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Properties

Label 14.0.41898815302...4987.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 7^{24}$
Root discriminant $48.67$
Ramified primes $3, 7$
Class number $203$ (GRH)
Class group $[203]$ (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![9409, -8148, 17920, -8246, 18151, -8701, 8869, -2696, 2184, -553, 350, -42, 21, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 21*x^12 - 42*x^11 + 350*x^10 - 553*x^9 + 2184*x^8 - 2696*x^7 + 8869*x^6 - 8701*x^5 + 18151*x^4 - 8246*x^3 + 17920*x^2 - 8148*x + 9409)
 
gp: K = bnfinit(x^14 + 21*x^12 - 42*x^11 + 350*x^10 - 553*x^9 + 2184*x^8 - 2696*x^7 + 8869*x^6 - 8701*x^5 + 18151*x^4 - 8246*x^3 + 17920*x^2 - 8148*x + 9409, 1)
 

Normalized defining polynomial

\( x^{14} + 21 x^{12} - 42 x^{11} + 350 x^{10} - 553 x^{9} + 2184 x^{8} - 2696 x^{7} + 8869 x^{6} - 8701 x^{5} + 18151 x^{4} - 8246 x^{3} + 17920 x^{2} - 8148 x + 9409 \)

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:  \(-418988153029298748294987=-\,3^{7}\cdot 7^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.67$
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:  $3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(147=3\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{147}(64,·)$, $\chi_{147}(1,·)$, $\chi_{147}(134,·)$, $\chi_{147}(71,·)$, $\chi_{147}(8,·)$, $\chi_{147}(106,·)$, $\chi_{147}(43,·)$, $\chi_{147}(113,·)$, $\chi_{147}(50,·)$, $\chi_{147}(85,·)$, $\chi_{147}(22,·)$, $\chi_{147}(92,·)$, $\chi_{147}(29,·)$, $\chi_{147}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{589} a^{10} - \frac{268}{589} a^{9} - \frac{125}{589} a^{8} + \frac{265}{589} a^{7} - \frac{156}{589} a^{6} + \frac{23}{589} a^{5} - \frac{273}{589} a^{4} - \frac{198}{589} a^{3} + \frac{159}{589} a^{2} + \frac{26}{589} a - \frac{233}{589}$, $\frac{1}{589} a^{11} - \frac{91}{589} a^{9} - \frac{251}{589} a^{8} + \frac{184}{589} a^{7} + \frac{34}{589} a^{6} + \frac{1}{589} a^{5} + \frac{263}{589} a^{4} + \frac{105}{589} a^{3} + \frac{230}{589} a^{2} + \frac{256}{589} a - \frac{10}{589}$, $\frac{1}{589} a^{12} + \frac{99}{589} a^{9} - \frac{59}{589} a^{6} - \frac{118}{589} a^{3} + \frac{1}{589}$, $\frac{1}{1895673790255648453} a^{13} + \frac{5828455076247}{19543028765522149} a^{12} - \frac{91476252389347}{1895673790255648453} a^{11} - \frac{1013754594507566}{1895673790255648453} a^{10} + \frac{774374721858168606}{1895673790255648453} a^{9} + \frac{176500704584479044}{1895673790255648453} a^{8} + \frac{97273106589424927}{1895673790255648453} a^{7} - \frac{95180973716813111}{1895673790255648453} a^{6} + \frac{296184478089752521}{1895673790255648453} a^{5} + \frac{54065034148456543}{1895673790255648453} a^{4} + \frac{618754878968356045}{1895673790255648453} a^{3} - \frac{27392730788016995}{1895673790255648453} a^{2} + \frac{49711488262123048}{1895673790255648453} a - \frac{4328955486703941}{19543028765522149}$

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

Class group and class number

$C_{203}$, which has order $203$ (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:  \( -\frac{161289742466524}{1895673790255648453} a^{13} - \frac{957311663154}{19543028765522149} a^{12} - \frac{3252983796414282}{1895673790255648453} a^{11} + \frac{4785500909859803}{1895673790255648453} a^{10} - \frac{50010109733421474}{1895673790255648453} a^{9} + \frac{50386722190134414}{1895673790255648453} a^{8} - \frac{257564795016183278}{1895673790255648453} a^{7} + \frac{157348402783570740}{1895673790255648453} a^{6} - \frac{958061773957764798}{1895673790255648453} a^{5} + \frac{288377486595742334}{1895673790255648453} a^{4} - \frac{1285265248884962772}{1895673790255648453} a^{3} - \frac{1523096170019655210}{1895673790255648453} a^{2} - \frac{1081764845716455943}{1895673790255648453} a + \frac{2712612543346663}{19543028765522149} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 35256.6897369 \) (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

\(\Q(\sqrt{-3}) \), 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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ R ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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}$$[\ ]_{2}^{7}$
$7$7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$
7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$