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Properties

Label 14.2.73569120049...9056.1
Degree $14$
Signature $[2, 6]$
Discriminant $2^{36}\cdot 3^{6}\cdot 7^{10}\cdot 151^{4}$
Root discriminant $160.24$
Ramified primes $2, 3, 7, 151$
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![-553441, 848106, -483259, 681940, -882119, 854042, -403081, 169992, -73967, 8382, 1155, 100, 87, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 87*x^12 + 100*x^11 + 1155*x^10 + 8382*x^9 - 73967*x^8 + 169992*x^7 - 403081*x^6 + 854042*x^5 - 882119*x^4 + 681940*x^3 - 483259*x^2 + 848106*x - 553441)
 
gp: K = bnfinit(x^14 - 2*x^13 + 87*x^12 + 100*x^11 + 1155*x^10 + 8382*x^9 - 73967*x^8 + 169992*x^7 - 403081*x^6 + 854042*x^5 - 882119*x^4 + 681940*x^3 - 483259*x^2 + 848106*x - 553441, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} + 87 x^{12} + 100 x^{11} + 1155 x^{10} + 8382 x^{9} - 73967 x^{8} + 169992 x^{7} - 403081 x^{6} + 854042 x^{5} - 882119 x^{4} + 681940 x^{3} - 483259 x^{2} + 848106 x - 553441 \)

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:  \(7356912004979662163042742829056=2^{36}\cdot 3^{6}\cdot 7^{10}\cdot 151^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $160.24$
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:  $2, 3, 7, 151$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{8} a^{8} - \frac{1}{2} a^{2} + \frac{3}{8}$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{7}{16} a + \frac{1}{16}$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{8} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} + \frac{3}{32} a^{2} - \frac{1}{4} a + \frac{1}{32}$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{10} - \frac{1}{64} a^{9} + \frac{1}{64} a^{8} - \frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{8} a^{5} + \frac{19}{64} a^{3} - \frac{11}{64} a^{2} + \frac{25}{64} a - \frac{17}{64}$, $\frac{1}{256} a^{12} - \frac{1}{128} a^{10} - \frac{11}{256} a^{8} + \frac{1}{32} a^{7} - \frac{3}{64} a^{6} + \frac{5}{32} a^{5} - \frac{13}{256} a^{4} - \frac{13}{32} a^{3} - \frac{9}{128} a^{2} + \frac{7}{32} a + \frac{55}{256}$, $\frac{1}{16194185309743834702119593278790656} a^{13} - \frac{5105147385562994895484611874521}{16194185309743834702119593278790656} a^{12} - \frac{53971074328518330018063098171317}{8097092654871917351059796639395328} a^{11} - \frac{2774259816253660868770147301707}{8097092654871917351059796639395328} a^{10} + \frac{375144169332548612711279649194429}{16194185309743834702119593278790656} a^{9} + \frac{164210564062066994162658792865155}{16194185309743834702119593278790656} a^{8} + \frac{184966103982712830805812372475475}{4048546327435958675529898319697664} a^{7} - \frac{409271728308082332125794427164339}{4048546327435958675529898319697664} a^{6} + \frac{2432186222541823924653587798393291}{16194185309743834702119593278790656} a^{5} - \frac{2981304726760925649753239523347299}{16194185309743834702119593278790656} a^{4} - \frac{598767652699263866477785927832529}{8097092654871917351059796639395328} a^{3} - \frac{3368284423034872755519109202745871}{8097092654871917351059796639395328} a^{2} - \frac{7706822713352933014288377361053897}{16194185309743834702119593278790656} a - \frac{4898131712863553413659714379913927}{16194185309743834702119593278790656}$

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:  \( 101787628164 \) (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

\(\Q(\sqrt{2}) \)

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.4.11.3$x^{4} + 4 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.3$x^{4} + 4 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.3$x^{4} + 4 x^{2} + 18$$4$$1$$11$$C_4$$[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}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.7.7.4$x^{7} + 14 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
151Data not computed