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Properties

Label 14.0.12731016142...6331.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{25}\cdot 37^{7}$
Root discriminant $196.43$
Ramified primes $7, 37$
Class number $1044572$ (GRH)
Class group $[1044572]$ (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![758859649, -722816269, 684084919, -521303860, 333745951, -146975164, 40933963, -8999703, 1941534, -176134, 43729, -847, 378, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 378*x^12 - 847*x^11 + 43729*x^10 - 176134*x^9 + 1941534*x^8 - 8999703*x^7 + 40933963*x^6 - 146975164*x^5 + 333745951*x^4 - 521303860*x^3 + 684084919*x^2 - 722816269*x + 758859649)
 
gp: K = bnfinit(x^14 + 378*x^12 - 847*x^11 + 43729*x^10 - 176134*x^9 + 1941534*x^8 - 8999703*x^7 + 40933963*x^6 - 146975164*x^5 + 333745951*x^4 - 521303860*x^3 + 684084919*x^2 - 722816269*x + 758859649, 1)
 

Normalized defining polynomial

\( x^{14} + 378 x^{12} - 847 x^{11} + 43729 x^{10} - 176134 x^{9} + 1941534 x^{8} - 8999703 x^{7} + 40933963 x^{6} - 146975164 x^{5} + 333745951 x^{4} - 521303860 x^{3} + 684084919 x^{2} - 722816269 x + 758859649 \)

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:  \(-127310161428861423711034656546331=-\,7^{25}\cdot 37^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $196.43$
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:  $7, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1813=7^{2}\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{1813}(1,·)$, $\chi_{1813}(258,·)$, $\chi_{1813}(260,·)$, $\chi_{1813}(517,·)$, $\chi_{1813}(519,·)$, $\chi_{1813}(776,·)$, $\chi_{1813}(778,·)$, $\chi_{1813}(1035,·)$, $\chi_{1813}(1037,·)$, $\chi_{1813}(1294,·)$, $\chi_{1813}(1296,·)$, $\chi_{1813}(1553,·)$, $\chi_{1813}(1555,·)$, $\chi_{1813}(1812,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{31} a^{11} - \frac{3}{31} a^{10} - \frac{4}{31} a^{9} + \frac{13}{31} a^{8} + \frac{8}{31} a^{7} + \frac{14}{31} a^{6} - \frac{13}{31} a^{5} - \frac{6}{31} a^{4} + \frac{6}{31} a^{3} + \frac{2}{31} a^{2} + \frac{14}{31} a - \frac{1}{31}$, $\frac{1}{8153} a^{12} + \frac{56}{8153} a^{11} + \frac{2671}{8153} a^{10} - \frac{3416}{8153} a^{9} + \frac{89}{263} a^{8} + \frac{300}{8153} a^{7} - \frac{3248}{8153} a^{6} - \frac{1207}{8153} a^{5} + \frac{613}{8153} a^{4} + \frac{1689}{8153} a^{3} - \frac{1635}{8153} a^{2} - \frac{1810}{8153} a - \frac{1671}{8153}$, $\frac{1}{291203384660975725422946323433516545887639841336134669293} a^{13} - \frac{7976200909805792964635800919724773645125808656668109}{291203384660975725422946323433516545887639841336134669293} a^{12} - \frac{26843101891096236536035405318238029052396695709941891}{9393657569708894368482139465597307931859349720520473203} a^{11} - \frac{116440963730945396486441428702936886150047311580242431072}{291203384660975725422946323433516545887639841336134669293} a^{10} - \frac{4564457244831564914157322990861954469720985648562530770}{9393657569708894368482139465597307931859349720520473203} a^{9} - \frac{8902675533212325351496867689091627581369328133863516724}{291203384660975725422946323433516545887639841336134669293} a^{8} + \frac{1037123952695865379256328516101290016562116124124306319}{4346319174044413812282780946768903669964773751285592079} a^{7} + \frac{81286956924582189478306283965356283376578853120401518465}{291203384660975725422946323433516545887639841336134669293} a^{6} - \frac{3200881342827743113245209490412402533083675913213007803}{291203384660975725422946323433516545887639841336134669293} a^{5} + \frac{101799487846618050333878884812331234735114596616566722672}{291203384660975725422946323433516545887639841336134669293} a^{4} + \frac{14666761870352059037680814124148647083145982400272987189}{291203384660975725422946323433516545887639841336134669293} a^{3} + \frac{46253870035747507813442378708190422271745597843221568258}{291203384660975725422946323433516545887639841336134669293} a^{2} - \frac{94920716018554925996794515087953059390782708245905144328}{291203384660975725422946323433516545887639841336134669293} a + \frac{49052416954586446188620989762894269394174800329277210031}{291203384660975725422946323433516545887639841336134669293}$

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

Class group and class number

$C_{1044572}$, which has order $1044572$ (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:  \( 35256.68973693789 \) (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

\(\Q(\sqrt{-259}) \), 7.7.13841287201.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{14}$ R ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$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}$$[2]_{2}$
$37$37.14.7.1$x^{14} - 405224 x^{8} + 41051622544 x^{2} - 2373296928325$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$