168347}$

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

Class group and class number

$C_{13538}$, which has order $13538$ (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:  \( 35256.68973693789 \) (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

\(\Q(\sqrt{-5}) \), 7.7.13841287201.1

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 ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ R R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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$2.14.14.15$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}$$[2]^{7}$
$5$5.14.7.1$x^{14} - 250 x^{8} + 15625 x^{2} - 312500$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$7$7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$
7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$