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Properties

Label 14.14.1496728370...8125.1
Degree $14$
Signature $[14, 0]$
Discriminant $5^{7}\cdot 7^{24}$
Root discriminant $62.84$
Ramified primes $5, 7$
Class number $1$ (GRH)
Class group Trivial (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![-589, -1876, 5201, 12390, -12474, -20622, 8197, 11383, -2149, -2415, 322, 217, -28, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 - 28*x^12 + 217*x^11 + 322*x^10 - 2415*x^9 - 2149*x^8 + 11383*x^7 + 8197*x^6 - 20622*x^5 - 12474*x^4 + 12390*x^3 + 5201*x^2 - 1876*x - 589)
 
gp: K = bnfinit(x^14 - 7*x^13 - 28*x^12 + 217*x^11 + 322*x^10 - 2415*x^9 - 2149*x^8 + 11383*x^7 + 8197*x^6 - 20622*x^5 - 12474*x^4 + 12390*x^3 + 5201*x^2 - 1876*x - 589, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} - 28 x^{12} + 217 x^{11} + 322 x^{10} - 2415 x^{9} - 2149 x^{8} + 11383 x^{7} + 8197 x^{6} - 20622 x^{5} - 12474 x^{4} + 12390 x^{3} + 5201 x^{2} - 1876 x - 589 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(14967283701606751125078125=5^{7}\cdot 7^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.84$
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:  $5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(245=5\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{245}(64,·)$, $\chi_{245}(1,·)$, $\chi_{245}(99,·)$, $\chi_{245}(36,·)$, $\chi_{245}(134,·)$, $\chi_{245}(71,·)$, $\chi_{245}(169,·)$, $\chi_{245}(106,·)$, $\chi_{245}(204,·)$, $\chi_{245}(141,·)$, $\chi_{245}(239,·)$, $\chi_{245}(176,·)$, $\chi_{245}(211,·)$, $\chi_{245}(29,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{589} a^{11} + \frac{13}{589} a^{10} + \frac{170}{589} a^{9} - \frac{204}{589} a^{8} + \frac{106}{589} a^{7} - \frac{136}{589} a^{6} - \frac{106}{589} a^{5} + \frac{262}{589} a^{4} + \frac{79}{589} a^{3} - \frac{183}{589} a^{2} + \frac{91}{589} a$, $\frac{1}{589} a^{12} + \frac{1}{589} a^{10} - \frac{58}{589} a^{9} - \frac{187}{589} a^{8} + \frac{253}{589} a^{7} - \frac{105}{589} a^{6} - \frac{127}{589} a^{5} + \frac{207}{589} a^{4} - \frac{32}{589} a^{3} + \frac{6}{31} a^{2} - \frac{5}{589} a$, $\frac{1}{146776297014413291} a^{13} + \frac{12772865764368}{146776297014413291} a^{12} - \frac{116180750389565}{146776297014413291} a^{11} + \frac{62510134126723885}{146776297014413291} a^{10} - \frac{16320162898847939}{146776297014413291} a^{9} + \frac{415965361032722}{7725068263916489} a^{8} + \frac{47537833566566550}{146776297014413291} a^{7} - \frac{69965445861419937}{146776297014413291} a^{6} - \frac{16635609971193554}{146776297014413291} a^{5} + \frac{14725369167854599}{146776297014413291} a^{4} - \frac{3375183488733797}{7725068263916489} a^{3} + \frac{53600853976526919}{146776297014413291} a^{2} - \frac{61809170052177473}{146776297014413291} a - \frac{7236534299523}{249195750448919}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $13$
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:  \( 182487917.959 \) (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

\(\Q(\sqrt{5}) \), 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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ R R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$5$5.14.7.1$x^{14} - 250 x^{8} + 15625 x^{2} - 312500$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
7Data not computed