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Properties

Label 14.0.48441564835...8523.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,467^{7}$
Root discriminant $21.61$
Ramified prime $467$
Class number $1$
Class group Trivial
Galois group $D_{7}$ (as 14T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, -6, 117, -250, 879, -736, 1066, -114, 155, -32, 34, -26, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 3*x^12 - 26*x^11 + 34*x^10 - 32*x^9 + 155*x^8 - 114*x^7 + 1066*x^6 - 736*x^5 + 879*x^4 - 250*x^3 + 117*x^2 - 6*x + 4)
 
gp: K = bnfinit(x^14 - 2*x^13 + 3*x^12 - 26*x^11 + 34*x^10 - 32*x^9 + 155*x^8 - 114*x^7 + 1066*x^6 - 736*x^5 + 879*x^4 - 250*x^3 + 117*x^2 - 6*x + 4, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} + 3 x^{12} - 26 x^{11} + 34 x^{10} - 32 x^{9} + 155 x^{8} - 114 x^{7} + 1066 x^{6} - 736 x^{5} + 879 x^{4} - 250 x^{3} + 117 x^{2} - 6 x + 4 \)

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:  \(-4844156483581198523=-\,467^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.61$
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:  $467$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{8} - \frac{1}{16} a^{7} + \frac{1}{16} a^{5} - \frac{1}{4} a^{4} - \frac{3}{16} a^{3} - \frac{1}{16} a^{2} + \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{7} + \frac{1}{16} a^{6} - \frac{3}{16} a^{5} + \frac{1}{16} a^{4} - \frac{1}{4} a^{3} + \frac{5}{16} a^{2} + \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{32} a^{10} - \frac{3}{32} a^{6} - \frac{3}{16} a^{5} - \frac{1}{4} a^{4} + \frac{1}{16} a^{3} - \frac{7}{32} a^{2} - \frac{3}{16} a - \frac{3}{8}$, $\frac{1}{32} a^{11} - \frac{3}{32} a^{7} + \frac{1}{16} a^{6} - \frac{1}{4} a^{5} - \frac{3}{16} a^{4} - \frac{7}{32} a^{3} + \frac{1}{16} a^{2} + \frac{1}{8} a$, $\frac{1}{640} a^{12} - \frac{1}{128} a^{11} - \frac{9}{640} a^{10} + \frac{1}{40} a^{9} - \frac{11}{640} a^{8} + \frac{49}{640} a^{7} + \frac{13}{128} a^{6} + \frac{11}{80} a^{5} + \frac{7}{640} a^{4} + \frac{147}{640} a^{3} - \frac{151}{640} a^{2} - \frac{63}{320} a - \frac{77}{160}$, $\frac{1}{927656320} a^{13} - \frac{30005}{46382816} a^{12} - \frac{6658857}{463828160} a^{11} - \frac{6843729}{927656320} a^{10} - \frac{13423891}{927656320} a^{9} + \frac{10903427}{463828160} a^{8} + \frac{3734431}{92765632} a^{7} - \frac{27275807}{927656320} a^{6} + \frac{45571447}{927656320} a^{5} - \frac{65125139}{463828160} a^{4} - \frac{23936147}{115957040} a^{3} - \frac{72352141}{927656320} a^{2} + \frac{180876911}{463828160} a + \frac{5853639}{46382816}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 14606.3296098 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_7$ (as 14T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 14
The 5 conjugacy class representatives for $D_{7}$
Character table for $D_{7}$

Intermediate fields

\(\Q(\sqrt{-467}) \), 7.1.101847563.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.101847563.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
467Data not computed