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Properties

Label 14.0.97059481264...7663.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{24}\cdot 47^{7}$
Root discriminant $192.66$
Ramified primes $7, 47$
Class number $280315$ (GRH)
Class group $[280315]$ (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![469806631, -208592853, 144088714, -44907863, 18981767, -4484578, 1385356, -250281, 64330, -8946, 2044, -245, 63, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 63*x^12 - 245*x^11 + 2044*x^10 - 8946*x^9 + 64330*x^8 - 250281*x^7 + 1385356*x^6 - 4484578*x^5 + 18981767*x^4 - 44907863*x^3 + 144088714*x^2 - 208592853*x + 469806631)
 
gp: K = bnfinit(x^14 - 7*x^13 + 63*x^12 - 245*x^11 + 2044*x^10 - 8946*x^9 + 64330*x^8 - 250281*x^7 + 1385356*x^6 - 4484578*x^5 + 18981767*x^4 - 44907863*x^3 + 144088714*x^2 - 208592853*x + 469806631, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} + 63 x^{12} - 245 x^{11} + 2044 x^{10} - 8946 x^{9} + 64330 x^{8} - 250281 x^{7} + 1385356 x^{6} - 4484578 x^{5} + 18981767 x^{4} - 44907863 x^{3} + 144088714 x^{2} - 208592853 x + 469806631 \)

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:  \(-97059481264166574360249800987663=-\,7^{24}\cdot 47^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $192.66$
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:  $7, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2303=7^{2}\cdot 47\)
Dirichlet character group:    $\lbrace$$\chi_{2303}(1,·)$, $\chi_{2303}(610,·)$, $\chi_{2303}(1317,·)$, $\chi_{2303}(1926,·)$, $\chi_{2303}(330,·)$, $\chi_{2303}(939,·)$, $\chi_{2303}(1646,·)$, $\chi_{2303}(2255,·)$, $\chi_{2303}(659,·)$, $\chi_{2303}(1268,·)$, $\chi_{2303}(1975,·)$, $\chi_{2303}(281,·)$, $\chi_{2303}(988,·)$, $\chi_{2303}(1597,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{589} a^{12} + \frac{105}{589} a^{11} + \frac{282}{589} a^{10} - \frac{110}{589} a^{9} - \frac{11}{589} a^{8} + \frac{50}{589} a^{7} + \frac{5}{19} a^{6} - \frac{96}{589} a^{5} - \frac{184}{589} a^{4} + \frac{102}{589} a^{3} - \frac{3}{19} a^{2} + \frac{268}{589} a + \frac{9}{589}$, $\frac{1}{10413986656602646180615364866441627559011} a^{13} - \frac{1854274064176721178862349401108198245}{10413986656602646180615364866441627559011} a^{12} - \frac{2314525745039289283162838428788043084075}{10413986656602646180615364866441627559011} a^{11} - \frac{258930758466322272965489293653413360177}{548104560873823483190282361391664608369} a^{10} - \frac{3150570781342734109988578281046070113265}{10413986656602646180615364866441627559011} a^{9} + \frac{386881653239190015804604994300125731051}{10413986656602646180615364866441627559011} a^{8} - \frac{4277945983695051088354368390377714847}{17680792286252370425492979399731116399} a^{7} + \frac{3380020136394427368662932298282270856127}{10413986656602646180615364866441627559011} a^{6} + \frac{4381252337295637595342890539709300535819}{10413986656602646180615364866441627559011} a^{5} + \frac{4257459149425364227809352930045238932273}{10413986656602646180615364866441627559011} a^{4} - \frac{144482694438275012680963634499955458555}{335935053438795038084366608594891211581} a^{3} + \frac{1306013957463100699692606857411344885722}{10413986656602646180615364866441627559011} a^{2} - \frac{2685372522148000261968488009028097615359}{10413986656602646180615364866441627559011} a + \frac{43694952010233105884002083227340520302}{335935053438795038084366608594891211581}$

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

Class group and class number

$C_{280315}$, which has order $280315$ (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:  \( 35256.68973693789 \) (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

\(\Q(\sqrt{-47}) \), 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ R ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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$7.7.12.1$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]$
$47$47.14.7.2$x^{14} - 10779215329 x^{2} + 3546361843241$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$