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Properties

Label 14.14.6372517516...9181.1
Degree $14$
Signature $[14, 0]$
Discriminant $3^{7}\cdot 7^{7}\cdot 29^{12}$
Root discriminant $82.15$
Ramified primes $3, 7, 29$
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![-2999, -34448, -6213, 104780, -7172, -86298, 12607, 28717, -5149, -4123, 798, 251, -50, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 - 50*x^12 + 251*x^11 + 798*x^10 - 4123*x^9 - 5149*x^8 + 28717*x^7 + 12607*x^6 - 86298*x^5 - 7172*x^4 + 104780*x^3 - 6213*x^2 - 34448*x - 2999)
 
gp: K = bnfinit(x^14 - 5*x^13 - 50*x^12 + 251*x^11 + 798*x^10 - 4123*x^9 - 5149*x^8 + 28717*x^7 + 12607*x^6 - 86298*x^5 - 7172*x^4 + 104780*x^3 - 6213*x^2 - 34448*x - 2999, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} - 50 x^{12} + 251 x^{11} + 798 x^{10} - 4123 x^{9} - 5149 x^{8} + 28717 x^{7} + 12607 x^{6} - 86298 x^{5} - 7172 x^{4} + 104780 x^{3} - 6213 x^{2} - 34448 x - 2999 \)

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:  \(637251751667769538275359181=3^{7}\cdot 7^{7}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.15$
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:  $3, 7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(609=3\cdot 7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{609}(1,·)$, $\chi_{609}(547,·)$, $\chi_{609}(484,·)$, $\chi_{609}(545,·)$, $\chi_{609}(169,·)$, $\chi_{609}(587,·)$, $\chi_{609}(335,·)$, $\chi_{609}(400,·)$, $\chi_{609}(146,·)$, $\chi_{609}(83,·)$, $\chi_{609}(20,·)$, $\chi_{609}(442,·)$, $\chi_{609}(314,·)$, $\chi_{609}(190,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{17} a^{9} - \frac{3}{17} a^{8} + \frac{2}{17} a^{7} + \frac{3}{17} a^{6} + \frac{2}{17} a^{4} + \frac{5}{17} a^{2} - \frac{3}{17} a - \frac{7}{17}$, $\frac{1}{17} a^{10} - \frac{7}{17} a^{8} - \frac{8}{17} a^{7} - \frac{8}{17} a^{6} + \frac{2}{17} a^{5} + \frac{6}{17} a^{4} + \frac{5}{17} a^{3} - \frac{5}{17} a^{2} + \frac{1}{17} a - \frac{4}{17}$, $\frac{1}{17} a^{11} + \frac{5}{17} a^{8} + \frac{6}{17} a^{7} + \frac{6}{17} a^{6} + \frac{6}{17} a^{5} + \frac{2}{17} a^{4} - \frac{5}{17} a^{3} + \frac{2}{17} a^{2} - \frac{8}{17} a + \frac{2}{17}$, $\frac{1}{17} a^{12} + \frac{4}{17} a^{8} - \frac{4}{17} a^{7} + \frac{8}{17} a^{6} + \frac{2}{17} a^{5} + \frac{2}{17} a^{4} + \frac{2}{17} a^{3} + \frac{1}{17} a^{2} + \frac{1}{17}$, $\frac{1}{1232835625520778137081} a^{13} + \frac{28603414008591630688}{1232835625520778137081} a^{12} + \frac{1684637727454007452}{72519742677692831593} a^{11} + \frac{28928200367871923919}{1232835625520778137081} a^{10} - \frac{27705805855482318916}{1232835625520778137081} a^{9} + \frac{607699978543142431899}{1232835625520778137081} a^{8} + \frac{339941692544916604568}{1232835625520778137081} a^{7} + \frac{207663909094719911191}{1232835625520778137081} a^{6} - \frac{19997592957337973949}{1232835625520778137081} a^{5} - \frac{382120119350555813318}{1232835625520778137081} a^{4} - \frac{494309510384887626770}{1232835625520778137081} a^{3} - \frac{194719883338532471799}{1232835625520778137081} a^{2} - \frac{421874647968653517114}{1232835625520778137081} a - \frac{35721925110514137126}{72519742677692831593}$

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:  \( 1313003537.670304 \) (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

\(\Q(\sqrt{21}) \), 7.7.594823321.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ R ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ R ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{14}$

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
$3$3.14.7.1$x^{14} - 54 x^{8} - 243 x^{4} - 729 x^{2} - 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$7$7.14.7.1$x^{14} - 117649 x^{2} + 1647086$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.14.12.1$x^{14} + 2407 x^{7} + 1839267$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$