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Properties

Label 14.0.87391712553...8987.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 43^{12}$
Root discriminant $43.52$
Ramified primes $3, 43$
Class number $203$
Class group $[203]$
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![2401, 343, 5145, -4452, 8835, -5324, 5112, -2372, 1819, -658, 321, -52, 19, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 19*x^12 - 52*x^11 + 321*x^10 - 658*x^9 + 1819*x^8 - 2372*x^7 + 5112*x^6 - 5324*x^5 + 8835*x^4 - 4452*x^3 + 5145*x^2 + 343*x + 2401)
 
gp: K = bnfinit(x^14 - x^13 + 19*x^12 - 52*x^11 + 321*x^10 - 658*x^9 + 1819*x^8 - 2372*x^7 + 5112*x^6 - 5324*x^5 + 8835*x^4 - 4452*x^3 + 5145*x^2 + 343*x + 2401, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 19 x^{12} - 52 x^{11} + 321 x^{10} - 658 x^{9} + 1819 x^{8} - 2372 x^{7} + 5112 x^{6} - 5324 x^{5} + 8835 x^{4} - 4452 x^{3} + 5145 x^{2} + 343 x + 2401 \)

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:  \(-87391712553613254588987=-\,3^{7}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.52$
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:  $3, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(129=3\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{129}(64,·)$, $\chi_{129}(1,·)$, $\chi_{129}(35,·)$, $\chi_{129}(4,·)$, $\chi_{129}(97,·)$, $\chi_{129}(41,·)$, $\chi_{129}(11,·)$, $\chi_{129}(44,·)$, $\chi_{129}(47,·)$, $\chi_{129}(16,·)$, $\chi_{129}(107,·)$, $\chi_{129}(121,·)$, $\chi_{129}(59,·)$, $\chi_{129}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{4}$, $\frac{1}{49} a^{11} + \frac{3}{49} a^{10} + \frac{3}{49} a^{9} + \frac{2}{49} a^{8} + \frac{2}{7} a^{6} + \frac{13}{49} a^{5} - \frac{17}{49} a^{4} - \frac{24}{49} a^{3} - \frac{23}{49} a^{2} + \frac{1}{7} a$, $\frac{1}{343} a^{12} + \frac{3}{343} a^{11} + \frac{17}{343} a^{10} + \frac{23}{343} a^{9} - \frac{3}{49} a^{8} + \frac{2}{49} a^{7} - \frac{85}{343} a^{6} - \frac{164}{343} a^{5} + \frac{158}{343} a^{4} - \frac{44}{343} a^{3} + \frac{18}{49} a^{2} + \frac{1}{7} a$, $\frac{1}{1935833521756099} a^{13} + \frac{652773871196}{1935833521756099} a^{12} + \frac{19229166875651}{1935833521756099} a^{11} + \frac{3403244616867}{1935833521756099} a^{10} + \frac{107304265046453}{1935833521756099} a^{9} + \frac{19484780208395}{276547645965157} a^{8} - \frac{93874409132582}{1935833521756099} a^{7} - \frac{202990318557557}{1935833521756099} a^{6} + \frac{436287258879593}{1935833521756099} a^{5} + \frac{478710712533833}{1935833521756099} a^{4} + \frac{56006832941221}{1935833521756099} a^{3} - \frac{12474108664587}{39506806566451} a^{2} - \frac{1696736584178}{5643829509493} a + \frac{118526112335}{5643829509493}$

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

Class group and class number

$C_{203}$, which has order $203$

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:  \( \frac{118826219574}{1935833521756099} a^{13} - \frac{622774099492}{1935833521756099} a^{12} + \frac{2722160359242}{1935833521756099} a^{11} - \frac{15436421411023}{1935833521756099} a^{10} + \frac{63052496719871}{1935833521756099} a^{9} - \frac{33269672178881}{276547645965157} a^{8} + \frac{516874305988720}{1935833521756099} a^{7} - \frac{1080676560329395}{1935833521756099} a^{6} + \frac{1501498263074528}{1935833521756099} a^{5} - \frac{2645229387205772}{1935833521756099} a^{4} + \frac{2942322682402753}{1935833521756099} a^{3} - \frac{85039226233463}{39506806566451} a^{2} + \frac{3700936318953}{5643829509493} a - \frac{863918778955}{5643829509493} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 35991.6418506 \)
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

\(\Q(\sqrt{-3}) \), 7.7.6321363049.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ R ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ R ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$3$3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$43$43.7.6.1$x^{7} - 43$$7$$1$$6$$C_7$$[\ ]_{7}$
43.7.6.1$x^{7} - 43$$7$$1$$6$$C_7$$[\ ]_{7}$