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Properties

Label 14.0.26602062621...0000.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{8}\cdot 3^{7}\cdot 5^{6}\cdot 7^{12}\cdot 13^{3}$
Root discriminant $47.12$
Ramified primes $2, 3, 5, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T31

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8308, 3390, -9859, 2920, 4874, -9215, 10263, -7040, 3959, -1910, 723, -185, 43, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 + 43*x^12 - 185*x^11 + 723*x^10 - 1910*x^9 + 3959*x^8 - 7040*x^7 + 10263*x^6 - 9215*x^5 + 4874*x^4 + 2920*x^3 - 9859*x^2 + 3390*x + 8308)
 
gp: K = bnfinit(x^14 - 5*x^13 + 43*x^12 - 185*x^11 + 723*x^10 - 1910*x^9 + 3959*x^8 - 7040*x^7 + 10263*x^6 - 9215*x^5 + 4874*x^4 + 2920*x^3 - 9859*x^2 + 3390*x + 8308, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} + 43 x^{12} - 185 x^{11} + 723 x^{10} - 1910 x^{9} + 3959 x^{8} - 7040 x^{7} + 10263 x^{6} - 9215 x^{5} + 4874 x^{4} + 2920 x^{3} - 9859 x^{2} + 3390 x + 8308 \)

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:  \(-266020626214262556000000=-\,2^{8}\cdot 3^{7}\cdot 5^{6}\cdot 7^{12}\cdot 13^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.12$
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:  $2, 3, 5, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{4} + \frac{1}{5}$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{5} + \frac{1}{5} a$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{11775} a^{12} + \frac{193}{2355} a^{11} - \frac{41}{2355} a^{10} - \frac{13}{2355} a^{9} + \frac{428}{11775} a^{8} - \frac{85}{471} a^{7} + \frac{802}{2355} a^{6} - \frac{202}{2355} a^{5} - \frac{3562}{11775} a^{4} + \frac{232}{2355} a^{3} - \frac{737}{2355} a^{2} + \frac{26}{2355} a + \frac{286}{11775}$, $\frac{1}{7064109017521334508150} a^{13} + \frac{1104400472080091}{31396040077872597814} a^{12} - \frac{12601689155806378921}{1412821803504266901630} a^{11} + \frac{20398341208307149663}{470940601168088967210} a^{10} - \frac{24134355418922801009}{2354703005840444836050} a^{9} + \frac{20826443551463670391}{706410901752133450815} a^{8} + \frac{216932478401579834897}{1412821803504266901630} a^{7} - \frac{185649652952942008}{235470300584044483605} a^{6} + \frac{753008719955960666851}{2354703005840444836050} a^{5} + \frac{61776390149008168351}{282564360700853380326} a^{4} + \frac{137583742481543290489}{706410901752133450815} a^{3} - \frac{37727157250002568984}{78490100194681494535} a^{2} + \frac{2576846602907753429081}{7064109017521334508150} a - \frac{219144175967836292786}{706410901752133450815}$

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:  \( \frac{10748854414773}{199974777566067502} a^{13} - \frac{50709221558035}{599924332698202506} a^{12} + \frac{953026312024039}{599924332698202506} a^{11} - \frac{1540763044321385}{599924332698202506} a^{10} + \frac{7053744110209127}{599924332698202506} a^{9} + \frac{2978280574835570}{299962166349101253} a^{8} - \frac{30910820691325811}{599924332698202506} a^{7} + \frac{55552700191159025}{299962166349101253} a^{6} - \frac{247425654100971415}{599924332698202506} a^{5} + \frac{474963519129551995}{599924332698202506} a^{4} - \frac{261159274344314716}{299962166349101253} a^{3} + \frac{281833856382215330}{299962166349101253} a^{2} + \frac{68395962991296959}{599924332698202506} a + \frac{1643147607476749}{299962166349101253} \) (order $6$)
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:  \( 9512858.46082 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T31:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1176
The 17 conjugacy class representatives for [D(7)^2:3_3]2
Character table for [D(7)^2:3_3]2

Intermediate fields

\(\Q(\sqrt{-3}) \)

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 R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ R ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.6.5.6$x^{6} + 224$$6$$1$$5$$C_6$$[\ ]_{6}$
7.7.7.2$x^{7} + 21 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.6.3.2$x^{6} - 338 x^{2} + 13182$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$