↑

Properties

Label 14.0.97343695970...7551.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,29^{12}\cdot 31^{7}$
Root discriminant $99.81$
Ramified primes $29, 31$
Class number $21696$ (GRH)
Class group $[4, 4, 1356]$ (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![17394953, -5698041, 8361848, -2350647, 1779327, -427080, 216460, -43043, 16990, -2810, 954, -139, 41, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 + 41*x^12 - 139*x^11 + 954*x^10 - 2810*x^9 + 16990*x^8 - 43043*x^7 + 216460*x^6 - 427080*x^5 + 1779327*x^4 - 2350647*x^3 + 8361848*x^2 - 5698041*x + 17394953)
 
gp: K = bnfinit(x^14 - 5*x^13 + 41*x^12 - 139*x^11 + 954*x^10 - 2810*x^9 + 16990*x^8 - 43043*x^7 + 216460*x^6 - 427080*x^5 + 1779327*x^4 - 2350647*x^3 + 8361848*x^2 - 5698041*x + 17394953, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} + 41 x^{12} - 139 x^{11} + 954 x^{10} - 2810 x^{9} + 16990 x^{8} - 43043 x^{7} + 216460 x^{6} - 427080 x^{5} + 1779327 x^{4} - 2350647 x^{3} + 8361848 x^{2} - 5698041 x + 17394953 \)

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:  \(-9734369597099193349790237551=-\,29^{12}\cdot 31^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $99.81$
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:  $29, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(899=29\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{899}(1,·)$, $\chi_{899}(836,·)$, $\chi_{899}(774,·)$, $\chi_{899}(807,·)$, $\chi_{899}(712,·)$, $\chi_{899}(745,·)$, $\chi_{899}(30,·)$, $\chi_{899}(683,·)$, $\chi_{899}(402,·)$, $\chi_{899}(371,·)$, $\chi_{899}(373,·)$, $\chi_{899}(342,·)$, $\chi_{899}(123,·)$, $\chi_{899}(94,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{17} a^{12} + \frac{7}{17} a^{11} - \frac{8}{17} a^{10} + \frac{7}{17} a^{9} - \frac{6}{17} a^{8} - \frac{5}{17} a^{7} - \frac{3}{17} a^{6} + \frac{1}{17} a^{5} + \frac{2}{17} a^{4} + \frac{4}{17} a^{3} + \frac{8}{17} a^{2} - \frac{3}{17}$, $\frac{1}{31946739135364682429647605954773723} a^{13} - \frac{340805860512036378651215194647326}{31946739135364682429647605954773723} a^{12} - \frac{13476615166672791742018670425637163}{31946739135364682429647605954773723} a^{11} - \frac{7452990903036364835470592872851822}{31946739135364682429647605954773723} a^{10} - \frac{137772938351471461123869884763493}{1879219949139098966449859173810219} a^{9} - \frac{4444966247333870756798768676564000}{31946739135364682429647605954773723} a^{8} + \frac{341989778346881108983623765972865}{1879219949139098966449859173810219} a^{7} - \frac{2072025254778299775962734060024337}{31946739135364682429647605954773723} a^{6} - \frac{15682542397847073298496148532697569}{31946739135364682429647605954773723} a^{5} + \frac{6767748991378371840896362828362615}{31946739135364682429647605954773723} a^{4} + \frac{10703755732515898605659261566528401}{31946739135364682429647605954773723} a^{3} + \frac{8329028850439458426849334887596046}{31946739135364682429647605954773723} a^{2} + \frac{6768598623273663086464176877816023}{31946739135364682429647605954773723} a + \frac{6919394250333463986519567408945888}{31946739135364682429647605954773723}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{1356}$, which has order $21696$ (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

\(\Q(\sqrt{-31}) \), 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ R R ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{14}$

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
$29$29.14.12.1$x^{14} + 2407 x^{7} + 1839267$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$
$31$31.14.7.2$x^{14} - 887503681 x^{2} + 495227053998$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$