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Properties

Label 14.2.16108477729...0000.1
Degree $14$
Signature $[2, 6]$
Discriminant $2^{8}\cdot 3^{6}\cdot 5^{10}\cdot 7^{14}\cdot 19^{4}$
Root discriminant $121.96$
Ramified primes $2, 3, 5, 7, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T36

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![42751, -72149, 158718, -156394, 168028, -85351, 47138, -3604, 742, 1239, 469, -133, -35, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 35*x^12 - 133*x^11 + 469*x^10 + 1239*x^9 + 742*x^8 - 3604*x^7 + 47138*x^6 - 85351*x^5 + 168028*x^4 - 156394*x^3 + 158718*x^2 - 72149*x + 42751)
 
gp: K = bnfinit(x^14 - 35*x^12 - 133*x^11 + 469*x^10 + 1239*x^9 + 742*x^8 - 3604*x^7 + 47138*x^6 - 85351*x^5 + 168028*x^4 - 156394*x^3 + 158718*x^2 - 72149*x + 42751, 1)
 

Normalized defining polynomial

\( x^{14} - 35 x^{12} - 133 x^{11} + 469 x^{10} + 1239 x^{9} + 742 x^{8} - 3604 x^{7} + 47138 x^{6} - 85351 x^{5} + 168028 x^{4} - 156394 x^{3} + 158718 x^{2} - 72149 x + 42751 \)

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:  \(161084777292385129102500000000=2^{8}\cdot 3^{6}\cdot 5^{10}\cdot 7^{14}\cdot 19^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $121.96$
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:  $2, 3, 5, 7, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{3}{7}$, $\frac{1}{7} a^{8} - \frac{3}{7} a$, $\frac{1}{7} a^{9} - \frac{3}{7} a^{2}$, $\frac{1}{35} a^{10} - \frac{2}{35} a^{9} + \frac{2}{35} a^{7} - \frac{1}{5} a^{5} - \frac{3}{35} a^{3} - \frac{3}{7} a^{2} - \frac{1}{5} a + \frac{1}{35}$, $\frac{1}{35} a^{11} + \frac{1}{35} a^{9} + \frac{2}{35} a^{8} - \frac{1}{35} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{3}{35} a^{4} + \frac{2}{5} a^{3} - \frac{17}{35} a^{2} - \frac{13}{35} a + \frac{17}{35}$, $\frac{1}{175} a^{12} - \frac{1}{175} a^{11} + \frac{2}{175} a^{10} + \frac{4}{175} a^{9} + \frac{12}{175} a^{8} - \frac{4}{175} a^{7} + \frac{4}{25} a^{6} + \frac{74}{175} a^{5} - \frac{53}{175} a^{4} - \frac{34}{175} a^{3} + \frac{79}{175} a^{2} - \frac{22}{175} a - \frac{16}{175}$, $\frac{1}{839488878644038480951450} a^{13} + \frac{267800995806619922267}{167897775728807696190290} a^{12} - \frac{1179810062991591166297}{419744439322019240475725} a^{11} + \frac{4671280133451385666841}{839488878644038480951450} a^{10} + \frac{2414743288497407419658}{419744439322019240475725} a^{9} + \frac{38903502788071035826533}{839488878644038480951450} a^{8} - \frac{2269147330583745869993}{119926982663434068707350} a^{7} - \frac{104965189890231100821723}{839488878644038480951450} a^{6} + \frac{1547820154349611294721}{839488878644038480951450} a^{5} - \frac{160870038221125191678731}{419744439322019240475725} a^{4} - \frac{33034306323184645796953}{83948887864403848095145} a^{3} + \frac{72260292408135357317096}{419744439322019240475725} a^{2} + \frac{174111490726676226846296}{419744439322019240475725} a + \frac{13442089234578072971517}{119926982663434068707350}$

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:  $7$
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:  \( 3352784777.82 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T36:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1764
The 22 conjugacy class representatives for 1/2[F_42(7)^2]2
Character table for 1/2[F_42(7)^2]2 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \)

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

Sibling fields

Degree 28 sibling: 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 ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ R ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.2.0.1$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$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.12.6.1$x^{12} - 243 x^{2} + 1458$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
7Data not computed
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$