↑

Properties

Label 14.14.1386571940...3056.1
Degree $14$
Signature $[14, 0]$
Discriminant $2^{12}\cdot 7^{21}\cdot 67^{7}$
Root discriminant $274.61$
Ramified primes $2, 7, 67$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 14T26

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-420, -8036, -7840, 87024, -57036, -108388, 103684, 14714, -26082, -350, 2380, 0, -84, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 84*x^12 + 2380*x^10 - 350*x^9 - 26082*x^8 + 14714*x^7 + 103684*x^6 - 108388*x^5 - 57036*x^4 + 87024*x^3 - 7840*x^2 - 8036*x - 420)
 
gp: K = bnfinit(x^14 - 84*x^12 + 2380*x^10 - 350*x^9 - 26082*x^8 + 14714*x^7 + 103684*x^6 - 108388*x^5 - 57036*x^4 + 87024*x^3 - 7840*x^2 - 8036*x - 420, 1)
 

Normalized defining polynomial

\( x^{14} - 84 x^{12} + 2380 x^{10} - 350 x^{9} - 26082 x^{8} + 14714 x^{7} + 103684 x^{6} - 108388 x^{5} - 57036 x^{4} + 87024 x^{3} - 7840 x^{2} - 8036 x - 420 \)

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:  \(13865719400672142875994939266093056=2^{12}\cdot 7^{21}\cdot 67^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $274.61$
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:  $2, 7, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{296129781200329005341354} a^{13} - \frac{6031709341610897309881}{148064890600164502670677} a^{12} - \frac{65287091703858482127733}{296129781200329005341354} a^{11} - \frac{46003508821229424352003}{296129781200329005341354} a^{10} + \frac{21284435553664920846183}{296129781200329005341354} a^{9} + \frac{455524897816979238706}{148064890600164502670677} a^{8} + \frac{45889414918825760157709}{296129781200329005341354} a^{7} + \frac{29431495717523057685085}{148064890600164502670677} a^{6} - \frac{1656177211744515939474}{148064890600164502670677} a^{5} + \frac{49969404666305280723233}{148064890600164502670677} a^{4} - \frac{56110634849835094400453}{148064890600164502670677} a^{3} - \frac{36803755843769245819594}{148064890600164502670677} a^{2} - \frac{27284577598016220396829}{148064890600164502670677} a + \frac{70407059988518717014582}{148064890600164502670677}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 3053601932800 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T26:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 882
The 20 conjugacy class representatives for 1/2[1/2.F_42(7)^2]2
Character table for 1/2[1/2.F_42(7)^2]2

Intermediate fields

\(\Q(\sqrt{469}) \)

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

Sibling fields

Degree 21 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 ${\href{/LocalNumberField/3.7.0.1}{7} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.14.12.1$x^{14} - 2 x^{7} + 4$$7$$2$$12$$(C_7:C_3) \times C_2$$[\ ]_{7}^{6}$
7Data not computed
$67$67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.6.3.2$x^{6} - 4489 x^{2} + 4812208$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
67.6.3.2$x^{6} - 4489 x^{2} + 4812208$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$