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Properties

Label 14.14.2197232283...7968.1
Degree $14$
Signature $[14, 0]$
Discriminant $2^{27}\cdot 7^{14}\cdot 17^{6}$
Root discriminant $89.74$
Ramified primes $2, 7, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T24

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-56066, 32368, 240737, -97104, -283220, 26656, 111622, -1832, -20090, 0, 1757, 0, -70, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 70*x^12 + 1757*x^10 - 20090*x^8 - 1832*x^7 + 111622*x^6 + 26656*x^5 - 283220*x^4 - 97104*x^3 + 240737*x^2 + 32368*x - 56066)
 
gp: K = bnfinit(x^14 - 70*x^12 + 1757*x^10 - 20090*x^8 - 1832*x^7 + 111622*x^6 + 26656*x^5 - 283220*x^4 - 97104*x^3 + 240737*x^2 + 32368*x - 56066, 1)
 

Normalized defining polynomial

\( x^{14} - 70 x^{12} + 1757 x^{10} - 20090 x^{8} - 1832 x^{7} + 111622 x^{6} + 26656 x^{5} - 283220 x^{4} - 97104 x^{3} + 240737 x^{2} + 32368 x - 56066 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2197232283487253091967827968=2^{27}\cdot 7^{14}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $89.74$
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:  $2, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{119} a^{11} - \frac{3}{7} a^{10} - \frac{2}{119} a^{9} + \frac{1}{7} a^{8} - \frac{45}{119} a^{7} + \frac{3}{17} a^{5} + \frac{55}{119} a^{4} + \frac{3}{7} a^{3} + \frac{3}{7} a^{2} - \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{119} a^{12} + \frac{15}{119} a^{10} + \frac{2}{7} a^{9} - \frac{11}{119} a^{8} - \frac{2}{7} a^{7} + \frac{3}{17} a^{6} + \frac{55}{119} a^{5} + \frac{2}{7} a^{3} - \frac{2}{7} a^{2} - \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{2975800110428626439} a^{13} + \frac{2850124926783415}{2975800110428626439} a^{12} + \frac{2070092553318855}{2975800110428626439} a^{11} + \frac{435140465461130483}{2975800110428626439} a^{10} + \frac{169860589103803895}{425114301489803777} a^{9} - \frac{64634022964121046}{425114301489803777} a^{8} + \frac{76453566871509546}{2975800110428626439} a^{7} + \frac{76354171663575948}{175047065319330967} a^{6} + \frac{627794739868906350}{2975800110428626439} a^{5} + \frac{1337997932458942312}{2975800110428626439} a^{4} - \frac{37008898854800884}{175047065319330967} a^{3} + \frac{7179299164644159}{25006723617047281} a^{2} + \frac{616872420465216}{25006723617047281} a - \frac{2529189127028344}{175047065319330967}$

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:  $13$
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:  \( 3947012497.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T24:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 588
The 19 conjugacy class representatives for [7^2:6]2
Character table for [7^2:6]2

Intermediate fields

\(\Q(\sqrt{2}) \)

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

Sibling fields

Degree 14 siblings: data not computed
Degree 28 siblings: data not computed
Degree 42 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.12.24.79$x^{12} - 4 x^{11} - 10 x^{10} + 16 x^{9} - 6 x^{8} + 16 x^{7} + 4 x^{6} - 8 x^{5} + 16 x^{4} + 16 x^{3} + 16 x^{2} + 8$$4$$3$$24$$C_6\times C_2$$[2, 3]^{3}$
$7$7.7.7.1$x^{7} + 42 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
7.7.7.1$x^{7} + 42 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
17.7.6.1$x^{7} - 17$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$