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Properties

Label 14.0.84639214494...4667.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,827^{13}$
Root discriminant $511.81$
Ramified prime $827$
Class number $175616$ (GRH)
Class group $[2, 2, 2, 4, 28, 196]$ (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![52386024151, 39964632396, 8180290855, -1593603637, -740982278, 77463303, 66541559, 8336957, -148079, -104324, -800, 464, 30, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 30*x^12 + 464*x^11 - 800*x^10 - 104324*x^9 - 148079*x^8 + 8336957*x^7 + 66541559*x^6 + 77463303*x^5 - 740982278*x^4 - 1593603637*x^3 + 8180290855*x^2 + 39964632396*x + 52386024151)
 
gp: K = bnfinit(x^14 - x^13 + 30*x^12 + 464*x^11 - 800*x^10 - 104324*x^9 - 148079*x^8 + 8336957*x^7 + 66541559*x^6 + 77463303*x^5 - 740982278*x^4 - 1593603637*x^3 + 8180290855*x^2 + 39964632396*x + 52386024151, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 30 x^{12} + 464 x^{11} - 800 x^{10} - 104324 x^{9} - 148079 x^{8} + 8336957 x^{7} + 66541559 x^{6} + 77463303 x^{5} - 740982278 x^{4} - 1593603637 x^{3} + 8180290855 x^{2} + 39964632396 x + 52386024151 \)

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:  \(-84639214494706556897052354063885994667=-\,827^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $511.81$
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:  $827$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(827\)
Dirichlet character group:    $\lbrace$$\chi_{827}(1,·)$, $\chi_{827}(389,·)$, $\chi_{827}(807,·)$, $\chi_{827}(490,·)$, $\chi_{827}(427,·)$, $\chi_{827}(557,·)$, $\chi_{827}(270,·)$, $\chi_{827}(400,·)$, $\chi_{827}(337,·)$, $\chi_{827}(20,·)$, $\chi_{827}(438,·)$, $\chi_{827}(826,·)$, $\chi_{827}(124,·)$, $\chi_{827}(703,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{379135} a^{12} + \frac{35622}{379135} a^{11} + \frac{46197}{379135} a^{10} + \frac{13697}{379135} a^{9} + \frac{124518}{379135} a^{8} - \frac{136948}{379135} a^{7} - \frac{13416}{75827} a^{6} + \frac{117699}{379135} a^{5} + \frac{84536}{379135} a^{4} + \frac{33850}{75827} a^{3} - \frac{185452}{379135} a^{2} - \frac{177243}{379135} a + \frac{83294}{379135}$, $\frac{1}{355321741676949325433666434636942266228557382629529953235895} a^{13} + \frac{180384961793371013026679778738867764181830432473582796}{355321741676949325433666434636942266228557382629529953235895} a^{12} - \frac{24086132866166756845778190320639498274537260979046148402236}{71064348335389865086733286927388453245711476525905990647179} a^{11} - \frac{31980745031283260157239421974747068527839683669219074806306}{71064348335389865086733286927388453245711476525905990647179} a^{10} - \frac{156915982370245975146295767934629541515648459857557044295164}{355321741676949325433666434636942266228557382629529953235895} a^{9} - \frac{172218118418117943407253967425284491288499056087571893757546}{355321741676949325433666434636942266228557382629529953235895} a^{8} - \frac{174261858024562740993705563180433156703378624829199170525952}{355321741676949325433666434636942266228557382629529953235895} a^{7} + \frac{123732765196689012873811733766229221523053874120799174468299}{355321741676949325433666434636942266228557382629529953235895} a^{6} + \frac{176324920772398293150947127466552304574357342061514925809637}{355321741676949325433666434636942266228557382629529953235895} a^{5} - \frac{78344712699039388887584119036120385050582519059044035531216}{355321741676949325433666434636942266228557382629529953235895} a^{4} - \frac{100112025425968086894367194540770792426325827739534794935112}{355321741676949325433666434636942266228557382629529953235895} a^{3} - \frac{15246384226820085536081780539935323329728853164315755961506}{355321741676949325433666434636942266228557382629529953235895} a^{2} - \frac{164786454480840841001624872111382458781775308899359233093063}{355321741676949325433666434636942266228557382629529953235895} a - \frac{133918189516950414965636523579847607658457024869688714445379}{355321741676949325433666434636942266228557382629529953235895}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{28}\times C_{196}$, which has order $175616$ (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:  \( 48540426.16205039 \) (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

\(\Q(\sqrt{-827}) \), 7.7.319913861015774089.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
827Data not computed