↑

Properties

Label 14.0.12879984763...9232.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{14}\cdot 7^{24}\cdot 17^{7}$
Root discriminant $231.74$
Ramified primes $2, 7, 17$
Class number $1515892$ (GRH)
Class group $[1515892]$ (GRH)
Galois group $C_{14}$ (as 14T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2769616973, 412653556, 584163825, 57285116, 57312920, 3494218, 3408482, 93620, 135919, 392, 3836, -42, 77, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 77*x^12 - 42*x^11 + 3836*x^10 + 392*x^9 + 135919*x^8 + 93620*x^7 + 3408482*x^6 + 3494218*x^5 + 57312920*x^4 + 57285116*x^3 + 584163825*x^2 + 412653556*x + 2769616973)
 
gp: K = bnfinit(x^14 + 77*x^12 - 42*x^11 + 3836*x^10 + 392*x^9 + 135919*x^8 + 93620*x^7 + 3408482*x^6 + 3494218*x^5 + 57312920*x^4 + 57285116*x^3 + 584163825*x^2 + 412653556*x + 2769616973, 1)
 

Normalized defining polynomial

\( x^{14} + 77 x^{12} - 42 x^{11} + 3836 x^{10} + 392 x^{9} + 135919 x^{8} + 93620 x^{7} + 3408482 x^{6} + 3494218 x^{5} + 57312920 x^{4} + 57285116 x^{3} + 584163825 x^{2} + 412653556 x + 2769616973 \)

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:  \(-1287998476392981798480681859039232=-\,2^{14}\cdot 7^{24}\cdot 17^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $231.74$
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:  $2, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3332=2^{2}\cdot 7^{2}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{3332}(1,·)$, $\chi_{3332}(2787,·)$, $\chi_{3332}(2311,·)$, $\chi_{3332}(2857,·)$, $\chi_{3332}(1835,·)$, $\chi_{3332}(2381,·)$, $\chi_{3332}(1359,·)$, $\chi_{3332}(1905,·)$, $\chi_{3332}(883,·)$, $\chi_{3332}(1429,·)$, $\chi_{3332}(407,·)$, $\chi_{3332}(953,·)$, $\chi_{3332}(477,·)$, $\chi_{3332}(3263,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{31} a^{10} - \frac{8}{31} a^{9} + \frac{13}{31} a^{8} - \frac{7}{31} a^{6} + \frac{7}{31} a^{5} + \frac{8}{31} a^{4} + \frac{9}{31} a^{3} - \frac{15}{31} a^{2} - \frac{8}{31} a$, $\frac{1}{31} a^{11} + \frac{11}{31} a^{9} + \frac{11}{31} a^{8} - \frac{7}{31} a^{7} + \frac{13}{31} a^{6} + \frac{2}{31} a^{5} + \frac{11}{31} a^{4} - \frac{5}{31} a^{3} - \frac{4}{31} a^{2} - \frac{2}{31} a$, $\frac{1}{589} a^{12} + \frac{3}{589} a^{11} + \frac{5}{589} a^{10} - \frac{156}{589} a^{9} + \frac{227}{589} a^{8} - \frac{256}{589} a^{7} - \frac{10}{589} a^{6} - \frac{149}{589} a^{5} - \frac{268}{589} a^{4} + \frac{268}{589} a^{3} - \frac{203}{589} a^{2} - \frac{237}{589} a - \frac{4}{19}$, $\frac{1}{6901204867413778152222591967337933522017} a^{13} - \frac{954698972932630362790260260493473265}{6901204867413778152222591967337933522017} a^{12} + \frac{907503075259764767488134807634917862}{222619511852057359749115869914126887807} a^{11} + \frac{14582609220026027539217298836004479355}{6901204867413778152222591967337933522017} a^{10} + \frac{2443389334481678946629547528104334785267}{6901204867413778152222591967337933522017} a^{9} - \frac{1168303279095547596418105498204767192609}{6901204867413778152222591967337933522017} a^{8} + \frac{2875904735228085536963324242876050155568}{6901204867413778152222591967337933522017} a^{7} + \frac{1732801725074849613521918211334267175983}{6901204867413778152222591967337933522017} a^{6} - \frac{1420269199520654297043988760605186384208}{6901204867413778152222591967337933522017} a^{5} + \frac{2703493035993931307968370457015995139855}{6901204867413778152222591967337933522017} a^{4} - \frac{601375195869602119600808976392086612185}{6901204867413778152222591967337933522017} a^{3} + \frac{1771523893423220129303542002473961175539}{6901204867413778152222591967337933522017} a^{2} - \frac{676898986926483021631007160395742415447}{6901204867413778152222591967337933522017} a + \frac{29991628422956612980637844090873862761}{222619511852057359749115869914126887807}$

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

Class group and class number

$C_{1515892}$, which has order $1515892$ (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

\(\Q(\sqrt{-17}) \), 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 ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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$2.14.14.38$x^{14} + 4 x^{13} + 3 x^{12} - 2 x^{11} + 2 x^{10} - 2 x^{8} + 4 x^{6} - 2 x^{5} + 4 x^{3} - 2 x^{2} + 2 x + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
$7$7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$
7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$
$17$17.14.7.1$x^{14} - 9826 x^{8} + 24137569 x^{2} - 3693048057$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$