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Properties

Label 14.14.1290953281...0000.1
Degree $14$
Signature $[14, 0]$
Discriminant $2^{12}\cdot 5^{7}\cdot 7^{14}\cdot 29^{6}$
Root discriminant $120.05$
Ramified primes $2, 5, 7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T24

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![134444, 938266, 1195061, -318304, -906598, 28014, 254359, -734, -33880, 0, 2303, 0, -77, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 77*x^12 + 2303*x^10 - 33880*x^8 - 734*x^7 + 254359*x^6 + 28014*x^5 - 906598*x^4 - 318304*x^3 + 1195061*x^2 + 938266*x + 134444)
 
gp: K = bnfinit(x^14 - 77*x^12 + 2303*x^10 - 33880*x^8 - 734*x^7 + 254359*x^6 + 28014*x^5 - 906598*x^4 - 318304*x^3 + 1195061*x^2 + 938266*x + 134444, 1)
 

Normalized defining polynomial

\( x^{14} - 77 x^{12} + 2303 x^{10} - 33880 x^{8} - 734 x^{7} + 254359 x^{6} + 28014 x^{5} - 906598 x^{4} - 318304 x^{3} + 1195061 x^{2} + 938266 x + 134444 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(129095328182677475689280000000=2^{12}\cdot 5^{7}\cdot 7^{14}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $120.05$
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:  $2, 5, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{14} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a + \frac{2}{7}$, $\frac{1}{14} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{3}{14} a$, $\frac{1}{14} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{3}{14} a^{2}$, $\frac{1}{14} a^{10} - \frac{1}{2} a^{4} + \frac{2}{7} a^{3} - \frac{1}{2} a$, $\frac{1}{1106} a^{11} + \frac{23}{1106} a^{10} + \frac{9}{1106} a^{9} - \frac{29}{1106} a^{8} - \frac{2}{553} a^{7} + \frac{49}{158} a^{6} + \frac{9}{158} a^{5} + \frac{529}{1106} a^{4} + \frac{242}{553} a^{3} - \frac{55}{1106} a^{2} - \frac{107}{553} a + \frac{230}{553}$, $\frac{1}{1106} a^{12} + \frac{33}{1106} a^{10} + \frac{1}{1106} a^{9} + \frac{31}{1106} a^{8} - \frac{39}{1106} a^{7} + \frac{67}{158} a^{6} - \frac{367}{1106} a^{5} + \frac{69}{158} a^{4} + \frac{213}{553} a^{3} + \frac{170}{553} a^{2} + \frac{89}{1106} a - \frac{155}{553}$, $\frac{1}{568225142426466} a^{13} - \frac{241506333913}{568225142426466} a^{12} + \frac{89379073432}{284112571213233} a^{11} - \frac{2617083981527}{81175020346638} a^{10} - \frac{11253227013905}{568225142426466} a^{9} + \frac{17767744135247}{568225142426466} a^{8} + \frac{3836706106361}{189408380808822} a^{7} + \frac{121758825387068}{284112571213233} a^{6} - \frac{13126507766489}{31568063468137} a^{5} + \frac{38120284672192}{94704190404411} a^{4} - \frac{10888442306083}{81175020346638} a^{3} + \frac{13324662817597}{31568063468137} a^{2} + \frac{14128789438528}{284112571213233} a + \frac{81371970808945}{284112571213233}$

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:  $13$
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:  \( 17896254443.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T24:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 588
The 19 conjugacy class representatives for [7^2:6]2
Character table for [7^2:6]2

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 14 siblings: data not computed
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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ R ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$$[\ ]^{2}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7Data not computed
$29$29.7.6.4$x^{7} - 1856$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.0.1$x^{7} - x + 3$$1$$7$$0$$C_7$$[\ ]^{7}$