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Properties

Label 14.0.61107168976...4976.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{21}\cdot 7^{7}\cdot 29^{12}$
Root discriminant $134.15$
Ramified primes $2, 7, 29$
Class number $125696$ (GRH)
Class group $[4, 4, 4, 1964]$ (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![441622847, -31573230, 133911229, -8526420, 18374426, -1077430, 1509222, -84712, 81987, -4256, 3042, -130, 75, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 75*x^12 - 130*x^11 + 3042*x^10 - 4256*x^9 + 81987*x^8 - 84712*x^7 + 1509222*x^6 - 1077430*x^5 + 18374426*x^4 - 8526420*x^3 + 133911229*x^2 - 31573230*x + 441622847)
 
gp: K = bnfinit(x^14 - 2*x^13 + 75*x^12 - 130*x^11 + 3042*x^10 - 4256*x^9 + 81987*x^8 - 84712*x^7 + 1509222*x^6 - 1077430*x^5 + 18374426*x^4 - 8526420*x^3 + 133911229*x^2 - 31573230*x + 441622847, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} + 75 x^{12} - 130 x^{11} + 3042 x^{10} - 4256 x^{9} + 81987 x^{8} - 84712 x^{7} + 1509222 x^{6} - 1077430 x^{5} + 18374426 x^{4} - 8526420 x^{3} + 133911229 x^{2} - 31573230 x + 441622847 \)

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:  \(-611071689763862013138201214976=-\,2^{21}\cdot 7^{7}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $134.15$
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:  $2, 7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1624=2^{3}\cdot 7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{1624}(1,·)$, $\chi_{1624}(1301,·)$, $\chi_{1624}(517,·)$, $\chi_{1624}(1009,·)$, $\chi_{1624}(393,·)$, $\chi_{1624}(1357,·)$, $\chi_{1624}(1457,·)$, $\chi_{1624}(741,·)$, $\chi_{1624}(181,·)$, $\chi_{1624}(169,·)$, $\chi_{1624}(953,·)$, $\chi_{1624}(281,·)$, $\chi_{1624}(349,·)$, $\chi_{1624}(629,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{17} a^{12} + \frac{4}{17} a^{11} + \frac{8}{17} a^{10} - \frac{4}{17} a^{9} - \frac{5}{17} a^{8} + \frac{5}{17} a^{7} + \frac{5}{17} a^{6} - \frac{1}{17} a^{5} - \frac{6}{17} a^{4} - \frac{5}{17} a^{2} + \frac{2}{17} a - \frac{7}{17}$, $\frac{1}{1318376291114455623192126756862704345073} a^{13} - \frac{20744512155954438374750905079008890804}{1318376291114455623192126756862704345073} a^{12} + \frac{135404735998830728161084077092235971994}{1318376291114455623192126756862704345073} a^{11} - \frac{347075195252540517985309201433623831760}{1318376291114455623192126756862704345073} a^{10} - \frac{18231017659813222129246778809175903932}{77551546536144448423066279815453196769} a^{9} + \frac{110855116743649024287708259401818212234}{1318376291114455623192126756862704345073} a^{8} - \frac{4585322574395273480325174995758945920}{22345360866346705477832656895978039747} a^{7} - \frac{155859224695704816603173725722862466047}{1318376291114455623192126756862704345073} a^{6} + \frac{619149398005478731416253616296737288173}{1318376291114455623192126756862704345073} a^{5} + \frac{37121319560502805031073807698948668291}{1318376291114455623192126756862704345073} a^{4} - \frac{65297502663264216877689982594221801017}{1318376291114455623192126756862704345073} a^{3} + \frac{150997661627346694799317373883207089930}{1318376291114455623192126756862704345073} a^{2} - \frac{443100289723156263841262864212689658444}{1318376291114455623192126756862704345073} a + \frac{7005310535650081296883382499222984685}{22345360866346705477832656895978039747}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{4}\times C_{1964}$, which has order $125696$ (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:  \( 6020.985100147561 \) (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

\(\Q(\sqrt{-14}) \), 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 R ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{14}$

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
$2$2.14.21.34$x^{14} + 4 x^{13} + 8 x^{12} + 4 x^{11} + 5 x^{10} + 8 x^{9} - 6 x^{8} - 6 x^{7} + x^{6} + 6 x^{5} + 2 x^{3} + 7 x^{2} + 6 x - 7$$2$$7$$21$$C_{14}$$[3]^{7}$
$7$7.14.7.1$x^{14} - 117649 x^{2} + 1647086$$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}$