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Properties

Label 14.0.21030741190...4343.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,743^{13}$
Root discriminant $463.36$
Ramified prime $743$
Class number $66003$ (GRH)
Class group $[66003]$ (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![41555683709, -44249771299, 19083041153, -4014600069, 412351459, -41378938, 9890696, 549511, -716739, 100640, 2856, -963, 27, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 27*x^12 - 963*x^11 + 2856*x^10 + 100640*x^9 - 716739*x^8 + 549511*x^7 + 9890696*x^6 - 41378938*x^5 + 412351459*x^4 - 4014600069*x^3 + 19083041153*x^2 - 44249771299*x + 41555683709)
 
gp: K = bnfinit(x^14 - x^13 + 27*x^12 - 963*x^11 + 2856*x^10 + 100640*x^9 - 716739*x^8 + 549511*x^7 + 9890696*x^6 - 41378938*x^5 + 412351459*x^4 - 4014600069*x^3 + 19083041153*x^2 - 44249771299*x + 41555683709, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 27 x^{12} - 963 x^{11} + 2856 x^{10} + 100640 x^{9} - 716739 x^{8} + 549511 x^{7} + 9890696 x^{6} - 41378938 x^{5} + 412351459 x^{4} - 4014600069 x^{3} + 19083041153 x^{2} - 44249771299 x + 41555683709 \)

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:  \(-21030741190818981798961548926499574343=-\,743^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $463.36$
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:  $743$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(743\)
Dirichlet character group:    $\lbrace$$\chi_{743}(1,·)$, $\chi_{743}(742,·)$, $\chi_{743}(328,·)$, $\chi_{743}(490,·)$, $\chi_{743}(111,·)$, $\chi_{743}(592,·)$, $\chi_{743}(433,·)$, $\chi_{743}(253,·)$, $\chi_{743}(310,·)$, $\chi_{743}(151,·)$, $\chi_{743}(632,·)$, $\chi_{743}(415,·)$, $\chi_{743}(232,·)$, $\chi_{743}(511,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{21493882165061105267230208137388625343046379704238886690530557} a^{13} - \frac{7929721882992088980518709900749431988222079202461916085735}{17896654592057539772881105859607514856824629229174759942157} a^{12} + \frac{5755347893736402053888654194982575355048797499305056194150761}{21493882165061105267230208137388625343046379704238886690530557} a^{11} + \frac{1075028727556568010271536052896428854959557961054852643453159}{21493882165061105267230208137388625343046379704238886690530557} a^{10} + \frac{9848883436561989156970237601188339474798679603148502639831848}{21493882165061105267230208137388625343046379704238886690530557} a^{9} + \frac{6033997444517289140181266374546151094466493993850808791104135}{21493882165061105267230208137388625343046379704238886690530557} a^{8} + \frac{6270673236303180832027592490056204461386389108267833991711301}{21493882165061105267230208137388625343046379704238886690530557} a^{7} + \frac{374602299538199629257119255337878573895674011809168219978220}{21493882165061105267230208137388625343046379704238886690530557} a^{6} - \frac{6242637404650974882196881308162468894757326862636350645538525}{21493882165061105267230208137388625343046379704238886690530557} a^{5} - \frac{4984451617731217230951134484799136624590522644421180321455525}{21493882165061105267230208137388625343046379704238886690530557} a^{4} + \frac{7412123319662073461287093937604085569350340231395743142735032}{21493882165061105267230208137388625343046379704238886690530557} a^{3} + \frac{531265760613663859094913858106495509305755657988518291954900}{21493882165061105267230208137388625343046379704238886690530557} a^{2} + \frac{8696591417973131637731865273760075317632966458687050807222452}{21493882165061105267230208137388625343046379704238886690530557} a + \frac{4554127237415181999561518608713312204749218727721634050282478}{21493882165061105267230208137388625343046379704238886690530557}$

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

Class group and class number

$C_{66003}$, which has order $66003$ (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:  \( 63739206.972093605 \) (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

\(\Q(\sqrt{-743}) \), 7.7.168241403464173649.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
743Data not computed