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Properties

Label 14.2.32557342958...7904.1
Degree $14$
Signature $[2, 6]$
Discriminant $2^{27}\cdot 7^{10}\cdot 97^{5}$
Root discriminant $78.30$
Ramified primes $2, 7, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T45

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9153, -3294, 11915, -484, -3310, 4052, -5118, 2984, 277, -442, 239, -120, 29, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 6*x^13 + 29*x^12 - 120*x^11 + 239*x^10 - 442*x^9 + 277*x^8 + 2984*x^7 - 5118*x^6 + 4052*x^5 - 3310*x^4 - 484*x^3 + 11915*x^2 - 3294*x + 9153)
 
gp: K = bnfinit(x^14 - 6*x^13 + 29*x^12 - 120*x^11 + 239*x^10 - 442*x^9 + 277*x^8 + 2984*x^7 - 5118*x^6 + 4052*x^5 - 3310*x^4 - 484*x^3 + 11915*x^2 - 3294*x + 9153, 1)
 

Normalized defining polynomial

\( x^{14} - 6 x^{13} + 29 x^{12} - 120 x^{11} + 239 x^{10} - 442 x^{9} + 277 x^{8} + 2984 x^{7} - 5118 x^{6} + 4052 x^{5} - 3310 x^{4} - 484 x^{3} + 11915 x^{2} - 3294 x + 9153 \)

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:  \(325573429585516975729147904=2^{27}\cdot 7^{10}\cdot 97^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.30$
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:  $2, 7, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{63} a^{11} - \frac{10}{63} a^{10} - \frac{4}{63} a^{9} - \frac{10}{63} a^{8} + \frac{1}{63} a^{7} + \frac{2}{9} a^{6} - \frac{23}{63} a^{5} + \frac{11}{63} a^{4} - \frac{2}{7} a^{3} - \frac{3}{7} a^{2} + \frac{20}{63} a - \frac{2}{7}$, $\frac{1}{1071} a^{12} + \frac{2}{357} a^{11} - \frac{122}{1071} a^{10} - \frac{32}{1071} a^{9} - \frac{6}{119} a^{8} - \frac{172}{357} a^{7} + \frac{48}{119} a^{6} + \frac{5}{51} a^{5} - \frac{115}{1071} a^{4} - \frac{5}{51} a^{3} + \frac{29}{1071} a^{2} + \frac{533}{1071} a + \frac{24}{119}$, $\frac{1}{1139639814560780469216} a^{13} - \frac{7386921230399239}{67037636150634145248} a^{12} - \frac{191355474932165423}{31656661515577235256} a^{11} + \frac{902120914603012843}{10552220505192411752} a^{10} + \frac{10119385908445218941}{162805687794397209888} a^{9} - \frac{105402717739929768629}{1139639814560780469216} a^{8} + \frac{135981613183791022717}{569819907280390234608} a^{7} - \frac{60030734574278271961}{569819907280390234608} a^{6} - \frac{29715680330348240147}{284909953640195117304} a^{5} - \frac{5822693172333495865}{11871248068341463221} a^{4} - \frac{132844087535912839175}{569819907280390234608} a^{3} - \frac{158738535227103240235}{569819907280390234608} a^{2} - \frac{21768512051919436981}{379879938186926823072} a + \frac{482998059642501029}{6029840288681378144}$

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:  \( 193942346.218 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T45:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 3528
The 35 conjugacy class representatives for [F_42(7)^2]2=F_42(7)wr2
Character table for [F_42(7)^2]2=F_42(7)wr2 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 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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$$[3]$
2.12.24.79$x^{12} - 4 x^{11} - 10 x^{10} + 16 x^{9} - 6 x^{8} + 16 x^{7} + 4 x^{6} - 8 x^{5} + 16 x^{4} + 16 x^{3} + 16 x^{2} + 8$$4$$3$$24$$C_6\times C_2$$[2, 3]^{3}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.7.7.6$x^{7} + 28 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
$97$$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
97.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
97.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
97.6.5.2$x^{6} - 2425$$6$$1$$5$$C_6$$[\ ]_{6}$