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Properties

Label 14.14.5366153233...5888.1
Degree $14$
Signature $[14, 0]$
Discriminant $2^{12}\cdot 7^{10}\cdot 173^{7}$
Root discriminant $95.65$
Ramified primes $2, 7, 173$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 14T40

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![35, 641, -17415, 87888, -70414, -118996, 88460, 39336, -21694, -4476, 2031, 199, -78, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 3*x^13 - 78*x^12 + 199*x^11 + 2031*x^10 - 4476*x^9 - 21694*x^8 + 39336*x^7 + 88460*x^6 - 118996*x^5 - 70414*x^4 + 87888*x^3 - 17415*x^2 + 641*x + 35)
 
gp: K = bnfinit(x^14 - 3*x^13 - 78*x^12 + 199*x^11 + 2031*x^10 - 4476*x^9 - 21694*x^8 + 39336*x^7 + 88460*x^6 - 118996*x^5 - 70414*x^4 + 87888*x^3 - 17415*x^2 + 641*x + 35, 1)
 

Normalized defining polynomial

\( x^{14} - 3 x^{13} - 78 x^{12} + 199 x^{11} + 2031 x^{10} - 4476 x^{9} - 21694 x^{8} + 39336 x^{7} + 88460 x^{6} - 118996 x^{5} - 70414 x^{4} + 87888 x^{3} - 17415 x^{2} + 641 x + 35 \)

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:  \(5366153233223785078672805888=2^{12}\cdot 7^{10}\cdot 173^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.65$
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:  $2, 7, 173$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7972798379241832892841464987} a^{13} - \frac{2780753635348133586485483317}{7972798379241832892841464987} a^{12} + \frac{1916801995301731606587413734}{7972798379241832892841464987} a^{11} - \frac{3294156461892422770282419777}{7972798379241832892841464987} a^{10} - \frac{2440049251651764325102753459}{7972798379241832892841464987} a^{9} + \frac{2104579665697830547924073938}{7972798379241832892841464987} a^{8} - \frac{2137674675156528354987189942}{7972798379241832892841464987} a^{7} - \frac{2657623231346747148540098649}{7972798379241832892841464987} a^{6} - \frac{2740558442062219575715738880}{7972798379241832892841464987} a^{5} + \frac{393145160154280931261192807}{7972798379241832892841464987} a^{4} - \frac{686200674007927418126992622}{7972798379241832892841464987} a^{3} - \frac{2074491821198599025217691380}{7972798379241832892841464987} a^{2} - \frac{1127474733162164048029467292}{7972798379241832892841464987} a - \frac{2591997087070079706408598494}{7972798379241832892841464987}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 2255205793.73 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T40:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2688
The 20 conjugacy class representatives for 1/2[2^7]F_42(7)
Character table for 1/2[2^7]F_42(7)

Intermediate fields

7.7.12431698517.1

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 16 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.12.12.4$x^{12} - 6 x^{10} + 15 x^{8} - 20 x^{6} + 15 x^{4} - 38 x^{2} - 31$$2$$6$$12$12T87$[2, 2, 2, 2, 2]^{6}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.12.10.5$x^{12} + 56 x^{6} + 1323$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
173Data not computed