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Properties

Label 14.0.32918715676...4443.5
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{24}\cdot 43^{13}$
Root discriminant $923.70$
Ramified primes $7, 43$
Class number $1662773$ (GRH)
Class group $[1662773]$ (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![38059805953523, 5856771467617, -2342604234320, -498852081686, 49838686632, 13714379322, -336846090, -156246735, 383474, 1147412, 26488, -5719, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5719*x^11 + 26488*x^10 + 1147412*x^9 + 383474*x^8 - 156246735*x^7 - 336846090*x^6 + 13714379322*x^5 + 49838686632*x^4 - 498852081686*x^3 - 2342604234320*x^2 + 5856771467617*x + 38059805953523)
 
gp: K = bnfinit(x^14 - 5719*x^11 + 26488*x^10 + 1147412*x^9 + 383474*x^8 - 156246735*x^7 - 336846090*x^6 + 13714379322*x^5 + 49838686632*x^4 - 498852081686*x^3 - 2342604234320*x^2 + 5856771467617*x + 38059805953523, 1)
 

Normalized defining polynomial

\( x^{14} - 5719 x^{11} + 26488 x^{10} + 1147412 x^{9} + 383474 x^{8} - 156246735 x^{7} - 336846090 x^{6} + 13714379322 x^{5} + 49838686632 x^{4} - 498852081686 x^{3} - 2342604234320 x^{2} + 5856771467617 x + 38059805953523 \)

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:  \(-329187156767051876908796215815371233484443=-\,7^{24}\cdot 43^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $923.70$
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:  $7, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2107=7^{2}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{2107}(1632,·)$, $\chi_{2107}(1,·)$, $\chi_{2107}(260,·)$, $\chi_{2107}(806,·)$, $\chi_{2107}(1478,·)$, $\chi_{2107}(967,·)$, $\chi_{2107}(680,·)$, $\chi_{2107}(1513,·)$, $\chi_{2107}(813,·)$, $\chi_{2107}(687,·)$, $\chi_{2107}(176,·)$, $\chi_{2107}(1688,·)$, $\chi_{2107}(624,·)$, $\chi_{2107}(1919,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{6451507722051953348843847330915128716049266798429745701330355255117542254578681} a^{13} + \frac{647894030438127505987091020123000730265554615926264497922337475416812360358611}{6451507722051953348843847330915128716049266798429745701330355255117542254578681} a^{12} + \frac{704734492031662813144892350562017564378534701223371591709216736355470352374556}{6451507722051953348843847330915128716049266798429745701330355255117542254578681} a^{11} + \frac{2350078794282807705207568984054406941233420079820969328208191479336523579364443}{6451507722051953348843847330915128716049266798429745701330355255117542254578681} a^{10} - \frac{7255128931950175209861076514800425239032518736830176446814373664079136907739}{81664654709518396820808194062216819190497048081389186092789307026804332336439} a^{9} - \frac{2386788567050700541104406977019049189204926858587436252982316126221856832270355}{6451507722051953348843847330915128716049266798429745701330355255117542254578681} a^{8} + \frac{2235912514299279421838873994902846035337025613310443440570960472008506759879389}{6451507722051953348843847330915128716049266798429745701330355255117542254578681} a^{7} + \frac{12013568622374304273767504711714695010800584086926544455274771648809874440040}{81664654709518396820808194062216819190497048081389186092789307026804332336439} a^{6} - \frac{1491675036077218981739890241771432361952355054396756529346095011621968693261739}{6451507722051953348843847330915128716049266798429745701330355255117542254578681} a^{5} + \frac{2578939667230996133369276128167696938120317688516933855773684340498405133684936}{6451507722051953348843847330915128716049266798429745701330355255117542254578681} a^{4} + \frac{2306655102329553384926776914789205613225830631305629305698567213312995960579707}{6451507722051953348843847330915128716049266798429745701330355255117542254578681} a^{3} + \frac{1643709333396630235074070460216691055802739410187451317633975915901422371091315}{6451507722051953348843847330915128716049266798429745701330355255117542254578681} a^{2} + \frac{857348566105158610335492280914035148298947875399298486716030401371140425926417}{6451507722051953348843847330915128716049266798429745701330355255117542254578681} a + \frac{151673267901401825771892046660299528611036819803961706769058553159180207169371}{6451507722051953348843847330915128716049266798429745701330355255117542254578681}$

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

Class group and class number

$C_{1662773}$, which has order $1662773$ (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:  \( 224277011.0596314 \) (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

\(\Q(\sqrt{-43}) \), 7.7.87495801462998035849.2

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$7$7.14.24.53$x^{14} + 931 x^{13} + 2310 x^{12} + 903 x^{11} + 392 x^{10} + 2198 x^{9} + 2296 x^{8} + 1485 x^{7} + 637 x^{6} + 1295 x^{5} + 2303 x^{4} + 1449 x^{3} + 1316 x^{2} + 2219 x + 2383$$7$$2$$24$$C_{14}$$[2]^{2}$
$43$43.14.13.8$x^{14} + 387$$14$$1$$13$$C_{14}$$[\ ]_{14}$