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Properties

Label 14.8.19460040067...3584.1
Degree $14$
Signature $[8, 3]$
Discriminant $-\,2^{16}\cdot 3^{8}\cdot 7^{10}\cdot 13^{7}\cdot 294467^{3}$
Root discriminant $889.66$
Ramified primes $2, 3, 7, 13, 294467$
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![-299096421324, -304854167870, -109890439183, -47478809070, -8943505081, 2124722810, 562633386, -32202766, -11363289, 211890, 109942, -518, -525, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 525*x^12 - 518*x^11 + 109942*x^10 + 211890*x^9 - 11363289*x^8 - 32202766*x^7 + 562633386*x^6 + 2124722810*x^5 - 8943505081*x^4 - 47478809070*x^3 - 109890439183*x^2 - 304854167870*x - 299096421324)
 
gp: K = bnfinit(x^14 - 525*x^12 - 518*x^11 + 109942*x^10 + 211890*x^9 - 11363289*x^8 - 32202766*x^7 + 562633386*x^6 + 2124722810*x^5 - 8943505081*x^4 - 47478809070*x^3 - 109890439183*x^2 - 304854167870*x - 299096421324, 1)
 

Normalized defining polynomial

\( x^{14} - 525 x^{12} - 518 x^{11} + 109942 x^{10} + 211890 x^{9} - 11363289 x^{8} - 32202766 x^{7} + 562633386 x^{6} + 2124722810 x^{5} - 8943505081 x^{4} - 47478809070 x^{3} - 109890439183 x^{2} - 304854167870 x - 299096421324 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-194600400675149389603579413527667469123584=-\,2^{16}\cdot 3^{8}\cdot 7^{10}\cdot 13^{7}\cdot 294467^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $889.66$
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:  $2, 3, 7, 13, 294467$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{3}{7} a^{5} - \frac{2}{7} a^{3} - \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{14} a^{7} - \frac{1}{14} a^{6} - \frac{3}{7} a^{5} + \frac{5}{14} a^{4} - \frac{2}{7} a^{3} - \frac{3}{14} a^{2} - \frac{1}{2} a - \frac{1}{7}$, $\frac{1}{14} a^{8} - \frac{1}{14} a^{6} - \frac{5}{14} a^{5} + \frac{1}{14} a^{4} - \frac{5}{14} a^{3} + \frac{2}{7} a^{2} + \frac{1}{14} a + \frac{3}{7}$, $\frac{1}{42} a^{9} + \frac{1}{42} a^{7} - \frac{1}{42} a^{6} - \frac{1}{42} a^{5} + \frac{5}{42} a^{4} - \frac{1}{21} a^{3} - \frac{19}{42} a^{2} + \frac{8}{21} a + \frac{3}{7}$, $\frac{1}{42} a^{10} + \frac{1}{42} a^{8} - \frac{1}{42} a^{7} - \frac{1}{42} a^{6} + \frac{5}{42} a^{5} - \frac{1}{21} a^{4} - \frac{19}{42} a^{3} + \frac{8}{21} a^{2} + \frac{3}{7} a$, $\frac{1}{126} a^{11} + \frac{1}{63} a^{8} - \frac{1}{63} a^{7} - \frac{1}{14} a^{6} + \frac{10}{63} a^{5} + \frac{1}{6} a^{4} - \frac{5}{42} a^{3} + \frac{1}{18} a^{2} + \frac{23}{126} a - \frac{5}{21}$, $\frac{1}{2646} a^{12} + \frac{4}{1323} a^{11} + \frac{1}{294} a^{10} - \frac{25}{2646} a^{9} + \frac{34}{1323} a^{8} + \frac{46}{1323} a^{7} + \frac{73}{1323} a^{6} + \frac{626}{1323} a^{5} - \frac{13}{49} a^{4} - \frac{185}{2646} a^{3} + \frac{79}{2646} a^{2} + \frac{626}{1323} a - \frac{148}{441}$, $\frac{1}{69258400437048130248544362507243679635773742} a^{13} - \frac{2687269746153511587681373256249530721149}{34629200218524065124272181253621839817886871} a^{12} + \frac{6662338506293638390591022817677087923801}{69258400437048130248544362507243679635773742} a^{11} - \frac{2490508320556335842827017166839653684126}{706718371806613573964738392931057955467079} a^{10} - \frac{153824057151405274159260300657235175016581}{23086133479016043416181454169081226545257914} a^{9} - \frac{87914703012497413054276954023509147714257}{23086133479016043416181454169081226545257914} a^{8} + \frac{86852303826044765071136208637845175225906}{3847688913169340569363575694846871090876319} a^{7} + \frac{1835356813816197808151952758806402486122763}{34629200218524065124272181253621839817886871} a^{6} - \frac{12186296507075495793932726800507178202490286}{34629200218524065124272181253621839817886871} a^{5} - \frac{1968021657759576761517521981475697266142165}{4947028602646295017753168750517405688269553} a^{4} - \frac{4102172395895522142569364456458978876362126}{11543066739508021708090727084540613272628957} a^{3} - \frac{11064305380504016326560284287658238394389641}{23086133479016043416181454169081226545257914} a^{2} - \frac{12807492446886271981351289705450614744808621}{34629200218524065124272181253621839817886871} a + \frac{4223192697253247229885059821991321574948319}{11543066739508021708090727084540613272628957}$

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:  $10$
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:  \( 30039155053300000 \) (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

\(\Q(\sqrt{13}) \)

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ R ${\href{/LocalNumberField/17.7.0.1}{7} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.12.16.11$x^{12} + 20 x^{10} - 44 x^{8} - 4 x^{6} - 16 x^{4} - 48$$6$$2$$16$$C_3\times (C_3 : C_4)$$[2]_{3}^{6}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.12.10.3$x^{12} - 49 x^{6} + 3969$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.12.6.1$x^{12} + 338 x^{8} + 8788 x^{6} + 28561 x^{4} + 19307236$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
294467Data not computed