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Properties

Label 14.0.21972068264...0000.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{21}\cdot 5^{7}\cdot 7^{25}$
Root discriminant $204.24$
Ramified primes $2, 5, 7$
Class number $1030684$ (GRH)
Class group $[2, 515342]$ (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![70000000, 0, 196000000, 0, 147000000, 0, 37730000, 0, 2401000, 0, 53900, 0, 420, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 420*x^12 + 53900*x^10 + 2401000*x^8 + 37730000*x^6 + 147000000*x^4 + 196000000*x^2 + 70000000)
 
gp: K = bnfinit(x^14 + 420*x^12 + 53900*x^10 + 2401000*x^8 + 37730000*x^6 + 147000000*x^4 + 196000000*x^2 + 70000000, 1)
 

Normalized defining polynomial

\( x^{14} + 420 x^{12} + 53900 x^{10} + 2401000 x^{8} + 37730000 x^{6} + 147000000 x^{4} + 196000000 x^{2} + 70000000 \)

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:  \(-219720682645744009348218880000000=-\,2^{21}\cdot 5^{7}\cdot 7^{25}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $204.24$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1960=2^{3}\cdot 5\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{1960}(1,·)$, $\chi_{1960}(1189,·)$, $\chi_{1960}(69,·)$, $\chi_{1960}(1121,·)$, $\chi_{1960}(841,·)$, $\chi_{1960}(909,·)$, $\chi_{1960}(349,·)$, $\chi_{1960}(561,·)$, $\chi_{1960}(629,·)$, $\chi_{1960}(281,·)$, $\chi_{1960}(1681,·)$, $\chi_{1960}(1401,·)$, $\chi_{1960}(1469,·)$, $\chi_{1960}(1749,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{10} a^{2}$, $\frac{1}{10} a^{3}$, $\frac{1}{100} a^{4}$, $\frac{1}{100} a^{5}$, $\frac{1}{1000} a^{6}$, $\frac{1}{1000} a^{7}$, $\frac{1}{10000} a^{8}$, $\frac{1}{10000} a^{9}$, $\frac{1}{150100000} a^{10} - \frac{151}{7505000} a^{8} - \frac{59}{375250} a^{6} - \frac{303}{150100} a^{4} - \frac{11}{3002} a^{2} + \frac{638}{1501}$, $\frac{1}{150100000} a^{11} - \frac{151}{7505000} a^{9} - \frac{59}{375250} a^{7} - \frac{303}{150100} a^{5} - \frac{11}{3002} a^{3} + \frac{638}{1501} a$, $\frac{1}{46531000000} a^{12} + \frac{9}{4653100000} a^{10} - \frac{2049}{232655000} a^{8} + \frac{2927}{23265500} a^{6} - \frac{2727}{4653100} a^{4} + \frac{10551}{465310} a^{2} - \frac{16225}{46531}$, $\frac{1}{46531000000} a^{13} + \frac{9}{4653100000} a^{11} - \frac{2049}{232655000} a^{9} + \frac{2927}{23265500} a^{7} - \frac{2727}{4653100} a^{5} + \frac{10551}{465310} a^{3} - \frac{16225}{46531} a$

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

Class group and class number

$C_{2}\times C_{515342}$, which has order $1030684$ (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

\(\Q(\sqrt{-70}) \), 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 R ${\href{/LocalNumberField/3.14.0.1}{14} }$ R R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$2$2.14.21.33$x^{14} + 4 x^{13} + 4 x^{12} + 4 x^{11} - 3 x^{10} + 4 x^{9} - 2 x^{7} - x^{6} - 2 x^{5} + 2 x^{4} - 2 x^{3} + 3 x^{2} + 2 x + 1$$2$$7$$21$$C_{14}$$[3]^{7}$
$5$5.14.7.1$x^{14} - 250 x^{8} + 15625 x^{2} - 312500$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$7$7.14.25.75$x^{14} + 112 x^{13} + 28 x^{12} - 98 x^{11} + 147 x^{10} + 98 x^{9} + 49 x^{8} - 161 x^{7} + 49 x^{6} + 147 x^{5} + 147 x^{2} - 147 x - 126$$14$$1$$25$$C_{14}$$[2]_{2}$