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Properties

Label 14.0.45401376875...1251.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,11^{7}\cdot 13^{12}$
Root discriminant $29.89$
Ramified primes $11, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![44, -396, 1565, -3595, 5518, -6205, 5352, -3635, 2060, -971, 412, -137, 38, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 38*x^12 - 137*x^11 + 412*x^10 - 971*x^9 + 2060*x^8 - 3635*x^7 + 5352*x^6 - 6205*x^5 + 5518*x^4 - 3595*x^3 + 1565*x^2 - 396*x + 44)
 
gp: K = bnfinit(x^14 - 7*x^13 + 38*x^12 - 137*x^11 + 412*x^10 - 971*x^9 + 2060*x^8 - 3635*x^7 + 5352*x^6 - 6205*x^5 + 5518*x^4 - 3595*x^3 + 1565*x^2 - 396*x + 44, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} + 38 x^{12} - 137 x^{11} + 412 x^{10} - 971 x^{9} + 2060 x^{8} - 3635 x^{7} + 5352 x^{6} - 6205 x^{5} + 5518 x^{4} - 3595 x^{3} + 1565 x^{2} - 396 x + 44 \)

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:  \(-454013768754343191251=-\,11^{7}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.89$
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:  $11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{112} a^{10} - \frac{5}{112} a^{9} + \frac{13}{112} a^{8} + \frac{3}{56} a^{7} - \frac{3}{56} a^{6} + \frac{9}{56} a^{5} - \frac{15}{112} a^{4} - \frac{53}{112} a^{3} - \frac{5}{16} a^{2} + \frac{5}{28} a - \frac{13}{28}$, $\frac{1}{112} a^{11} - \frac{3}{28} a^{9} - \frac{13}{112} a^{8} - \frac{1}{28} a^{7} - \frac{3}{28} a^{6} + \frac{19}{112} a^{5} - \frac{1}{7} a^{4} - \frac{5}{28} a^{3} - \frac{15}{112} a^{2} - \frac{9}{28} a + \frac{5}{28}$, $\frac{1}{448} a^{12} - \frac{1}{224} a^{11} + \frac{1}{448} a^{10} + \frac{1}{224} a^{9} - \frac{5}{448} a^{8} - \frac{5}{224} a^{7} + \frac{7}{64} a^{6} - \frac{1}{28} a^{5} - \frac{99}{448} a^{4} - \frac{47}{112} a^{3} - \frac{13}{448} a^{2} + \frac{9}{56} a - \frac{39}{112}$, $\frac{1}{12992} a^{13} + \frac{1}{1624} a^{12} + \frac{13}{12992} a^{11} - \frac{197}{12992} a^{9} - \frac{121}{1624} a^{8} - \frac{251}{12992} a^{7} - \frac{423}{6496} a^{6} + \frac{323}{1856} a^{5} - \frac{587}{6496} a^{4} - \frac{207}{1856} a^{3} + \frac{501}{6496} a^{2} - \frac{571}{3248} a + \frac{47}{1624}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 234208.307826 \) (assuming GRH)
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

\(\Q(\sqrt{-11}) \), 7.1.6424482779.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.6424482779.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ R R ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
13Data not computed