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Properties

Label 14.0.57969014080...7744.2
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{14}\cdot 29^{12}$
Root discriminant $35.85$
Ramified primes $2, 29$
Class number $7$ (GRH)
Class group $[7]$ (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![26244, 0, 4212, 0, -1935, 0, 1369, 0, -238, 0, 50, 0, -7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^12 + 50*x^10 - 238*x^8 + 1369*x^6 - 1935*x^4 + 4212*x^2 + 26244)
 
gp: K = bnfinit(x^14 - 7*x^12 + 50*x^10 - 238*x^8 + 1369*x^6 - 1935*x^4 + 4212*x^2 + 26244, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{12} + 50 x^{10} - 238 x^{8} + 1369 x^{6} - 1935 x^{4} + 4212 x^{2} + 26244 \)

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:  \(-5796901408038404767744=-\,2^{14}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.85$
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:  $2, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{6} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{24} a^{7} - \frac{1}{12} a^{5} + \frac{7}{24} a^{3} - \frac{1}{2} a^{2} + \frac{5}{12} a - \frac{1}{2}$, $\frac{1}{24} a^{8} - \frac{1}{12} a^{6} - \frac{5}{24} a^{4} - \frac{1}{2} a^{3} - \frac{1}{12} a^{2} - \frac{1}{2} a$, $\frac{1}{48} a^{9} - \frac{1}{48} a^{8} - \frac{1}{12} a^{6} + \frac{1}{16} a^{5} + \frac{5}{48} a^{4} - \frac{1}{2} a^{3} + \frac{5}{12} a^{2} + \frac{5}{12} a + \frac{1}{4}$, $\frac{1}{288} a^{10} + \frac{5}{288} a^{8} + \frac{35}{288} a^{6} - \frac{1}{288} a^{4} + \frac{19}{72} a^{2} + \frac{3}{8}$, $\frac{1}{5184} a^{11} - \frac{1}{576} a^{10} + \frac{29}{5184} a^{9} - \frac{5}{576} a^{8} - \frac{13}{5184} a^{7} - \frac{35}{576} a^{6} + \frac{455}{5184} a^{5} + \frac{1}{576} a^{4} - \frac{497}{1296} a^{3} + \frac{53}{144} a^{2} + \frac{11}{144} a + \frac{5}{16}$, $\frac{1}{8802432} a^{12} + \frac{169}{1100304} a^{10} + \frac{1789}{4401216} a^{8} + \frac{103477}{1100304} a^{6} + \frac{859393}{8802432} a^{4} - \frac{1}{2} a^{3} + \frac{28877}{61128} a^{2} - \frac{1}{2} a + \frac{2219}{9056}$, $\frac{1}{52814592} a^{13} - \frac{1}{17604864} a^{12} + \frac{169}{6601824} a^{11} - \frac{169}{2200608} a^{10} - \frac{181595}{26407296} a^{9} - \frac{1789}{8802432} a^{8} - \frac{79907}{6601824} a^{7} + \frac{171599}{2200608} a^{6} - \frac{6109199}{52814592} a^{5} + \frac{3541823}{17604864} a^{4} - \frac{42439}{366768} a^{3} - \frac{44159}{122256} a^{2} - \frac{6807}{18112} a - \frac{6747}{18112}$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  \( \frac{253}{13203648} a^{13} + \frac{379}{6601824} a^{11} - \frac{103}{206307} a^{9} + \frac{24289}{6601824} a^{7} - \frac{398105}{13203648} a^{5} + \frac{12515}{61128} a^{3} - \frac{9817}{40752} a \) (order $4$)
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:  \( 4434400.28009 \) (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

\(\Q(\sqrt{-1}) \), 7.1.38068692544.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.38068692544.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$2$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
$29$29.7.6.1$x^{7} + 232$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.1$x^{7} + 232$$7$$1$$6$$C_7$$[\ ]_{7}$