↑

Properties

Label 14.2.24385268673...4761.1
Degree $14$
Signature $[2, 6]$
Discriminant $29^{12}\cdot 41^{3}$
Root discriminant $39.73$
Ramified primes $29, 41$
Class number $12$
Class group $[2, 6]$
Galois group $C_2 \wr C_7$ (as 14T29)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2477, 5836, -8890, 5464, -237, -5712, 7180, -6097, 3682, -1741, 665, -191, 47, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 47*x^12 - 191*x^11 + 665*x^10 - 1741*x^9 + 3682*x^8 - 6097*x^7 + 7180*x^6 - 5712*x^5 - 237*x^4 + 5464*x^3 - 8890*x^2 + 5836*x - 2477)
 
gp: K = bnfinit(x^14 - 7*x^13 + 47*x^12 - 191*x^11 + 665*x^10 - 1741*x^9 + 3682*x^8 - 6097*x^7 + 7180*x^6 - 5712*x^5 - 237*x^4 + 5464*x^3 - 8890*x^2 + 5836*x - 2477, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} + 47 x^{12} - 191 x^{11} + 665 x^{10} - 1741 x^{9} + 3682 x^{8} - 6097 x^{7} + 7180 x^{6} - 5712 x^{5} - 237 x^{4} + 5464 x^{3} - 8890 x^{2} + 5836 x - 2477 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(24385268673304131774761=29^{12}\cdot 41^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.73$
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:  $29, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{41} a^{10} - \frac{5}{41} a^{9} + \frac{15}{41} a^{8} + \frac{11}{41} a^{7} - \frac{12}{41} a^{6} - \frac{3}{41} a^{5} + \frac{6}{41} a^{4} + \frac{9}{41} a^{3} + \frac{3}{41} a^{2} + \frac{16}{41} a + \frac{15}{41}$, $\frac{1}{697} a^{11} + \frac{3}{697} a^{10} + \frac{303}{697} a^{9} + \frac{8}{697} a^{8} + \frac{199}{697} a^{7} + \frac{106}{697} a^{6} + \frac{105}{697} a^{5} + \frac{57}{697} a^{4} - \frac{253}{697} a^{3} - \frac{124}{697} a^{2} + \frac{225}{697} a + \frac{161}{697}$, $\frac{1}{4486589} a^{12} - \frac{6}{4486589} a^{11} - \frac{14463}{4486589} a^{10} + \frac{72370}{4486589} a^{9} - \frac{1025296}{4486589} a^{8} - \frac{819691}{4486589} a^{7} + \frac{1521229}{4486589} a^{6} + \frac{852546}{4486589} a^{5} + \frac{1999012}{4486589} a^{4} + \frac{1705400}{4486589} a^{3} + \frac{373233}{4486589} a^{2} - \frac{177746}{4486589} a + \frac{1754855}{4486589}$, $\frac{1}{4486589} a^{13} - \frac{1625}{4486589} a^{11} + \frac{24214}{4486589} a^{10} - \frac{1176843}{4486589} a^{9} + \frac{2104703}{4486589} a^{8} - \frac{834991}{4486589} a^{7} - \frac{2115203}{4486589} a^{6} - \frac{507120}{4486589} a^{5} + \frac{973523}{4486589} a^{4} - \frac{1624667}{4486589} a^{3} + \frac{465276}{4486589} a^{2} - \frac{901560}{4486589} a - \frac{857923}{4486589}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 22485.4239303 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2 \wr C_7$ (as 14T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 896
The 32 conjugacy class representatives for $C_2 \wr C_7$
Character table for $C_2 \wr C_7$ is not computed

Intermediate fields

7.7.594823321.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 14 siblings: data not computed
Degree 28 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{10}$

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}$
41Data not computed