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Properties

Label 14.0.53588606915...5607.4
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{25}\cdot 43^{12}$
Root discriminant $811.37$
Ramified primes $7, 43$
Class number $942613$ (GRH)
Class group $[7, 134659]$ (GRH)
Galois group $C_{14}$ (as 14T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![62055692847703, 17279409997031, 7166415442578, 1267459647895, 262979976931, 29686323660, 3646509048, 215168302, 13779479, -556850, -50267, -7224, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7224*x^11 - 50267*x^10 - 556850*x^9 + 13779479*x^8 + 215168302*x^7 + 3646509048*x^6 + 29686323660*x^5 + 262979976931*x^4 + 1267459647895*x^3 + 7166415442578*x^2 + 17279409997031*x + 62055692847703)
 
gp: K = bnfinit(x^14 - 7224*x^11 - 50267*x^10 - 556850*x^9 + 13779479*x^8 + 215168302*x^7 + 3646509048*x^6 + 29686323660*x^5 + 262979976931*x^4 + 1267459647895*x^3 + 7166415442578*x^2 + 17279409997031*x + 62055692847703, 1)
 

Normalized defining polynomial

\( x^{14} - 7224 x^{11} - 50267 x^{10} - 556850 x^{9} + 13779479 x^{8} + 215168302 x^{7} + 3646509048 x^{6} + 29686323660 x^{5} + 262979976931 x^{4} + 1267459647895 x^{3} + 7166415442578 x^{2} + 17279409997031 x + 62055692847703 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-53588606915566584613059849086223224055607=-\,7^{25}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $811.37$
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:  $7, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2107=7^{2}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{2107}(64,·)$, $\chi_{2107}(1,·)$, $\chi_{2107}(1282,·)$, $\chi_{2107}(1989,·)$, $\chi_{2107}(1000,·)$, $\chi_{2107}(1420,·)$, $\chi_{2107}(428,·)$, $\chi_{2107}(876,·)$, $\chi_{2107}(944,·)$, $\chi_{2107}(2099,·)$, $\chi_{2107}(790,·)$, $\chi_{2107}(279,·)$, $\chi_{2107}(1595,·)$, $\chi_{2107}(1982,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{43} a^{7}$, $\frac{1}{43} a^{8}$, $\frac{1}{43} a^{9}$, $\frac{1}{1849} a^{10} - \frac{8}{43} a^{6} - \frac{7}{43} a^{5} + \frac{17}{43} a^{4} + \frac{4}{43} a^{3}$, $\frac{1}{1849} a^{11} - \frac{7}{43} a^{6} + \frac{17}{43} a^{5} + \frac{4}{43} a^{4}$, $\frac{1}{2305703} a^{12} + \frac{10}{53621} a^{11} + \frac{4}{53621} a^{10} - \frac{168}{53621} a^{9} - \frac{524}{53621} a^{8} + \frac{122}{53621} a^{7} + \frac{9133}{53621} a^{6} + \frac{14022}{53621} a^{5} + \frac{194}{1247} a^{4} + \frac{403}{1247} a^{3} - \frac{14}{29} a^{2} + \frac{6}{29} a$, $\frac{1}{256189314510912092928476801717033814137138204568383205197359} a^{13} - \frac{43945392586784993576189818276698039760473642609501222}{256189314510912092928476801717033814137138204568383205197359} a^{12} + \frac{1181191593691300873123765857778291171088226249014456968}{5957891035137490533220390737605437538072981501590307097613} a^{11} + \frac{1512247964026348674115908715777891318460070096772374855}{5957891035137490533220390737605437538072981501590307097613} a^{10} - \frac{43199388984974467287925414391796108171413827543181901522}{5957891035137490533220390737605437538072981501590307097613} a^{9} + \frac{62070310174906807254600485894549172909488774642002893879}{5957891035137490533220390737605437538072981501590307097613} a^{8} + \frac{17954204613947564913914779392286805650604593173110776313}{5957891035137490533220390737605437538072981501590307097613} a^{7} - \frac{2656140983031982903889778457828233895303435372391822136538}{5957891035137490533220390737605437538072981501590307097613} a^{6} + \frac{916372129775773954126756298124928073078024597557570044673}{5957891035137490533220390737605437538072981501590307097613} a^{5} - \frac{31496826915223757880902708060950514270465208491363295443}{138555605468313733330706761339661338094720500036983885991} a^{4} - \frac{6078208027149069965258698805161837088753936023208686806}{138555605468313733330706761339661338094720500036983885991} a^{3} - \frac{1346055263265016303907887905989848323500417730914685605}{3222223382984040310016436310224682281272569768301950837} a^{2} + \frac{260364352291118843415990312919556124753497908922656974}{3222223382984040310016436310224682281272569768301950837} a - \frac{470821847730536424490168426248162930868218345950732}{983284523339652215445967748008752603378873899390281}$

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

Class group and class number

$C_{7}\times C_{134659}$, which has order $942613$ (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:  $6$
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:  \( 534504501.8334233 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{14}$ (as 14T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 14
The 14 conjugacy class representatives for $C_{14}$
Character table for $C_{14}$

Intermediate fields

\(\Q(\sqrt{-7}) \), 7.7.87495801462998035849.4

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ R ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$7$7.14.25.59$x^{14} - 168 x^{13} + 70 x^{12} - 147 x^{11} + 147 x^{10} - 98 x^{9} + 49 x^{8} + 168 x^{7} - 49 x^{4} - 147 x^{3} - 49 x^{2} + 98 x + 126$$14$$1$$25$$C_{14}$$[2]_{2}$
$43$43.7.6.7$x^{7} - 22851963$$7$$1$$6$$C_7$$[\ ]_{7}$
43.7.6.7$x^{7} - 22851963$$7$$1$$6$$C_7$$[\ ]_{7}$