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Properties

Label 14.0.39806962627...9375.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 5^{7}\cdot 13^{12}$
Root discriminant $34.90$
Ramified primes $3, 5, 13$
Class number $2$ (GRH)
Class group $[2]$ (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![10869, -4737, -4601, -1841, 3281, 1359, 193, -2083, 1051, -97, -8, -21, 14, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 + 14*x^12 - 21*x^11 - 8*x^10 - 97*x^9 + 1051*x^8 - 2083*x^7 + 193*x^6 + 1359*x^5 + 3281*x^4 - 1841*x^3 - 4601*x^2 - 4737*x + 10869)
 
gp: K = bnfinit(x^14 - 5*x^13 + 14*x^12 - 21*x^11 - 8*x^10 - 97*x^9 + 1051*x^8 - 2083*x^7 + 193*x^6 + 1359*x^5 + 3281*x^4 - 1841*x^3 - 4601*x^2 - 4737*x + 10869, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} + 14 x^{12} - 21 x^{11} - 8 x^{10} - 97 x^{9} + 1051 x^{8} - 2083 x^{7} + 193 x^{6} + 1359 x^{5} + 3281 x^{4} - 1841 x^{3} - 4601 x^{2} - 4737 x + 10869 \)

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:  \(-3980696262723902109375=-\,3^{7}\cdot 5^{7}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.90$
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:  $3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{15} a^{7} - \frac{1}{15} a^{6} + \frac{1}{15} a^{5} - \frac{1}{15} a^{4} + \frac{4}{15} a^{3} - \frac{4}{15} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{45} a^{8} - \frac{4}{15} a^{4} - \frac{2}{9} a^{2} + \frac{7}{15}$, $\frac{1}{45} a^{9} - \frac{4}{15} a^{5} - \frac{2}{9} a^{3} + \frac{7}{15} a$, $\frac{1}{225} a^{10} - \frac{1}{225} a^{9} - \frac{2}{225} a^{8} - \frac{2}{75} a^{7} - \frac{2}{75} a^{6} - \frac{28}{75} a^{5} + \frac{4}{45} a^{4} - \frac{14}{225} a^{3} + \frac{4}{45} a^{2} + \frac{4}{15} a + \frac{4}{75}$, $\frac{1}{675} a^{11} + \frac{1}{675} a^{10} - \frac{4}{675} a^{9} + \frac{1}{135} a^{8} - \frac{1}{225} a^{7} - \frac{37}{225} a^{6} - \frac{133}{675} a^{5} + \frac{56}{675} a^{4} - \frac{173}{675} a^{3} + \frac{23}{135} a^{2} + \frac{14}{225} a + \frac{68}{225}$, $\frac{1}{2025} a^{12} - \frac{1}{2025} a^{11} - \frac{1}{675} a^{10} - \frac{4}{405} a^{9} - \frac{4}{2025} a^{8} - \frac{11}{675} a^{7} + \frac{206}{2025} a^{6} + \frac{104}{405} a^{5} + \frac{1}{5} a^{4} - \frac{271}{2025} a^{3} + \frac{937}{2025} a^{2} + \frac{32}{135} a - \frac{289}{675}$, $\frac{1}{2776661582625} a^{13} + \frac{34385809}{308517953625} a^{12} + \frac{66351809}{111066463305} a^{11} - \frac{5283260726}{2776661582625} a^{10} - \frac{9260006633}{925553860875} a^{9} - \frac{3318700873}{396665940375} a^{8} + \frac{4615506724}{555332316525} a^{7} + \frac{18366513752}{132221980125} a^{6} + \frac{110084857979}{555332316525} a^{5} + \frac{157353925949}{2776661582625} a^{4} - \frac{3014167022}{61703590725} a^{3} + \frac{1052781465244}{2776661582625} a^{2} - \frac{135874589128}{308517953625} a - \frac{17753948668}{925553860875}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 1713561.51513 \) (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

\(\Q(\sqrt{-15}) \), 7.1.16290480375.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.16290480375.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
13Data not computed