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Properties

Label 14.0.40177496255...5952.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{21}\cdot 7^{24}$
Root discriminant $79.48$
Ramified primes $2, 7$
Class number $953$ (GRH)
Class group $[953]$ (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![420353, 334516, 123060, 6146, 35840, 35938, 8057, -4240, -791, 1022, 371, -42, -28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 28*x^12 - 42*x^11 + 371*x^10 + 1022*x^9 - 791*x^8 - 4240*x^7 + 8057*x^6 + 35938*x^5 + 35840*x^4 + 6146*x^3 + 123060*x^2 + 334516*x + 420353)
 
gp: K = bnfinit(x^14 - 28*x^12 - 42*x^11 + 371*x^10 + 1022*x^9 - 791*x^8 - 4240*x^7 + 8057*x^6 + 35938*x^5 + 35840*x^4 + 6146*x^3 + 123060*x^2 + 334516*x + 420353, 1)
 

Normalized defining polynomial

\( x^{14} - 28 x^{12} - 42 x^{11} + 371 x^{10} + 1022 x^{9} - 791 x^{8} - 4240 x^{7} + 8057 x^{6} + 35938 x^{5} + 35840 x^{4} + 6146 x^{3} + 123060 x^{2} + 334516 x + 420353 \)

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:  \(-401774962552217617093885952=-\,2^{21}\cdot 7^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.48$
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:  $2, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(392=2^{3}\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{392}(1,·)$, $\chi_{392}(323,·)$, $\chi_{392}(211,·)$, $\chi_{392}(225,·)$, $\chi_{392}(169,·)$, $\chi_{392}(267,·)$, $\chi_{392}(337,·)$, $\chi_{392}(113,·)$, $\chi_{392}(99,·)$, $\chi_{392}(155,·)$, $\chi_{392}(43,·)$, $\chi_{392}(281,·)$, $\chi_{392}(57,·)$, $\chi_{392}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{19} a^{11} - \frac{2}{19} a^{10} + \frac{1}{19} a^{9} - \frac{6}{19} a^{8} + \frac{4}{19} a^{7} + \frac{2}{19} a^{6} - \frac{7}{19} a^{5} - \frac{5}{19} a^{4} - \frac{7}{19} a^{3} + \frac{4}{19} a^{2} - \frac{9}{19} a + \frac{5}{19}$, $\frac{1}{589} a^{12} + \frac{13}{589} a^{11} + \frac{275}{589} a^{10} - \frac{181}{589} a^{9} - \frac{162}{589} a^{8} - \frac{166}{589} a^{7} - \frac{167}{589} a^{6} + \frac{175}{589} a^{5} + \frac{13}{589} a^{4} - \frac{25}{589} a^{3} - \frac{253}{589} a^{2} + \frac{3}{589} a + \frac{227}{589}$, $\frac{1}{6407930378631117233228957503} a^{13} + \frac{334771912130471124321347}{6407930378631117233228957503} a^{12} - \frac{48189930837159862032045949}{6407930378631117233228957503} a^{11} - \frac{1410185484965713761141740588}{6407930378631117233228957503} a^{10} - \frac{2474033987243003630328522221}{6407930378631117233228957503} a^{9} - \frac{68377282251526347344568781}{337259493612164064906787237} a^{8} - \frac{25432803619058540481680753}{206707431568745717200934113} a^{7} + \frac{2636092861200192703340991410}{6407930378631117233228957503} a^{6} - \frac{2000945476535487009403776934}{6407930378631117233228957503} a^{5} + \frac{895681454583815203006891575}{6407930378631117233228957503} a^{4} - \frac{1936718645880634061395275977}{6407930378631117233228957503} a^{3} - \frac{685558687867534553071292}{10879338503618195642154427} a^{2} + \frac{522731669669079189646111518}{6407930378631117233228957503} a - \frac{1802674105049301663760871550}{6407930378631117233228957503}$

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

Class group and class number

$C_{953}$, which has order $953$ (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

\(\Q(\sqrt{-2}) \), 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 ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$2$2.14.21.6$x^{14} + 4 x^{11} - 3 x^{10} + 4 x^{9} + 2 x^{8} + 2 x^{7} - 3 x^{6} + 2 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} - 2 x + 1$$2$$7$$21$$C_{14}$$[3]^{7}$
$7$7.14.24.53$x^{14} + 931 x^{13} + 2310 x^{12} + 903 x^{11} + 392 x^{10} + 2198 x^{9} + 2296 x^{8} + 1485 x^{7} + 637 x^{6} + 1295 x^{5} + 2303 x^{4} + 1449 x^{3} + 1316 x^{2} + 2219 x + 2383$$7$$2$$24$$C_{14}$$[2]^{2}$