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Properties

Label 14.2.19263774627...1648.1
Degree $14$
Signature $[2, 6]$
Discriminant $2^{27}\cdot 3^{6}\cdot 7^{11}\cdot 263^{3}\cdot 3529^{5}$
Root discriminant $1716.42$
Ramified primes $2, 3, 7, 263, 3529$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T45

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-613625842441849, 237150347682498, -121702249507305, 11976481209516, 435254400205, -238430330210, 11370375683, 2500711652, -203574775, -8273862, 916925, 10232, -1657, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 - 1657*x^12 + 10232*x^11 + 916925*x^10 - 8273862*x^9 - 203574775*x^8 + 2500711652*x^7 + 11370375683*x^6 - 238430330210*x^5 + 435254400205*x^4 + 11976481209516*x^3 - 121702249507305*x^2 + 237150347682498*x - 613625842441849)
 
gp: K = bnfinit(x^14 - 2*x^13 - 1657*x^12 + 10232*x^11 + 916925*x^10 - 8273862*x^9 - 203574775*x^8 + 2500711652*x^7 + 11370375683*x^6 - 238430330210*x^5 + 435254400205*x^4 + 11976481209516*x^3 - 121702249507305*x^2 + 237150347682498*x - 613625842441849, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} - 1657 x^{12} + 10232 x^{11} + 916925 x^{10} - 8273862 x^{9} - 203574775 x^{8} + 2500711652 x^{7} + 11370375683 x^{6} - 238430330210 x^{5} + 435254400205 x^{4} + 11976481209516 x^{3} - 121702249507305 x^{2} + 237150347682498 x - 613625842441849 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1926377462756066120847727193419105645413531648=2^{27}\cdot 3^{6}\cdot 7^{11}\cdot 263^{3}\cdot 3529^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1716.42$
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:  $2, 3, 7, 263, 3529$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{18} a^{8} - \frac{1}{9} a^{5} + \frac{4}{9} a^{4} + \frac{2}{9} a^{3} - \frac{1}{9} a^{2} - \frac{1}{9} a - \frac{7}{18}$, $\frac{1}{18} a^{9} - \frac{1}{9} a^{6} + \frac{1}{9} a^{5} - \frac{1}{9} a^{4} + \frac{2}{9} a^{3} - \frac{1}{9} a^{2} - \frac{7}{18} a + \frac{1}{3}$, $\frac{1}{378} a^{10} + \frac{5}{189} a^{9} + \frac{5}{189} a^{8} + \frac{1}{378} a^{7} + \frac{1}{18} a^{6} + \frac{7}{54} a^{5} - \frac{23}{54} a^{4} + \frac{23}{126} a^{3} - \frac{61}{189} a^{2} + \frac{41}{126} a - \frac{1}{378}$, $\frac{1}{378} a^{11} - \frac{1}{63} a^{9} + \frac{1}{63} a^{8} + \frac{11}{378} a^{7} + \frac{7}{54} a^{6} - \frac{1}{6} a^{5} - \frac{43}{378} a^{4} + \frac{5}{27} a^{3} - \frac{169}{378} a^{2} - \frac{139}{378} a - \frac{95}{378}$, $\frac{1}{378} a^{12} + \frac{1}{126} a^{9} + \frac{4}{189} a^{8} - \frac{4}{189} a^{7} + \frac{31}{189} a^{5} + \frac{7}{54} a^{4} + \frac{4}{27} a^{3} - \frac{89}{189} a^{2} + \frac{13}{378} a - \frac{1}{63}$, $\frac{1}{75896461470243013352784637035058033519631180709004741350665187596597265458166} a^{13} - \frac{19701988338843235593570092254008325423709091785840886647661139511555515025}{25298820490081004450928212345019344506543726903001580450221729198865755152722} a^{12} - \frac{38730083214625676594298712337163705894194796773005941081659710874466423149}{37948230735121506676392318517529016759815590354502370675332593798298632729083} a^{11} + \frac{2139981079364683719877874796583143204072468779682572214931953222103086553}{25298820490081004450928212345019344506543726903001580450221729198865755152722} a^{10} - \frac{17321087905113843942530847470997581309857410516888627749908933173846968744}{37948230735121506676392318517529016759815590354502370675332593798298632729083} a^{9} + \frac{101791464663696055343617201909884958338139411711406280202372529777534465162}{5421175819303072382341759788218430965687941479214624382190370542614090389869} a^{8} - \frac{855930056137907696340563621745185067731905583709873486709517103500271184981}{37948230735121506676392318517529016759815590354502370675332593798298632729083} a^{7} - \frac{4899705981126493971219736738006988919983834173951023686937813668463806777563}{37948230735121506676392318517529016759815590354502370675332593798298632729083} a^{6} + \frac{4642518756501317872322325665866294573838214272961059031817004743566630485623}{75896461470243013352784637035058033519631180709004741350665187596597265458166} a^{5} + \frac{17648694967336985239402654729863584944459754413776473786852029178566514599}{8432940163360334816976070781673114835514575634333860150073909732955251717574} a^{4} - \frac{4471740584307986321486573716018162860559488655284335068470871343847697541680}{37948230735121506676392318517529016759815590354502370675332593798298632729083} a^{3} - \frac{14284593189928531005466131060247277052937320978642354000981223043307281605483}{75896461470243013352784637035058033519631180709004741350665187596597265458166} a^{2} + \frac{1326466678528935145586248657652209293576508453735443679705870779770118617870}{5421175819303072382341759788218430965687941479214624382190370542614090389869} a + \frac{1832529774340110223401147922081612545387481208439953039777500368093549676168}{37948230735121506676392318517529016759815590354502370675332593798298632729083}$

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:  $7$
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:  \( 393913205273000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T45:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 3528
The 35 conjugacy class representatives for [F_42(7)^2]2=F_42(7)wr2
Character table for [F_42(7)^2]2=F_42(7)wr2 is not computed

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 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.14.27.213$x^{14} + 2 x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{8} + 4 x^{6} + 4 x^{5} + 4 x^{2} - 2$$14$$1$$27$$(C_7:C_3) \times C_2$$[3]_{7}^{3}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.7.7.1$x^{7} + 42 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
263Data not computed
3529Data not computed