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Properties

Label 14.0.40337227579...0279.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{7}\cdot 113^{13}$
Root discriminant $213.29$
Ramified primes $7, 113$
Class number $59392$ (GRH)
Class group $[4, 4, 4, 928]$ (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![401609231, -44623278, 212182491, -30681299, 33375503, -7462084, 2331639, -755470, 139645, -29768, 8592, -195, 174, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 174*x^12 - 195*x^11 + 8592*x^10 - 29768*x^9 + 139645*x^8 - 755470*x^7 + 2331639*x^6 - 7462084*x^5 + 33375503*x^4 - 30681299*x^3 + 212182491*x^2 - 44623278*x + 401609231)
 
gp: K = bnfinit(x^14 - x^13 + 174*x^12 - 195*x^11 + 8592*x^10 - 29768*x^9 + 139645*x^8 - 755470*x^7 + 2331639*x^6 - 7462084*x^5 + 33375503*x^4 - 30681299*x^3 + 212182491*x^2 - 44623278*x + 401609231, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 174 x^{12} - 195 x^{11} + 8592 x^{10} - 29768 x^{9} + 139645 x^{8} - 755470 x^{7} + 2331639 x^{6} - 7462084 x^{5} + 33375503 x^{4} - 30681299 x^{3} + 212182491 x^{2} - 44623278 x + 401609231 \)

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:  \(-403372275797631336682377398030279=-\,7^{7}\cdot 113^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $213.29$
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:  $7, 113$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(791=7\cdot 113\)
Dirichlet character group:    $\lbrace$$\chi_{791}(1,·)$, $\chi_{791}(162,·)$, $\chi_{791}(708,·)$, $\chi_{791}(694,·)$, $\chi_{791}(230,·)$, $\chi_{791}(97,·)$, $\chi_{791}(106,·)$, $\chi_{791}(141,·)$, $\chi_{791}(685,·)$, $\chi_{791}(561,·)$, $\chi_{791}(83,·)$, $\chi_{791}(629,·)$, $\chi_{791}(790,·)$, $\chi_{791}(650,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{281668520004835275272705459064378110203362656806341967} a^{13} - \frac{25116276994565449230442037104417400455058900030011972}{281668520004835275272705459064378110203362656806341967} a^{12} + \frac{75484509371253108100052878387578290185043668855650599}{281668520004835275272705459064378110203362656806341967} a^{11} - \frac{6321909327254516455214692281540852527185241689320879}{281668520004835275272705459064378110203362656806341967} a^{10} - \frac{123382074329486485956063846836182132499497657430717822}{281668520004835275272705459064378110203362656806341967} a^{9} + \frac{17891328691432836110076900102087233550758584503576668}{281668520004835275272705459064378110203362656806341967} a^{8} - \frac{130860372962716755605669047481523653737504162900847360}{281668520004835275272705459064378110203362656806341967} a^{7} + \frac{18237141304248982523673803617813951129695580942183489}{281668520004835275272705459064378110203362656806341967} a^{6} - \frac{134921995185089252512853123613088160342675839131876492}{281668520004835275272705459064378110203362656806341967} a^{5} - \frac{109687254499135394032637809370241379159991003940189060}{281668520004835275272705459064378110203362656806341967} a^{4} + \frac{5201543187595319961990960776612585851272443019408875}{281668520004835275272705459064378110203362656806341967} a^{3} + \frac{3553592537274645419418620267076295072217494548795420}{281668520004835275272705459064378110203362656806341967} a^{2} + \frac{140167619629391830919388663524828501008949779356081528}{281668520004835275272705459064378110203362656806341967} a + \frac{19705648091807962822033983707607213795494293078892862}{281668520004835275272705459064378110203362656806341967}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{4}\times C_{928}$, which has order $59392$ (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:  \( 222748.97284811488 \) (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

\(\Q(\sqrt{-791}) \), 7.7.2081951752609.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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$7.14.7.1$x^{14} - 117649 x^{2} + 1647086$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$113$113.14.13.1$x^{14} - 113$$14$$1$$13$$C_{14}$$[\ ]_{14}$