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Properties

Label 14.0.30947458246...0463.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 7^{7}\cdot 43^{13}$
Root discriminant $150.63$
Ramified primes $3, 7, 43$
Class number $115712$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 904]$ (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![356178109, 377213368, 308477505, 183273867, 71615373, 26563459, 7469613, 1691539, 447988, 42167, 15033, 239, 217, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 217*x^12 + 239*x^11 + 15033*x^10 + 42167*x^9 + 447988*x^8 + 1691539*x^7 + 7469613*x^6 + 26563459*x^5 + 71615373*x^4 + 183273867*x^3 + 308477505*x^2 + 377213368*x + 356178109)
 
gp: K = bnfinit(x^14 - x^13 + 217*x^12 + 239*x^11 + 15033*x^10 + 42167*x^9 + 447988*x^8 + 1691539*x^7 + 7469613*x^6 + 26563459*x^5 + 71615373*x^4 + 183273867*x^3 + 308477505*x^2 + 377213368*x + 356178109, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 217 x^{12} + 239 x^{11} + 15033 x^{10} + 42167 x^{9} + 447988 x^{8} + 1691539 x^{7} + 7469613 x^{6} + 26563459 x^{5} + 71615373 x^{4} + 183273867 x^{3} + 308477505 x^{2} + 377213368 x + 356178109 \)

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:  \(-3094745824656233782531059200463=-\,3^{7}\cdot 7^{7}\cdot 43^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $150.63$
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:  $3, 7, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(903=3\cdot 7\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{903}(64,·)$, $\chi_{903}(1,·)$, $\chi_{903}(419,·)$, $\chi_{903}(484,·)$, $\chi_{903}(902,·)$, $\chi_{903}(839,·)$, $\chi_{903}(776,·)$, $\chi_{903}(778,·)$, $\chi_{903}(524,·)$, $\chi_{903}(274,·)$, $\chi_{903}(629,·)$, $\chi_{903}(379,·)$, $\chi_{903}(125,·)$, $\chi_{903}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} - \frac{2}{7} a^{9} - \frac{2}{7} a^{7} - \frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{2}{7} a^{3} + \frac{2}{7} a^{2}$, $\frac{1}{7} a^{11} + \frac{3}{7} a^{9} - \frac{2}{7} a^{8} + \frac{2}{7} a^{7} + \frac{2}{7} a^{6} + \frac{1}{7} a^{5} + \frac{2}{7} a^{4} - \frac{1}{7} a^{3} - \frac{3}{7} a^{2}$, $\frac{1}{553} a^{12} + \frac{2}{553} a^{11} + \frac{8}{553} a^{10} - \frac{167}{553} a^{9} - \frac{65}{553} a^{8} + \frac{220}{553} a^{7} - \frac{34}{79} a^{6} - \frac{228}{553} a^{5} + \frac{10}{553} a^{4} - \frac{142}{553} a^{3} - \frac{150}{553} a^{2} + \frac{24}{79} a + \frac{13}{79}$, $\frac{1}{18555403797243500097531069858961719089609129244277} a^{13} + \frac{10757308804754315830403189204988537301801796116}{18555403797243500097531069858961719089609129244277} a^{12} - \frac{440151715441990050065542369544699911796262952260}{18555403797243500097531069858961719089609129244277} a^{11} + \frac{65673265833883108203229934157510652942113888794}{18555403797243500097531069858961719089609129244277} a^{10} + \frac{8420481694972864803053543582786600492619940518524}{18555403797243500097531069858961719089609129244277} a^{9} - \frac{530025173343918505239565164940301367091940696749}{2650771971034785728218724265565959869944161320611} a^{8} + \frac{2311790652717317212307981367027110051177121564475}{18555403797243500097531069858961719089609129244277} a^{7} - \frac{43615700385127420398831204571256313283964297942}{2650771971034785728218724265565959869944161320611} a^{6} + \frac{3424206168977502705531088811569214630313130803788}{18555403797243500097531069858961719089609129244277} a^{5} + \frac{7703248872053133174718399690127271476600474251430}{18555403797243500097531069858961719089609129244277} a^{4} - \frac{75113412563873166105473171640010925545071188696}{234878529079031646804190757708376191007710496763} a^{3} + \frac{4171980549801950934947982379282541543593330982461}{18555403797243500097531069858961719089609129244277} a^{2} - \frac{228630348339212579272338447854850847610351009476}{2650771971034785728218724265565959869944161320611} a + \frac{826971830792764982747094068071122144708628787709}{2650771971034785728218724265565959869944161320611}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{904}$, which has order $115712$ (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:  \( 35991.64185055774 \) (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

\(\Q(\sqrt{-903}) \), 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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}$$[\ ]_{2}^{7}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
$43$43.14.13.11$x^{14} + 205667667$$14$$1$$13$$C_{14}$$[\ ]_{14}$