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Properties

Label 14.0.47448990293...6087.5
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{25}\cdot 29^{12}$
Root discriminant $578.88$
Ramified primes $7, 29$
Class number $452123$ (GRH)
Class group $[7, 64589]$ (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![3854284665359, 251900249993, -423674050058, -19413083053, 19700239139, 329836836, -437392326, -4215266, 5325705, 91350, -25375, -812, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 812*x^11 - 25375*x^10 + 91350*x^9 + 5325705*x^8 - 4215266*x^7 - 437392326*x^6 + 329836836*x^5 + 19700239139*x^4 - 19413083053*x^3 - 423674050058*x^2 + 251900249993*x + 3854284665359)
 
gp: K = bnfinit(x^14 - 812*x^11 - 25375*x^10 + 91350*x^9 + 5325705*x^8 - 4215266*x^7 - 437392326*x^6 + 329836836*x^5 + 19700239139*x^4 - 19413083053*x^3 - 423674050058*x^2 + 251900249993*x + 3854284665359, 1)
 

Normalized defining polynomial

\( x^{14} - 812 x^{11} - 25375 x^{10} + 91350 x^{9} + 5325705 x^{8} - 4215266 x^{7} - 437392326 x^{6} + 329836836 x^{5} + 19700239139 x^{4} - 19413083053 x^{3} - 423674050058 x^{2} + 251900249993 x + 3854284665359 \)

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:  \(-474489902930063357496193840307474416087=-\,7^{25}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $578.88$
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:  $7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1421=7^{2}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{1421}(1,·)$, $\chi_{1421}(1154,·)$, $\chi_{1421}(547,·)$, $\chi_{1421}(132,·)$, $\chi_{1421}(1238,·)$, $\chi_{1421}(806,·)$, $\chi_{1421}(239,·)$, $\chi_{1421}(146,·)$, $\chi_{1421}(372,·)$, $\chi_{1421}(790,·)$, $\chi_{1421}(281,·)$, $\chi_{1421}(314,·)$, $\chi_{1421}(286,·)$, $\chi_{1421}(799,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{29} a^{7}$, $\frac{1}{29} a^{8}$, $\frac{1}{29} a^{9}$, $\frac{1}{841} a^{10} - \frac{5}{29} a^{6} - \frac{11}{29} a^{5} - \frac{12}{29} a^{4} - \frac{6}{29} a^{3}$, $\frac{1}{841} a^{11} - \frac{11}{29} a^{6} - \frac{12}{29} a^{5} - \frac{6}{29} a^{4}$, $\frac{1}{24389} a^{12} + \frac{1}{841} a^{9} - \frac{5}{841} a^{8} - \frac{11}{841} a^{7} + \frac{307}{841} a^{6} + \frac{139}{841} a^{5}$, $\frac{1}{195142919051373439279459117932797288748921520762843846896001} a^{13} - \frac{241650040653053868685418047625830583436665150687267351}{195142919051373439279459117932797288748921520762843846896001} a^{12} - \frac{1180019245301788668280285171991919872575672670956614433}{6729066174185291009636521308027492715480052440098063686069} a^{11} - \frac{3847614554251987088712530930130786020949931481097284775}{6729066174185291009636521308027492715480052440098063686069} a^{10} + \frac{63508228147056035633903462671418453873719586075908053}{232036764627079000332293838207844576395863877244760816761} a^{9} + \frac{105012288111530895259801538225161547815660325063072703387}{6729066174185291009636521308027492715480052440098063686069} a^{8} - \frac{114146814622721771417985738366345111979772939637658172401}{6729066174185291009636521308027492715480052440098063686069} a^{7} - \frac{2043802973590776686706448448889570956539616224623373764009}{6729066174185291009636521308027492715480052440098063686069} a^{6} - \frac{3209634257232915462196193300034468117421783581172759119529}{6729066174185291009636521308027492715480052440098063686069} a^{5} + \frac{24232422628305456036888780215799916349538133107627151740}{232036764627079000332293838207844576395863877244760816761} a^{4} + \frac{102437154463867486571455305620476922556664502133233794068}{232036764627079000332293838207844576395863877244760816761} a^{3} - \frac{1993487903392135506751143737445174726084348325608462942}{8001267745761344839044615110615330220547030249819338509} a^{2} - \frac{2268256460482049187720690328822144940192688580097861697}{8001267745761344839044615110615330220547030249819338509} a - \frac{3122413072091369347002948647494264916835785666287735627}{8001267745761344839044615110615330220547030249819338509}$

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

Class group and class number

$C_{7}\times C_{64589}$, which has order $452123$ (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:  \( 78882035.29112078 \) (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

\(\Q(\sqrt{-7}) \), 7.7.8233120419813614521.4

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 ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{14}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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$7.14.25.89$x^{14} - 14 x^{13} + 21 x^{12} + 14 x^{7} - 21$$14$$1$$25$$C_{14}$$[2]_{2}$
$29$29.7.6.4$x^{7} - 1856$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.4$x^{7} - 1856$$7$$1$$6$$C_7$$[\ ]_{7}$