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Properties

Label 14.2.12275806904...1376.1
Degree $14$
Signature $[2, 6]$
Discriminant $2^{12}\cdot 7^{16}\cdot 11^{8}\cdot 29^{10}$
Root discriminant $730.30$
Ramified primes $2, 7, 11, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T22

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![723639232, -1497646080, 802837840, -490172256, 124211584, -48638016, 6832756, -2601696, 235648, -77728, 4473, -840, 42, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 42*x^12 - 840*x^11 + 4473*x^10 - 77728*x^9 + 235648*x^8 - 2601696*x^7 + 6832756*x^6 - 48638016*x^5 + 124211584*x^4 - 490172256*x^3 + 802837840*x^2 - 1497646080*x + 723639232)
 
gp: K = bnfinit(x^14 + 42*x^12 - 840*x^11 + 4473*x^10 - 77728*x^9 + 235648*x^8 - 2601696*x^7 + 6832756*x^6 - 48638016*x^5 + 124211584*x^4 - 490172256*x^3 + 802837840*x^2 - 1497646080*x + 723639232, 1)
 

Normalized defining polynomial

\( x^{14} + 42 x^{12} - 840 x^{11} + 4473 x^{10} - 77728 x^{9} + 235648 x^{8} - 2601696 x^{7} + 6832756 x^{6} - 48638016 x^{5} + 124211584 x^{4} - 490172256 x^{3} + 802837840 x^{2} - 1497646080 x + 723639232 \)

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:  \(12275806904099446920772541656987945701376=2^{12}\cdot 7^{16}\cdot 11^{8}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $730.30$
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:  $2, 7, 11, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{112} a^{10} + \frac{1}{56} a^{9} + \frac{3}{56} a^{8} + \frac{3}{56} a^{7} + \frac{1}{16} a^{6} + \frac{1}{8} a^{4} - \frac{1}{7} a^{3} + \frac{13}{28} a^{2} - \frac{5}{14} a + \frac{1}{7}$, $\frac{1}{560} a^{11} - \frac{3}{140} a^{9} + \frac{1}{70} a^{8} - \frac{19}{560} a^{7} - \frac{1}{10} a^{6} + \frac{7}{40} a^{5} + \frac{6}{35} a^{4} - \frac{3}{20} a^{3} - \frac{16}{35} a^{2} + \frac{13}{35} a - \frac{9}{35}$, $\frac{1}{32480} a^{12} + \frac{1}{1624} a^{11} - \frac{61}{16240} a^{10} - \frac{109}{4060} a^{9} + \frac{741}{32480} a^{8} + \frac{41}{8120} a^{7} - \frac{79}{1160} a^{6} + \frac{3}{1015} a^{5} + \frac{457}{4060} a^{4} + \frac{643}{4060} a^{3} - \frac{1469}{4060} a^{2} + \frac{351}{2030} a + \frac{69}{203}$, $\frac{1}{158740475241014801215847701264200159680} a^{13} - \frac{21104245965570178449726280084291}{3968511881025370030396192531605003992} a^{12} + \frac{50482960723804109208862478906212569}{79370237620507400607923850632100079840} a^{11} - \frac{15193459724810726838302064709109669}{19842559405126850151980962658025019960} a^{10} - \frac{4373526217639303096031790916080010299}{158740475241014801215847701264200159680} a^{9} + \frac{2364287122969734439095943870133843301}{39685118810253700303961925316050039920} a^{8} + \frac{77392334025137287687191492385644827}{39685118810253700303961925316050039920} a^{7} + \frac{3325419984509160044570270768848280269}{39685118810253700303961925316050039920} a^{6} - \frac{8914390361787184230152729697388628791}{39685118810253700303961925316050039920} a^{5} + \frac{916992445882028134966257863564564987}{4960639851281712537995240664506254990} a^{4} + \frac{21319573644202261246512430196065002}{2480319925640856268997620332253127495} a^{3} - \frac{762162393385837690406375810012183606}{2480319925640856268997620332253127495} a^{2} + \frac{409275768224386968296297946819494801}{1984255940512685015198096265802501996} a + \frac{255429879387827569075484680139630759}{992127970256342507599048132901250998}$

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:  \( 1824540926050000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T22:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 588
The 10 conjugacy class representatives for [1/6_-.F_42(7)^2]2_2
Character table for [1/6_-.F_42(7)^2]2_2

Intermediate fields

\(\Q(\sqrt{29}) \)

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

Sibling fields

Degree 28 sibling: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R R ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.6.5.1$x^{6} - 28$$6$$1$$5$$C_6$$[\ ]_{6}$
7.7.11.1$x^{7} + 7 x^{5} + 7$$7$$1$$11$$F_7$$[11/6]_{6}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.12.8.1$x^{12} - 33 x^{9} + 363 x^{6} - 1331 x^{3} + 117128$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.3.2$x^{4} - 116$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.2$x^{4} - 116$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.2$x^{4} - 116$$4$$1$$3$$C_4$$[\ ]_{4}$