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Properties

Label 14.0.10201866453...0931.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{8}\cdot 7^{20}\cdot 11^{7}$
Root discriminant $100.14$
Ramified primes $3, 7, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T26

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![75141, 119196, 36855, -3192, 40348, 33908, -735, 2273, 5061, 567, 280, -28, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 28*x^11 + 280*x^10 + 567*x^9 + 5061*x^8 + 2273*x^7 - 735*x^6 + 33908*x^5 + 40348*x^4 - 3192*x^3 + 36855*x^2 + 119196*x + 75141)
 
gp: K = bnfinit(x^14 - 28*x^11 + 280*x^10 + 567*x^9 + 5061*x^8 + 2273*x^7 - 735*x^6 + 33908*x^5 + 40348*x^4 - 3192*x^3 + 36855*x^2 + 119196*x + 75141, 1)
 

Normalized defining polynomial

\( x^{14} - 28 x^{11} + 280 x^{10} + 567 x^{9} + 5061 x^{8} + 2273 x^{7} - 735 x^{6} + 33908 x^{5} + 40348 x^{4} - 3192 x^{3} + 36855 x^{2} + 119196 x + 75141 \)

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:  \(-10201866453631127927668100931=-\,3^{8}\cdot 7^{20}\cdot 11^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $100.14$
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:  $3, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{11} a^{9} - \frac{5}{11} a^{8} - \frac{3}{11} a^{7} - \frac{3}{11} a^{6} + \frac{1}{11} a^{5} + \frac{5}{11} a^{4} + \frac{1}{11} a^{3} + \frac{1}{11} a$, $\frac{1}{11} a^{10} + \frac{5}{11} a^{8} + \frac{4}{11} a^{7} - \frac{3}{11} a^{6} - \frac{1}{11} a^{5} + \frac{4}{11} a^{4} + \frac{5}{11} a^{3} + \frac{1}{11} a^{2} + \frac{5}{11} a$, $\frac{1}{55} a^{11} + \frac{2}{55} a^{9} - \frac{14}{55} a^{8} - \frac{16}{55} a^{7} - \frac{3}{55} a^{6} + \frac{23}{55} a^{5} + \frac{23}{55} a^{4} + \frac{9}{55} a^{3} + \frac{16}{55} a^{2} + \frac{8}{55} a + \frac{1}{5}$, $\frac{1}{3795} a^{12} + \frac{3}{1265} a^{11} - \frac{41}{1265} a^{10} + \frac{134}{3795} a^{9} + \frac{13}{3795} a^{8} - \frac{144}{1265} a^{7} - \frac{263}{1265} a^{6} + \frac{31}{759} a^{5} + \frac{397}{1265} a^{4} - \frac{1828}{3795} a^{3} + \frac{1732}{3795} a^{2} + \frac{211}{1265} a + \frac{1}{5}$, $\frac{1}{25563663039009255360297075} a^{13} + \frac{690038396426603644399}{8521221013003085120099025} a^{12} + \frac{2633948333157298611156}{2840407004334361706699675} a^{11} - \frac{144960599962001562679676}{5112732607801851072059415} a^{10} + \frac{30575672584274409509642}{5112732607801851072059415} a^{9} - \frac{1914047582751283093689941}{8521221013003085120099025} a^{8} - \frac{545585950097481979744447}{1704244202600617024019805} a^{7} + \frac{6972466934880564448363718}{25563663039009255360297075} a^{6} - \frac{40952881661009194292937}{123495956710189639421725} a^{5} - \frac{58956443314146169231012}{1022546521560370214411883} a^{4} - \frac{8957304669309061292922917}{25563663039009255360297075} a^{3} - \frac{2034934862227000365744052}{8521221013003085120099025} a^{2} + \frac{569250131159582880285802}{2840407004334361706699675} a + \frac{463518376362646569296}{11226905155471785401975}$

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:  $6$
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:  \( 2528603759.06 \) (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

\(\Q(\sqrt{-11}) \)

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
7Data not computed
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$