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Properties

Label 14.0.25709745488...6875.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,5^{7}\cdot 7^{7}\cdot 43^{12}$
Root discriminant $148.64$
Ramified primes $5, 7, 43$
Class number $95104$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 1486]$ (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![58736881, 7744478, 19889249, 602378, 3204020, -89140, 328903, -27219, 22811, -2707, 1160, -165, 36, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 + 36*x^12 - 165*x^11 + 1160*x^10 - 2707*x^9 + 22811*x^8 - 27219*x^7 + 328903*x^6 - 89140*x^5 + 3204020*x^4 + 602378*x^3 + 19889249*x^2 + 7744478*x + 58736881)
 
gp: K = bnfinit(x^14 - 5*x^13 + 36*x^12 - 165*x^11 + 1160*x^10 - 2707*x^9 + 22811*x^8 - 27219*x^7 + 328903*x^6 - 89140*x^5 + 3204020*x^4 + 602378*x^3 + 19889249*x^2 + 7744478*x + 58736881, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} + 36 x^{12} - 165 x^{11} + 1160 x^{10} - 2707 x^{9} + 22811 x^{8} - 27219 x^{7} + 328903 x^{6} - 89140 x^{5} + 3204020 x^{4} + 602378 x^{3} + 19889249 x^{2} + 7744478 x + 58736881 \)

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:  \(-2570974548880469840391308046875=-\,5^{7}\cdot 7^{7}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $148.64$
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:  $5, 7, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1505=5\cdot 7\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{1505}(1,·)$, $\chi_{1505}(944,·)$, $\chi_{1505}(489,·)$, $\chi_{1505}(876,·)$, $\chi_{1505}(1294,·)$, $\chi_{1505}(176,·)$, $\chi_{1505}(594,·)$, $\chi_{1505}(1331,·)$, $\chi_{1505}(981,·)$, $\chi_{1505}(279,·)$, $\chi_{1505}(666,·)$, $\chi_{1505}(699,·)$, $\chi_{1505}(1086,·)$, $\chi_{1505}(1119,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{9} + \frac{2}{7} a^{8} + \frac{2}{7} a^{7} - \frac{2}{7} a^{6} - \frac{3}{7} a^{5} + \frac{1}{7} a^{4} - \frac{1}{7} a^{3} + \frac{2}{7} a^{2}$, $\frac{1}{7} a^{11} + \frac{1}{7} a^{9} - \frac{3}{7} a^{8} + \frac{2}{7} a^{6} - \frac{2}{7} a^{5} + \frac{1}{7} a^{3} + \frac{2}{7} a^{2}$, $\frac{1}{3871} a^{12} - \frac{276}{3871} a^{11} - \frac{11}{3871} a^{10} - \frac{43}{3871} a^{9} + \frac{1056}{3871} a^{8} + \frac{1784}{3871} a^{7} - \frac{1664}{3871} a^{6} - \frac{179}{553} a^{5} + \frac{1382}{3871} a^{4} + \frac{844}{3871} a^{3} - \frac{303}{3871} a^{2} + \frac{225}{553} a - \frac{216}{553}$, $\frac{1}{854850487169014583419643313527041} a^{13} + \frac{27792647679646175619844414720}{854850487169014583419643313527041} a^{12} - \frac{15454560501976630094003142251417}{854850487169014583419643313527041} a^{11} - \frac{15433940142791307767970747344879}{854850487169014583419643313527041} a^{10} - \frac{52698039885838686865648844998961}{122121498167002083345663330503863} a^{9} - \frac{3859198281457626836347983016532}{17445928309571726192237618643409} a^{8} - \frac{61363465433871805245036996274019}{854850487169014583419643313527041} a^{7} + \frac{171715375658263338341181437471585}{854850487169014583419643313527041} a^{6} + \frac{237238520887200349706348885140845}{854850487169014583419643313527041} a^{5} + \frac{357698445908724224035496842182442}{854850487169014583419643313527041} a^{4} + \frac{316993035711987590065956564032343}{854850487169014583419643313527041} a^{3} - \frac{31586472827178466988674791505211}{854850487169014583419643313527041} a^{2} - \frac{7109720654580281543676301233591}{17445928309571726192237618643409} a - \frac{4990488867841665960146584855815}{122121498167002083345663330503863}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{1486}$, which has order $95104$ (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

\(\Q(\sqrt{-35}) \), 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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ R R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ R ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$5$5.14.7.2$x^{14} - 15625 x^{2} + 156250$$2$$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$43.14.12.1$x^{14} + 3569 x^{7} + 4043763$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$