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Properties

Label 14.0.13605556779...6247.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,23^{7}\cdot 43^{12}$
Root discriminant $120.50$
Ramified primes $23, 43$
Class number $7899$ (GRH)
Class group $[7899]$ (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![7742617, 2157359, 3152636, 323699, 605069, 24230, 79540, -4227, 6620, -682, 512, -75, 15, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 + 15*x^12 - 75*x^11 + 512*x^10 - 682*x^9 + 6620*x^8 - 4227*x^7 + 79540*x^6 + 24230*x^5 + 605069*x^4 + 323699*x^3 + 3152636*x^2 + 2157359*x + 7742617)
 
gp: K = bnfinit(x^14 - 5*x^13 + 15*x^12 - 75*x^11 + 512*x^10 - 682*x^9 + 6620*x^8 - 4227*x^7 + 79540*x^6 + 24230*x^5 + 605069*x^4 + 323699*x^3 + 3152636*x^2 + 2157359*x + 7742617, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} + 15 x^{12} - 75 x^{11} + 512 x^{10} - 682 x^{9} + 6620 x^{8} - 4227 x^{7} + 79540 x^{6} + 24230 x^{5} + 605069 x^{4} + 323699 x^{3} + 3152636 x^{2} + 2157359 x + 7742617 \)

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:  \(-136055567791244518070906476247=-\,23^{7}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $120.50$
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:  $23, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(989=23\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{989}(1,·)$, $\chi_{989}(643,·)$, $\chi_{989}(484,·)$, $\chi_{989}(551,·)$, $\chi_{989}(231,·)$, $\chi_{989}(47,·)$, $\chi_{989}(944,·)$, $\chi_{989}(689,·)$, $\chi_{989}(852,·)$, $\chi_{989}(967,·)$, $\chi_{989}(919,·)$, $\chi_{989}(666,·)$, $\chi_{989}(183,·)$, $\chi_{989}(735,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{9} + \frac{1}{7} a^{8} + \frac{1}{7} a^{6} - \frac{2}{7} a^{5} + \frac{2}{7} a^{4} - \frac{1}{7} a^{3} + \frac{1}{7} a^{2} - \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{11} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} - \frac{1}{7} a^{6} + \frac{1}{7} a^{4} - \frac{2}{7} a^{2} + \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{12} + \frac{1}{7} a^{9} + \frac{1}{7} a^{8} - \frac{1}{7} a^{7} + \frac{1}{7} a^{5} - \frac{2}{7} a^{3} + \frac{3}{7} a^{2} - \frac{1}{7} a$, $\frac{1}{561500671985851382906695774393333} a^{13} + \frac{20349145735541512301847375748553}{561500671985851382906695774393333} a^{12} - \frac{71829090040691733892962701746}{80214381712264483272385110627619} a^{11} + \frac{2759519346421436073555511249534}{561500671985851382906695774393333} a^{10} - \frac{148691603022091229748161332192601}{561500671985851382906695774393333} a^{9} - \frac{257299274835507504099169794234687}{561500671985851382906695774393333} a^{8} + \frac{149548727390520508428547044327392}{561500671985851382906695774393333} a^{7} + \frac{270405017605868781194086651326835}{561500671985851382906695774393333} a^{6} - \frac{179693787939131494614812935229393}{561500671985851382906695774393333} a^{5} + \frac{32907212582709195798967655242285}{80214381712264483272385110627619} a^{4} - \frac{134876354957357297989860258976088}{561500671985851382906695774393333} a^{3} - \frac{84143135598569272708179407946384}{561500671985851382906695774393333} a^{2} + \frac{20277714167975725554057495622186}{80214381712264483272385110627619} a + \frac{89574473398554143486283182785314}{561500671985851382906695774393333}$

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

Class group and class number

$C_{7899}$, which has order $7899$ (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:  \( 35991.64185055774 \) (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

\(\Q(\sqrt{-23}) \), 7.7.6321363049.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ R ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$23$23.14.7.2$x^{14} - 148035889 x^{2} + 27238603576$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$43$43.14.12.1$x^{14} + 3569 x^{7} + 4043763$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$