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Properties

Label 14.0.17531246090...1875.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 5^{7}\cdot 29^{13}$
Root discriminant $88.31$
Ramified primes $3, 5, 29$
Class number $7424$ (GRH)
Class group $[2, 2, 4, 4, 116]$ (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![3832639, -5241351, 5495644, -1801888, 1099322, -695198, 283946, -69132, 50063, -3651, 3546, -104, 103, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 103*x^12 - 104*x^11 + 3546*x^10 - 3651*x^9 + 50063*x^8 - 69132*x^7 + 283946*x^6 - 695198*x^5 + 1099322*x^4 - 1801888*x^3 + 5495644*x^2 - 5241351*x + 3832639)
 
gp: K = bnfinit(x^14 - x^13 + 103*x^12 - 104*x^11 + 3546*x^10 - 3651*x^9 + 50063*x^8 - 69132*x^7 + 283946*x^6 - 695198*x^5 + 1099322*x^4 - 1801888*x^3 + 5495644*x^2 - 5241351*x + 3832639, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 103 x^{12} - 104 x^{11} + 3546 x^{10} - 3651 x^{9} + 50063 x^{8} - 69132 x^{7} + 283946 x^{6} - 695198 x^{5} + 1099322 x^{4} - 1801888 x^{3} + 5495644 x^{2} - 5241351 x + 3832639 \)

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:  \(-1753124609003161170886171875=-\,3^{7}\cdot 5^{7}\cdot 29^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.31$
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:  $3, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(435=3\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{435}(256,·)$, $\chi_{435}(1,·)$, $\chi_{435}(226,·)$, $\chi_{435}(419,·)$, $\chi_{435}(136,·)$, $\chi_{435}(299,·)$, $\chi_{435}(16,·)$, $\chi_{435}(209,·)$, $\chi_{435}(434,·)$, $\chi_{435}(179,·)$, $\chi_{435}(149,·)$, $\chi_{435}(286,·)$, $\chi_{435}(254,·)$, $\chi_{435}(181,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{389} a^{12} + \frac{194}{389} a^{11} - \frac{87}{389} a^{10} - \frac{4}{389} a^{9} + \frac{116}{389} a^{8} - \frac{111}{389} a^{7} + \frac{183}{389} a^{6} - \frac{34}{389} a^{5} - \frac{47}{389} a^{4} + \frac{149}{389} a^{3} + \frac{151}{389} a^{2} - \frac{132}{389} a + \frac{17}{389}$, $\frac{1}{385966054973843044100668475176334603530619} a^{13} + \frac{198930711274255779348144515478883878417}{385966054973843044100668475176334603530619} a^{12} - \frac{173301320146143312480869537474343316995153}{385966054973843044100668475176334603530619} a^{11} - \frac{101680277031260824070602353248872638864043}{385966054973843044100668475176334603530619} a^{10} - \frac{175123274834820584259645725248528740863631}{385966054973843044100668475176334603530619} a^{9} - \frac{118054098557022703825319027652847517406589}{385966054973843044100668475176334603530619} a^{8} - \frac{98667754647322045551244778807503442176601}{385966054973843044100668475176334603530619} a^{7} - \frac{4662470437590699067209430472831489071880}{9413806218874220587821182321374014720259} a^{6} + \frac{10840268652883345740901870049997881124832}{385966054973843044100668475176334603530619} a^{5} + \frac{55077561553031651417220558926417478683609}{385966054973843044100668475176334603530619} a^{4} + \frac{111080832941249359772416929559409567023090}{385966054973843044100668475176334603530619} a^{3} + \frac{91942558836099944507929628328511913048914}{385966054973843044100668475176334603530619} a^{2} - \frac{158535262321230753216570906262250712159910}{385966054973843044100668475176334603530619} a + \frac{2243196992003415395258201079128148343880}{9413806218874220587821182321374014720259}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{116}$, which has order $7424$ (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:  \( 6020.985100147561 \) (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

\(\Q(\sqrt{-435}) \), 7.7.594823321.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ R R ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$3$3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$5$5.14.7.2$x^{14} - 15625 x^{2} + 156250$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.14.13.11$x^{14} + 3712$$14$$1$$13$$C_{14}$$[\ ]_{14}$