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Properties

Label 14.0.64132141729...7703.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{7}\cdot 211^{12}$
Root discriminant $259.89$
Ramified primes $7, 211$
Class number $44303$ (GRH)
Class group $[44303]$ (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![73732261, 85068297, 52095720, 1156817, -9544515, -3813892, 1063886, 542193, -32628, -54802, 7020, 973, -157, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 - 157*x^12 + 973*x^11 + 7020*x^10 - 54802*x^9 - 32628*x^8 + 542193*x^7 + 1063886*x^6 - 3813892*x^5 - 9544515*x^4 + 1156817*x^3 + 52095720*x^2 + 85068297*x + 73732261)
 
gp: K = bnfinit(x^14 - 5*x^13 - 157*x^12 + 973*x^11 + 7020*x^10 - 54802*x^9 - 32628*x^8 + 542193*x^7 + 1063886*x^6 - 3813892*x^5 - 9544515*x^4 + 1156817*x^3 + 52095720*x^2 + 85068297*x + 73732261, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} - 157 x^{12} + 973 x^{11} + 7020 x^{10} - 54802 x^{9} - 32628 x^{8} + 542193 x^{7} + 1063886 x^{6} - 3813892 x^{5} - 9544515 x^{4} + 1156817 x^{3} + 52095720 x^{2} + 85068297 x + 73732261 \)

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:  \(-6413214172956389772541350262917703=-\,7^{7}\cdot 211^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $259.89$
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:  $7, 211$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1477=7\cdot 211\)
Dirichlet character group:    $\lbrace$$\chi_{1477}(1056,·)$, $\chi_{1477}(1,·)$, $\chi_{1477}(804,·)$, $\chi_{1477}(902,·)$, $\chi_{1477}(1254,·)$, $\chi_{1477}(545,·)$, $\chi_{1477}(1226,·)$, $\chi_{1477}(967,·)$, $\chi_{1477}(1324,·)$, $\chi_{1477}(1203,·)$, $\chi_{1477}(148,·)$, $\chi_{1477}(566,·)$, $\chi_{1477}(988,·)$, $\chi_{1477}(832,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{23} a^{9} - \frac{1}{23} a^{8} + \frac{4}{23} a^{7} + \frac{3}{23} a^{6} + \frac{4}{23} a^{5} - \frac{9}{23} a^{4} - \frac{7}{23} a^{3} - \frac{4}{23} a^{2} + \frac{4}{23} a - \frac{6}{23}$, $\frac{1}{437} a^{10} + \frac{6}{437} a^{9} + \frac{43}{437} a^{8} - \frac{2}{23} a^{7} + \frac{117}{437} a^{6} + \frac{1}{23} a^{5} + \frac{183}{437} a^{4} - \frac{168}{437} a^{3} + \frac{137}{437} a^{2} - \frac{185}{437} a - \frac{203}{437}$, $\frac{1}{59869} a^{11} + \frac{16}{59869} a^{10} + \frac{1034}{59869} a^{9} + \frac{23496}{59869} a^{8} + \frac{27496}{59869} a^{7} + \frac{8789}{59869} a^{6} - \frac{21686}{59869} a^{5} + \frac{24310}{59869} a^{4} + \frac{8546}{59869} a^{3} + \frac{26303}{59869} a^{2} + \frac{23084}{59869} a + \frac{11612}{59869}$, $\frac{1}{1137511} a^{12} - \frac{9}{1137511} a^{11} + \frac{771}{1137511} a^{10} - \frac{11944}{1137511} a^{9} - \frac{423863}{1137511} a^{8} - \frac{246513}{1137511} a^{7} - \frac{196749}{1137511} a^{6} - \frac{430489}{1137511} a^{5} + \frac{297872}{1137511} a^{4} - \frac{197348}{1137511} a^{3} + \frac{204223}{1137511} a^{2} - \frac{512743}{1137511} a - \frac{435246}{1137511}$, $\frac{1}{4398185456937173580085739594353799} a^{13} - \frac{708597886760637419578309647}{4398185456937173580085739594353799} a^{12} + \frac{16901440114893995352989804165}{4398185456937173580085739594353799} a^{11} + \frac{52593429546585702700273508190}{231483445101956504215038926018621} a^{10} - \frac{35097617309908084027455493762620}{4398185456937173580085739594353799} a^{9} - \frac{59252676647488697192364298330562}{231483445101956504215038926018621} a^{8} + \frac{2058599544748064435029298805355361}{4398185456937173580085739594353799} a^{7} - \frac{71602291491883446613206788779982}{231483445101956504215038926018621} a^{6} + \frac{1128359817220099246401682357923039}{4398185456937173580085739594353799} a^{5} + \frac{101095756027359204872582260384544}{4398185456937173580085739594353799} a^{4} - \frac{1683958596313664814862757974691384}{4398185456937173580085739594353799} a^{3} + \frac{1225902285013575015494523019587063}{4398185456937173580085739594353799} a^{2} - \frac{237976145592082963506862710606381}{4398185456937173580085739594353799} a + \frac{1539736817714672186860098920263973}{4398185456937173580085739594353799}$

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

Class group and class number

$C_{44303}$, which has order $44303$ (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:  \( 2943138.716633699 \) (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

\(\Q(\sqrt{-7}) \), 7.7.88245939632761.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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$7.14.7.1$x^{14} - 117649 x^{2} + 1647086$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
211Data not computed