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Properties

Label 14.0.16227613925...4384.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{21}\cdot 3^{7}\cdot 29^{12}$
Root discriminant $87.82$
Ramified primes $2, 3, 29$
Class number $4816$ (GRH)
Class group $[2, 2, 2, 602]$ (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![4387351, -533070, 2200477, -218804, 492458, -36022, 67046, -4968, 6227, -528, 402, -34, 19, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 19*x^12 - 34*x^11 + 402*x^10 - 528*x^9 + 6227*x^8 - 4968*x^7 + 67046*x^6 - 36022*x^5 + 492458*x^4 - 218804*x^3 + 2200477*x^2 - 533070*x + 4387351)
 
gp: K = bnfinit(x^14 - 2*x^13 + 19*x^12 - 34*x^11 + 402*x^10 - 528*x^9 + 6227*x^8 - 4968*x^7 + 67046*x^6 - 36022*x^5 + 492458*x^4 - 218804*x^3 + 2200477*x^2 - 533070*x + 4387351, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} + 19 x^{12} - 34 x^{11} + 402 x^{10} - 528 x^{9} + 6227 x^{8} - 4968 x^{7} + 67046 x^{6} - 36022 x^{5} + 492458 x^{4} - 218804 x^{3} + 2200477 x^{2} - 533070 x + 4387351 \)

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:  \(-1622761392560638877063184384=-\,2^{21}\cdot 3^{7}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.82$
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:  $2, 3, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(696=2^{3}\cdot 3\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{696}(1,·)$, $\chi_{696}(197,·)$, $\chi_{696}(529,·)$, $\chi_{696}(169,·)$, $\chi_{696}(581,·)$, $\chi_{696}(413,·)$, $\chi_{696}(49,·)$, $\chi_{696}(25,·)$, $\chi_{696}(629,·)$, $\chi_{696}(625,·)$, $\chi_{696}(313,·)$, $\chi_{696}(509,·)$, $\chi_{696}(605,·)$, $\chi_{696}(53,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{17} a^{12} + \frac{4}{17} a^{11} - \frac{6}{17} a^{10} + \frac{6}{17} a^{9} + \frac{1}{17} a^{8} + \frac{7}{17} a^{6} + \frac{4}{17} a^{5} + \frac{2}{17} a^{4} + \frac{4}{17} a^{3} - \frac{4}{17} a^{2} + \frac{4}{17} a - \frac{4}{17}$, $\frac{1}{419988929819654430499338688778897} a^{13} - \frac{10043643114432151418746863289560}{419988929819654430499338688778897} a^{12} - \frac{64824155734448411270109495882359}{419988929819654430499338688778897} a^{11} - \frac{173717908008986785025067021030757}{419988929819654430499338688778897} a^{10} + \frac{177968809396149124629242886508945}{419988929819654430499338688778897} a^{9} + \frac{70016859701505110226917440319743}{419988929819654430499338688778897} a^{8} - \frac{153012262790381439469749721579415}{419988929819654430499338688778897} a^{7} - \frac{6262791489640752509304396260279}{419988929819654430499338688778897} a^{6} - \frac{75774091381063508508935237272887}{419988929819654430499338688778897} a^{5} - \frac{45290027157849178422798995683969}{419988929819654430499338688778897} a^{4} - \frac{183119741187524629430307695873674}{419988929819654430499338688778897} a^{3} + \frac{206858777878910591641707034538147}{419988929819654430499338688778897} a^{2} + \frac{185976786353810990929008783202027}{419988929819654430499338688778897} a - \frac{140352514631919683556854967644833}{419988929819654430499338688778897}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{602}$, which has order $4816$ (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

\(\Q(\sqrt{-6}) \), 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 ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{14}$

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$2.14.21.33$x^{14} + 4 x^{13} + 4 x^{12} + 4 x^{11} - 3 x^{10} + 4 x^{9} - 2 x^{7} - x^{6} - 2 x^{5} + 2 x^{4} - 2 x^{3} + 3 x^{2} + 2 x + 1$$2$$7$$21$$C_{14}$$[3]^{7}$
$3$3.14.7.1$x^{14} - 54 x^{8} - 243 x^{4} - 729 x^{2} - 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$