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Properties

Label 14.0.37333762163...9571.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{24}\cdot 11^{7}$
Root discriminant $93.20$
Ramified primes $7, 11$
Class number $2059$ (GRH)
Class group $[2059]$ (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![766979, 401044, 298305, 4662, 16814, 2002, 15785, 1135, -497, -623, 238, 49, 0, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 49*x^11 + 238*x^10 - 623*x^9 - 497*x^8 + 1135*x^7 + 15785*x^6 + 2002*x^5 + 16814*x^4 + 4662*x^3 + 298305*x^2 + 401044*x + 766979)
 
gp: K = bnfinit(x^14 - 7*x^13 + 49*x^11 + 238*x^10 - 623*x^9 - 497*x^8 + 1135*x^7 + 15785*x^6 + 2002*x^5 + 16814*x^4 + 4662*x^3 + 298305*x^2 + 401044*x + 766979, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} + 49 x^{11} + 238 x^{10} - 623 x^{9} - 497 x^{8} + 1135 x^{7} + 15785 x^{6} + 2002 x^{5} + 16814 x^{4} + 4662 x^{3} + 298305 x^{2} + 401044 x + 766979 \)

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:  \(-3733376216303663794289149571=-\,7^{24}\cdot 11^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.20$
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:  $7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(539=7^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{539}(1,·)$, $\chi_{539}(386,·)$, $\chi_{539}(197,·)$, $\chi_{539}(232,·)$, $\chi_{539}(43,·)$, $\chi_{539}(428,·)$, $\chi_{539}(78,·)$, $\chi_{539}(463,·)$, $\chi_{539}(274,·)$, $\chi_{539}(309,·)$, $\chi_{539}(120,·)$, $\chi_{539}(505,·)$, $\chi_{539}(155,·)$, $\chi_{539}(351,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{589} a^{12} - \frac{117}{589} a^{11} + \frac{271}{589} a^{10} + \frac{94}{589} a^{9} + \frac{15}{589} a^{8} + \frac{256}{589} a^{7} + \frac{288}{589} a^{6} + \frac{103}{589} a^{5} + \frac{60}{589} a^{4} - \frac{16}{589} a^{3} + \frac{2}{19} a^{2} - \frac{245}{589} a + \frac{3}{589}$, $\frac{1}{5091852304558735383173334700159} a^{13} - \frac{3118893354598208133416205109}{5091852304558735383173334700159} a^{12} - \frac{1837829669639544739365345106403}{5091852304558735383173334700159} a^{11} + \frac{686917799324541479354902357772}{5091852304558735383173334700159} a^{10} + \frac{2254194687004934979592153213707}{5091852304558735383173334700159} a^{9} + \frac{129457506573540102243559526083}{5091852304558735383173334700159} a^{8} - \frac{630609887702506758874404646057}{5091852304558735383173334700159} a^{7} + \frac{2134925105654713592537357515554}{5091852304558735383173334700159} a^{6} + \frac{913198203012587047774228371206}{5091852304558735383173334700159} a^{5} + \frac{1068591306128636528307039978617}{5091852304558735383173334700159} a^{4} + \frac{2332784027184601524291457837552}{5091852304558735383173334700159} a^{3} - \frac{1215303132322036910718713458892}{5091852304558735383173334700159} a^{2} - \frac{1265608410406222791085414069297}{5091852304558735383173334700159} a - \frac{2458126697157098077000887094071}{5091852304558735383173334700159}$

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

Class group and class number

$C_{2059}$, which has order $2059$ (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

\(\Q(\sqrt{-11}) \), 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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R R ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$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}$$[2]^{2}$
$11$11.14.7.2$x^{14} - 1771561 x^{2} + 77948684$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$