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Properties

Label 14.14.1881356147...5824.2
Degree $14$
Signature $[14, 0]$
Discriminant $2^{26}\cdot 809^{6}$
Root discriminant $63.87$
Ramified primes $2, 809$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T33

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-557, -48355, 9870, 97817, -19464, -68809, 17647, 20998, -6791, -2603, 1004, 119, -58, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 - 58*x^12 + 119*x^11 + 1004*x^10 - 2603*x^9 - 6791*x^8 + 20998*x^7 + 17647*x^6 - 68809*x^5 - 19464*x^4 + 97817*x^3 + 9870*x^2 - 48355*x - 557)
 
gp: K = bnfinit(x^14 - x^13 - 58*x^12 + 119*x^11 + 1004*x^10 - 2603*x^9 - 6791*x^8 + 20998*x^7 + 17647*x^6 - 68809*x^5 - 19464*x^4 + 97817*x^3 + 9870*x^2 - 48355*x - 557, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} - 58 x^{12} + 119 x^{11} + 1004 x^{10} - 2603 x^{9} - 6791 x^{8} + 20998 x^{7} + 17647 x^{6} - 68809 x^{5} - 19464 x^{4} + 97817 x^{3} + 9870 x^{2} - 48355 x - 557 \)

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:  \(18813561479041924489805824=2^{26}\cdot 809^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.87$
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:  $2, 809$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{236} a^{12} - \frac{21}{118} a^{11} + \frac{13}{236} a^{10} + \frac{4}{59} a^{9} - \frac{111}{236} a^{8} + \frac{19}{59} a^{7} - \frac{53}{118} a^{6} - \frac{17}{59} a^{5} + \frac{33}{236} a^{4} + \frac{49}{118} a^{3} - \frac{85}{236} a^{2} - \frac{20}{59} a + \frac{85}{236}$, $\frac{1}{20872491374896556} a^{13} + \frac{14003688741827}{20872491374896556} a^{12} + \frac{2281495400100571}{20872491374896556} a^{11} + \frac{1381497353448573}{20872491374896556} a^{10} + \frac{8483452053332541}{20872491374896556} a^{9} + \frac{10122178373618761}{20872491374896556} a^{8} + \frac{4523267940355331}{10436245687448278} a^{7} + \frac{11492972022909}{10436245687448278} a^{6} + \frac{10277461658577753}{20872491374896556} a^{5} + \frac{9246558568081047}{20872491374896556} a^{4} - \frac{2706220916067515}{20872491374896556} a^{3} + \frac{5685227766245075}{20872491374896556} a^{2} + \frac{2756683370040657}{20872491374896556} a + \frac{6873033541055389}{20872491374896556}$

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:  \( 280744951.536 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T33:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1344
The 11 conjugacy class representatives for 2^3`L_7(14)
Character table for 2^3`L_7(14)

Intermediate fields

7.7.670188544.2

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

Sibling fields

Degree 14 sibling: 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 ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.8.22.135$x^{8} + 4 x^{7} + 4 x^{2} + 6$$8$$1$$22$$\textrm{GL(2,3)}$$[8/3, 8/3, 7/2]_{3}^{2}$
809Data not computed