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Properties

Label 14.2.10951333129...9712.1
Degree $14$
Signature $[2, 6]$
Discriminant $2^{12}\cdot 7^{8}\cdot 173^{7}$
Root discriminant $72.44$
Ramified primes $2, 7, 173$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 14T40

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19454, 31229, -39095, 24802, -3646, -6675, 7779, -3249, 363, -262, 244, -45, -3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 3*x^13 - 3*x^12 - 45*x^11 + 244*x^10 - 262*x^9 + 363*x^8 - 3249*x^7 + 7779*x^6 - 6675*x^5 - 3646*x^4 + 24802*x^3 - 39095*x^2 + 31229*x - 19454)
 
gp: K = bnfinit(x^14 - 3*x^13 - 3*x^12 - 45*x^11 + 244*x^10 - 262*x^9 + 363*x^8 - 3249*x^7 + 7779*x^6 - 6675*x^5 - 3646*x^4 + 24802*x^3 - 39095*x^2 + 31229*x - 19454, 1)
 

Normalized defining polynomial

\( x^{14} - 3 x^{13} - 3 x^{12} - 45 x^{11} + 244 x^{10} - 262 x^{9} + 363 x^{8} - 3249 x^{7} + 7779 x^{6} - 6675 x^{5} - 3646 x^{4} + 24802 x^{3} - 39095 x^{2} + 31229 x - 19454 \)

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:  \(109513331290281328136179712=2^{12}\cdot 7^{8}\cdot 173^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.44$
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:  $2, 7, 173$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{72853015565865714351005513} a^{13} + \frac{11151665343730520535292138}{72853015565865714351005513} a^{12} + \frac{5714905633114191498760559}{72853015565865714351005513} a^{11} - \frac{9665009029873728654627921}{72853015565865714351005513} a^{10} - \frac{12383081597837693751013829}{72853015565865714351005513} a^{9} - \frac{16712759799471050891026364}{72853015565865714351005513} a^{8} + \frac{31789236815886198874705450}{72853015565865714351005513} a^{7} - \frac{13807873788144807798130295}{72853015565865714351005513} a^{6} + \frac{16725433533469213330047315}{72853015565865714351005513} a^{5} - \frac{20310698694086648980757337}{72853015565865714351005513} a^{4} + \frac{25442253948905241103102878}{72853015565865714351005513} a^{3} + \frac{25622688837534584183465069}{72853015565865714351005513} a^{2} - \frac{20631862415420460281805056}{72853015565865714351005513} a - \frac{2604952982085141271814680}{72853015565865714351005513}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 17859161.8391 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T40:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2688
The 20 conjugacy class representatives for 1/2[2^7]F_42(7)
Character table for 1/2[2^7]F_42(7)

Intermediate fields

7.7.12431698517.1

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

Sibling fields

Degree 14 sibling: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.12.12.2$x^{12} - 6 x^{10} - 13 x^{8} - 28 x^{6} + 15 x^{4} - 30 x^{2} - 3$$2$$6$$12$12T105$[2, 2, 2, 2, 2]^{6}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
173Data not computed