↑

Properties

Label 14.6.27210071794...0000.1
Degree $14$
Signature $[6, 4]$
Discriminant $2^{28}\cdot 3^{14}\cdot 5^{12}\cdot 7^{2}\cdot 11^{6}$
Root discriminant $175.93$
Ramified primes $2, 3, 5, 7, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T62

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1000188, 2905308, 1375542, -2482812, -447273, 191106, -129651, -178104, -31226, -744, 756, 404, -9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 6*x^13 - 9*x^12 + 404*x^11 + 756*x^10 - 744*x^9 - 31226*x^8 - 178104*x^7 - 129651*x^6 + 191106*x^5 - 447273*x^4 - 2482812*x^3 + 1375542*x^2 + 2905308*x - 1000188)
 
gp: K = bnfinit(x^14 - 6*x^13 - 9*x^12 + 404*x^11 + 756*x^10 - 744*x^9 - 31226*x^8 - 178104*x^7 - 129651*x^6 + 191106*x^5 - 447273*x^4 - 2482812*x^3 + 1375542*x^2 + 2905308*x - 1000188, 1)
 

Normalized defining polynomial

\( x^{14} - 6 x^{13} - 9 x^{12} + 404 x^{11} + 756 x^{10} - 744 x^{9} - 31226 x^{8} - 178104 x^{7} - 129651 x^{6} + 191106 x^{5} - 447273 x^{4} - 2482812 x^{3} + 1375542 x^{2} + 2905308 x - 1000188 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(27210071794374475776000000000000=2^{28}\cdot 3^{14}\cdot 5^{12}\cdot 7^{2}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $175.93$
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:  $2, 3, 5, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{8} - \frac{1}{6} a^{5} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{6} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3}$, $\frac{1}{36} a^{10} - \frac{1}{12} a^{8} + \frac{2}{9} a^{7} + \frac{1}{12} a^{6} + \frac{1}{6} a^{5} + \frac{13}{36} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{108} a^{11} - \frac{1}{12} a^{9} + \frac{2}{27} a^{8} + \frac{7}{36} a^{7} - \frac{2}{9} a^{6} - \frac{5}{108} a^{5} - \frac{7}{18} a^{4} + \frac{4}{9} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{22680} a^{12} - \frac{2}{945} a^{11} + \frac{17}{1260} a^{10} + \frac{29}{567} a^{9} - \frac{7}{108} a^{8} + \frac{179}{945} a^{7} - \frac{467}{5670} a^{6} - \frac{59}{630} a^{5} + \frac{479}{1512} a^{4} - \frac{5}{63} a^{3} - \frac{331}{1260} a^{2} + \frac{13}{210} a - \frac{1}{10}$, $\frac{1}{950225502892867563488596606022266800} a^{13} - \frac{399963964688194192547720149609}{316741834297622521162865535340755600} a^{12} + \frac{45909931304557176789988473359021}{52790305716270420193810922556792600} a^{11} - \frac{3057980302827025524670348044874247}{475112751446433781744298303011133400} a^{10} - \frac{6232590739640965763283759765137}{129282381345968375984843075649288} a^{9} - \frac{9807233805154252506079739257139299}{158370917148811260581432767670377800} a^{8} - \frac{8902638884241778141358404097595169}{118778187861608445436074575752783350} a^{7} + \frac{101318381658890997383785599908789}{13197576429067605048452730639198150} a^{6} + \frac{56570405584724090496799470374865607}{316741834297622521162865535340755600} a^{5} - \frac{7337042126917768388255428403993947}{21116122286508168077524369022717040} a^{4} + \frac{21938096611008338640690118506114719}{52790305716270420193810922556792600} a^{3} + \frac{7171835192622285681801380762127089}{17596768572090140064603640852264200} a^{2} - \frac{27729001309925949785537299847393}{69828446714643412954776352588350} a - \frac{9814072517801555214911727022421}{19950984775612403701364672168100}$

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:  $9$
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:  \( 1461564547240 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T62:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 43589145600
The 72 conjugacy class representatives for A14 are not computed
Character table for A14 is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

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 R R R ${\href{/LocalNumberField/13.13.0.1}{13} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.13.0.1}{13} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/29.7.0.1}{7} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.11.0.1}{11} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.8.22.95$x^{8} + 184 x^{4} + 400$$8$$1$$22$$D_4\times C_2$$[2, 3, 7/2]^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.3.2$x^{4} - 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.6.11.6$x^{6} + 6 x^{3} + 15$$6$$1$$11$$S_3^2$$[2, 5/2]_{2}^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.10.10.6$x^{10} + 10 x^{6} + 10 x^{5} + 75 x^{2} + 50 x + 25$$5$$2$$10$$C_5^2 : C_8$$[5/4, 5/4]_{4}^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.7.0.1$x^{7} - x + 2$$1$$7$$0$$C_7$$[\ ]^{7}$
$11$11.3.2.1$x^{3} - 11$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$