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Properties

Label 14.6.11302682860...1488.1
Degree $14$
Signature $[6, 4]$
Discriminant $2^{20}\cdot 3^{7}\cdot 13^{2}\cdot 1707745189^{2}$
Root discriminant $140.17$
Ramified primes $2, 3, 13, 1707745189$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T58

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-668, 4260, -27415, 55216, -32984, -14260, 17648, 2144, -4098, -576, 552, 92, -36, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 - 36*x^12 + 92*x^11 + 552*x^10 - 576*x^9 - 4098*x^8 + 2144*x^7 + 17648*x^6 - 14260*x^5 - 32984*x^4 + 55216*x^3 - 27415*x^2 + 4260*x - 668)
 
gp: K = bnfinit(x^14 - 4*x^13 - 36*x^12 + 92*x^11 + 552*x^10 - 576*x^9 - 4098*x^8 + 2144*x^7 + 17648*x^6 - 14260*x^5 - 32984*x^4 + 55216*x^3 - 27415*x^2 + 4260*x - 668, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} - 36 x^{12} + 92 x^{11} + 552 x^{10} - 576 x^{9} - 4098 x^{8} + 2144 x^{7} + 17648 x^{6} - 14260 x^{5} - 32984 x^{4} + 55216 x^{3} - 27415 x^{2} + 4260 x - 668 \)

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:  \(1130268286027719787328320831488=2^{20}\cdot 3^{7}\cdot 13^{2}\cdot 1707745189^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $140.17$
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:  $2, 3, 13, 1707745189$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{3}{8} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{6} - \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{24} a^{10} + \frac{1}{24} a^{9} + \frac{1}{24} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{24} a^{4} - \frac{7}{24} a^{3} + \frac{1}{6} a^{2} - \frac{5}{24} a - \frac{1}{12}$, $\frac{1}{1248} a^{11} + \frac{7}{624} a^{10} - \frac{17}{312} a^{9} + \frac{43}{1248} a^{8} + \frac{3}{104} a^{7} + \frac{5}{26} a^{6} - \frac{15}{416} a^{5} + \frac{155}{624} a^{4} - \frac{2}{13} a^{3} + \frac{449}{1248} a^{2} - \frac{37}{78} a + \frac{151}{312}$, $\frac{1}{1248} a^{12} - \frac{1}{312} a^{10} + \frac{7}{1248} a^{9} + \frac{29}{624} a^{8} - \frac{1}{312} a^{7} - \frac{77}{1248} a^{6} + \frac{9}{104} a^{5} + \frac{25}{156} a^{4} - \frac{185}{416} a^{3} + \frac{67}{208} a^{2} + \frac{1}{12} a - \frac{5}{26}$, $\frac{1}{185798496} a^{13} + \frac{3727}{46449624} a^{12} + \frac{34699}{185798496} a^{11} + \frac{1585}{3505632} a^{10} + \frac{313771}{92899248} a^{9} + \frac{7983497}{185798496} a^{8} + \frac{856747}{185798496} a^{7} - \frac{213151}{1935401} a^{6} - \frac{2203913}{61932832} a^{5} + \frac{1876993}{61932832} a^{4} - \frac{20040479}{92899248} a^{3} - \frac{45634193}{185798496} a^{2} + \frac{707617}{46449624} a + \frac{7503689}{15483208}$

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:  \( 37323484827.4 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T58:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 12700800
The 54 conjugacy class representatives for [A(7)^2]2=A(7)wr2 are not computed
Character table for [A(7)^2]2=A(7)wr2 is not computed

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 30 siblings: data not computed
Degree 42 sibling: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$2$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.4.6.9$x^{4} + 2 x^{3} + 6$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.8.12.20$x^{8} + 8 x^{6} + 12 x^{4} + 80$$4$$2$$12$$C_2^3: C_4$$[2, 2, 2]^{4}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
1707745189Data not computed