↑

Properties

Label 14.0.29138168800...2263.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{7}\cdot 29^{12}$
Root discriminant $47.43$
Ramified primes $7, 29$
Class number $56$ (GRH)
Class group $[2, 2, 14]$ (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![22271, -13581, 20954, -10251, 12267, -5784, 3346, -1271, 682, -266, 0, 41, -1, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 - x^12 + 41*x^11 - 266*x^9 + 682*x^8 - 1271*x^7 + 3346*x^6 - 5784*x^5 + 12267*x^4 - 10251*x^3 + 20954*x^2 - 13581*x + 22271)
 
gp: K = bnfinit(x^14 - 5*x^13 - x^12 + 41*x^11 - 266*x^9 + 682*x^8 - 1271*x^7 + 3346*x^6 - 5784*x^5 + 12267*x^4 - 10251*x^3 + 20954*x^2 - 13581*x + 22271, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} - x^{12} + 41 x^{11} - 266 x^{9} + 682 x^{8} - 1271 x^{7} + 3346 x^{6} - 5784 x^{5} + 12267 x^{4} - 10251 x^{3} + 20954 x^{2} - 13581 x + 22271 \)

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:  \(-291381688005381590432263=-\,7^{7}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.43$
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:  $7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(203=7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{203}(1,·)$, $\chi_{203}(36,·)$, $\chi_{203}(197,·)$, $\chi_{203}(169,·)$, $\chi_{203}(139,·)$, $\chi_{203}(141,·)$, $\chi_{203}(78,·)$, $\chi_{203}(111,·)$, $\chi_{203}(146,·)$, $\chi_{203}(83,·)$, $\chi_{203}(20,·)$, $\chi_{203}(181,·)$, $\chi_{203}(132,·)$, $\chi_{203}(190,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{17} a^{12} + \frac{7}{17} a^{11} + \frac{7}{17} a^{10} + \frac{1}{17} a^{9} + \frac{7}{17} a^{8} - \frac{3}{17} a^{7} - \frac{5}{17} a^{6} + \frac{2}{17} a^{5} - \frac{7}{17} a^{4} - \frac{2}{17} a^{3} + \frac{8}{17} a^{2} - \frac{7}{17} a - \frac{2}{17}$, $\frac{1}{7128637193852842930567619} a^{13} - \frac{37155731070647142154817}{7128637193852842930567619} a^{12} + \frac{2661868916722742561928735}{7128637193852842930567619} a^{11} - \frac{2582694832433361145157711}{7128637193852842930567619} a^{10} + \frac{2485176271848327902126325}{7128637193852842930567619} a^{9} - \frac{2553674143582852068965825}{7128637193852842930567619} a^{8} + \frac{1606673425022783983741185}{7128637193852842930567619} a^{7} - \frac{1233526684624262182172077}{7128637193852842930567619} a^{6} - \frac{954693908923000428684579}{7128637193852842930567619} a^{5} + \frac{1837579390230127068539526}{7128637193852842930567619} a^{4} + \frac{1244147559999672282462241}{7128637193852842930567619} a^{3} - \frac{1530263496431078504431487}{7128637193852842930567619} a^{2} - \frac{2395592671139232881680747}{7128637193852842930567619} a - \frac{2946418936156182282268782}{7128637193852842930567619}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{14}$, which has order $56$ (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:  \( 6020.98510015 \) (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

\(\Q(\sqrt{-7}) \), 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
7Data not computed
$29$29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$