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Properties

Label 14.0.26290885243...4279.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 7^{24}\cdot 13^{7}$
Root discriminant $175.50$
Ramified primes $3, 7, 13$
Class number $109316$ (GRH)
Class group $[109316]$ (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![191861773, -95761729, 65267748, -22503019, 9634359, -2507386, 774270, -153373, 39270, -5810, 1246, -161, 49, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 49*x^12 - 161*x^11 + 1246*x^10 - 5810*x^9 + 39270*x^8 - 153373*x^7 + 774270*x^6 - 2507386*x^5 + 9634359*x^4 - 22503019*x^3 + 65267748*x^2 - 95761729*x + 191861773)
 
gp: K = bnfinit(x^14 - 7*x^13 + 49*x^12 - 161*x^11 + 1246*x^10 - 5810*x^9 + 39270*x^8 - 153373*x^7 + 774270*x^6 - 2507386*x^5 + 9634359*x^4 - 22503019*x^3 + 65267748*x^2 - 95761729*x + 191861773, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} + 49 x^{12} - 161 x^{11} + 1246 x^{10} - 5810 x^{9} + 39270 x^{8} - 153373 x^{7} + 774270 x^{6} - 2507386 x^{5} + 9634359 x^{4} - 22503019 x^{3} + 65267748 x^{2} - 95761729 x + 191861773 \)

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:  \(-26290885243157554005466712784279=-\,3^{7}\cdot 7^{24}\cdot 13^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $175.50$
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:  $3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1911=3\cdot 7^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{1911}(1,·)$, $\chi_{1911}(547,·)$, $\chi_{1911}(1093,·)$, $\chi_{1911}(1793,·)$, $\chi_{1911}(1639,·)$, $\chi_{1911}(428,·)$, $\chi_{1911}(974,·)$, $\chi_{1911}(1520,·)$, $\chi_{1911}(274,·)$, $\chi_{1911}(820,·)$, $\chi_{1911}(1366,·)$, $\chi_{1911}(155,·)$, $\chi_{1911}(701,·)$, $\chi_{1911}(1247,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{116033} a^{12} - \frac{22866}{116033} a^{11} + \frac{13817}{116033} a^{10} - \frac{38267}{116033} a^{9} - \frac{33565}{116033} a^{8} - \frac{55353}{116033} a^{7} + \frac{21450}{116033} a^{6} - \frac{42584}{116033} a^{5} - \frac{13701}{116033} a^{4} - \frac{25515}{116033} a^{3} + \frac{21431}{116033} a^{2} - \frac{44896}{116033} a + \frac{26445}{116033}$, $\frac{1}{2318873371322340420268557729193668671} a^{13} + \frac{9839117746578261848717945815524}{2318873371322340420268557729193668671} a^{12} - \frac{688479836548713289166562940612165672}{2318873371322340420268557729193668671} a^{11} + \frac{515459117299177924740965131644633752}{2318873371322340420268557729193668671} a^{10} + \frac{1112327092852066220186658584202119399}{2318873371322340420268557729193668671} a^{9} - \frac{1156097092707310134250958978878416212}{2318873371322340420268557729193668671} a^{8} - \frac{506725320741064019644028203745247378}{2318873371322340420268557729193668671} a^{7} + \frac{569581439272902426953627855392773651}{2318873371322340420268557729193668671} a^{6} - \frac{1020067841775192683247300891522374475}{2318873371322340420268557729193668671} a^{5} + \frac{995524169434591887882830928402420543}{2318873371322340420268557729193668671} a^{4} + \frac{945213681771136823360361416797093416}{2318873371322340420268557729193668671} a^{3} - \frac{136780292809947963610160261440324799}{2318873371322340420268557729193668671} a^{2} - \frac{765237890292507198326562214093985816}{2318873371322340420268557729193668671} a - \frac{343462145190031613064986919389400614}{2318873371322340420268557729193668671}$

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

Class group and class number

$C_{109316}$, which has order $109316$ (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

\(\Q(\sqrt{-39}) \), 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$3$3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$7$7.14.24.53$x^{14} + 931 x^{13} + 2310 x^{12} + 903 x^{11} + 392 x^{10} + 2198 x^{9} + 2296 x^{8} + 1485 x^{7} + 637 x^{6} + 1295 x^{5} + 2303 x^{4} + 1449 x^{3} + 1316 x^{2} + 2219 x + 2383$$7$$2$$24$$C_{14}$$[2]^{2}$
$13$13.14.7.1$x^{14} - 43940 x^{8} + 482680900 x^{2} - 250994068$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$