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Properties

Label 14.0.12710233464...0096.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{21}\cdot 67^{7}$
Root discriminant $23.15$
Ramified primes $2, 67$
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![256, -640, 1840, -2384, 868, -776, 1612, -932, 293, -138, 95, -32, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 3*x^12 - 32*x^11 + 95*x^10 - 138*x^9 + 293*x^8 - 932*x^7 + 1612*x^6 - 776*x^5 + 868*x^4 - 2384*x^3 + 1840*x^2 - 640*x + 256)
 
gp: K = bnfinit(x^14 - 2*x^13 + 3*x^12 - 32*x^11 + 95*x^10 - 138*x^9 + 293*x^8 - 932*x^7 + 1612*x^6 - 776*x^5 + 868*x^4 - 2384*x^3 + 1840*x^2 - 640*x + 256, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} + 3 x^{12} - 32 x^{11} + 95 x^{10} - 138 x^{9} + 293 x^{8} - 932 x^{7} + 1612 x^{6} - 776 x^{5} + 868 x^{4} - 2384 x^{3} + 1840 x^{2} - 640 x + 256 \)

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:  \(-12710233464526340096=-\,2^{21}\cdot 67^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.15$
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:  $2, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{8} - \frac{1}{16} a^{7} + \frac{1}{16} a^{6} - \frac{3}{16} a^{5} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{9} - \frac{1}{8} a^{6} - \frac{3}{16} a^{5} - \frac{1}{8} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{9} - \frac{3}{32} a^{6} + \frac{3}{32} a^{5} + \frac{3}{16} a^{4} - \frac{7}{16} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{9} + \frac{1}{32} a^{7} - \frac{1}{8} a^{6} - \frac{3}{32} a^{5} - \frac{1}{8} a^{4} - \frac{7}{16} a^{3} - \frac{1}{4} a$, $\frac{1}{448} a^{12} + \frac{1}{224} a^{11} + \frac{3}{448} a^{10} + \frac{3}{224} a^{9} - \frac{3}{448} a^{8} + \frac{13}{224} a^{7} - \frac{27}{448} a^{6} + \frac{41}{224} a^{5} - \frac{1}{32} a^{4} + \frac{55}{112} a^{3} + \frac{5}{56} a^{2} - \frac{11}{28} a - \frac{3}{7}$, $\frac{1}{2986540265408} a^{13} - \frac{1778266907}{2986540265408} a^{12} - \frac{20468059859}{2986540265408} a^{11} - \frac{21108000611}{2986540265408} a^{10} + \frac{9688110757}{2986540265408} a^{9} - \frac{71479919411}{2986540265408} a^{8} + \frac{27889291107}{2986540265408} a^{7} - \frac{363673146341}{2986540265408} a^{6} - \frac{158619179111}{1493270132704} a^{5} + \frac{343869087455}{1493270132704} a^{4} - \frac{118384256843}{373317533176} a^{3} + \frac{152668379889}{373317533176} a^{2} - \frac{1962836789}{46664691647} a - \frac{16414594122}{46664691647}$

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:  \( 174291.696675 \) (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

\(\Q(\sqrt{-134}) \), 7.1.153990656.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.153990656.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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$2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
$67$67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.1$x^{2} - 67$$2$$1$$1$$C_2$$[\ ]_{2}$