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Properties

Label 14.0.28341078543...2176.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{21}\cdot 7^{7}\cdot 71^{12}$
Root discriminant $288.99$
Ramified primes $2, 7, 71$
Class number $7568372$ (GRH)
Class group $[7568372]$ (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![2163359375, 349151250, 395200525, 34059520, 31057338, 479562, 1547230, -46196, 65883, -1600, 2254, -114, 39, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 39*x^12 - 114*x^11 + 2254*x^10 - 1600*x^9 + 65883*x^8 - 46196*x^7 + 1547230*x^6 + 479562*x^5 + 31057338*x^4 + 34059520*x^3 + 395200525*x^2 + 349151250*x + 2163359375)
 
gp: K = bnfinit(x^14 - 2*x^13 + 39*x^12 - 114*x^11 + 2254*x^10 - 1600*x^9 + 65883*x^8 - 46196*x^7 + 1547230*x^6 + 479562*x^5 + 31057338*x^4 + 34059520*x^3 + 395200525*x^2 + 349151250*x + 2163359375, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} + 39 x^{12} - 114 x^{11} + 2254 x^{10} - 1600 x^{9} + 65883 x^{8} - 46196 x^{7} + 1547230 x^{6} + 479562 x^{5} + 31057338 x^{4} + 34059520 x^{3} + 395200525 x^{2} + 349151250 x + 2163359375 \)

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:  \(-28341078543880537818112953392562176=-\,2^{21}\cdot 7^{7}\cdot 71^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $288.99$
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:  $2, 7, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3976=2^{3}\cdot 7\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{3976}(897,·)$, $\chi_{3976}(2309,·)$, $\chi_{3976}(1,·)$, $\chi_{3976}(517,·)$, $\chi_{3976}(1805,·)$, $\chi_{3976}(3653,·)$, $\chi_{3976}(1681,·)$, $\chi_{3976}(1749,·)$, $\chi_{3976}(853,·)$, $\chi_{3976}(1457,·)$, $\chi_{3976}(953,·)$, $\chi_{3976}(2801,·)$, $\chi_{3976}(3641,·)$, $\chi_{3976}(2533,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{4}$, $\frac{1}{25} a^{9} + \frac{1}{25} a^{8} + \frac{1}{25} a^{7} + \frac{1}{25} a^{6} - \frac{1}{25} a^{5} + \frac{9}{25} a^{4} + \frac{9}{25} a^{3} + \frac{9}{25} a^{2} + \frac{2}{5} a$, $\frac{1}{25} a^{10} - \frac{2}{25} a^{6} + \frac{1}{25} a^{2}$, $\frac{1}{25} a^{11} - \frac{2}{25} a^{7} + \frac{1}{25} a^{3}$, $\frac{1}{2125} a^{12} + \frac{36}{2125} a^{11} + \frac{12}{2125} a^{10} + \frac{1}{125} a^{9} - \frac{31}{425} a^{8} - \frac{19}{425} a^{7} + \frac{208}{2125} a^{6} + \frac{53}{2125} a^{5} + \frac{999}{2125} a^{4} - \frac{1046}{2125} a^{3} + \frac{1}{17} a^{2} + \frac{13}{85} a + \frac{8}{17}$, $\frac{1}{9430067864540801943453152318862293125} a^{13} - \frac{1478431134123157038978569093246222}{9430067864540801943453152318862293125} a^{12} - \frac{78490459870494641287697413110894531}{9430067864540801943453152318862293125} a^{11} - \frac{75568600557997116888986861565673804}{9430067864540801943453152318862293125} a^{10} + \frac{47849235270775050990710225201081339}{9430067864540801943453152318862293125} a^{9} + \frac{21043277176519738606461595066780043}{377202714581632077738126092754491725} a^{8} + \frac{398129693781898264532792935727966783}{9430067864540801943453152318862293125} a^{7} - \frac{557776671441884063195977011910533656}{9430067864540801943453152318862293125} a^{6} - \frac{95524777431045945600840694629818501}{1886013572908160388690630463772458625} a^{5} - \frac{2269861635817608845503929786953875793}{9430067864540801943453152318862293125} a^{4} - \frac{1687775297835293926474014367814108267}{9430067864540801943453152318862293125} a^{3} - \frac{290160209060451756383422786330894021}{1886013572908160388690630463772458625} a^{2} - \frac{46966472380047430227036058492332424}{377202714581632077738126092754491725} a - \frac{7102938590422184687288126648170134}{15088108583265283109525043710179669}$

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

Class group and class number

$C_{7568372}$, which has order $7568372$ (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:  \( 315114.6966253571 \) (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

\(\Q(\sqrt{-14}) \), 7.7.128100283921.1

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 R ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{14}$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$2$2.14.21.34$x^{14} + 4 x^{13} + 8 x^{12} + 4 x^{11} + 5 x^{10} + 8 x^{9} - 6 x^{8} - 6 x^{7} + x^{6} + 6 x^{5} + 2 x^{3} + 7 x^{2} + 6 x - 7$$2$$7$$21$$C_{14}$$[3]^{7}$
$7$7.14.7.1$x^{14} - 117649 x^{2} + 1647086$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$71$71.7.6.1$x^{7} - 71$$7$$1$$6$$C_7$$[\ ]_{7}$
71.7.6.1$x^{7} - 71$$7$$1$$6$$C_7$$[\ ]_{7}$