↑

Properties

Label 14.0.96396085363...8171.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,491^{13}$
Root discriminant $315.40$
Ramified prime $491$
Class number $133632$ (GRH)
Class group $[2, 2, 2, 4, 4, 1044]$ (GRH)
Galois group $C_{14}$ (as 14T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![979862339, 1080367394, 734418581, 364486968, 159906733, 56179943, 13603571, 1847851, 63369, -26794, -4914, -426, 18, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 18*x^12 - 426*x^11 - 4914*x^10 - 26794*x^9 + 63369*x^8 + 1847851*x^7 + 13603571*x^6 + 56179943*x^5 + 159906733*x^4 + 364486968*x^3 + 734418581*x^2 + 1080367394*x + 979862339)
 
gp: K = bnfinit(x^14 - x^13 + 18*x^12 - 426*x^11 - 4914*x^10 - 26794*x^9 + 63369*x^8 + 1847851*x^7 + 13603571*x^6 + 56179943*x^5 + 159906733*x^4 + 364486968*x^3 + 734418581*x^2 + 1080367394*x + 979862339, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 18 x^{12} - 426 x^{11} - 4914 x^{10} - 26794 x^{9} + 63369 x^{8} + 1847851 x^{7} + 13603571 x^{6} + 56179943 x^{5} + 159906733 x^{4} + 364486968 x^{3} + 734418581 x^{2} + 1080367394 x + 979862339 \)

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:  \(-96396085363089927687061485995798171=-\,491^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $315.40$
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:  $491$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(491\)
Dirichlet character group:    $\lbrace$$\chi_{491}(1,·)$, $\chi_{491}(386,·)$, $\chi_{491}(353,·)$, $\chi_{491}(159,·)$, $\chi_{491}(105,·)$, $\chi_{491}(138,·)$, $\chi_{491}(268,·)$, $\chi_{491}(240,·)$, $\chi_{491}(338,·)$, $\chi_{491}(332,·)$, $\chi_{491}(153,·)$, $\chi_{491}(251,·)$, $\chi_{491}(490,·)$, $\chi_{491}(223,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{41} a^{12} - \frac{16}{41} a^{11} + \frac{11}{41} a^{10} - \frac{1}{41} a^{9} + \frac{10}{41} a^{8} - \frac{6}{41} a^{7} - \frac{19}{41} a^{6} - \frac{15}{41} a^{5} + \frac{15}{41} a^{4} + \frac{16}{41} a^{3} - \frac{14}{41} a + \frac{18}{41}$, $\frac{1}{214991445195987433587116827589012947229383519062960551} a^{13} + \frac{81264590574785247415319166978101438353886709126853}{214991445195987433587116827589012947229383519062960551} a^{12} + \frac{12844568170542644070070481273860657614244834715999154}{214991445195987433587116827589012947229383519062960551} a^{11} - \frac{751692085147860108592457795450472419774313677179838}{5243693785267986185051629941195437737302037050316111} a^{10} - \frac{44912478692626255823202177922141096818406055741886446}{214991445195987433587116827589012947229383519062960551} a^{9} + \frac{49966652179153723554491160706552336172068435504169857}{214991445195987433587116827589012947229383519062960551} a^{8} + \frac{55306507836293378845179393448983747963904266446124603}{214991445195987433587116827589012947229383519062960551} a^{7} + \frac{53165508912504766054147301549713849107597827343337749}{214991445195987433587116827589012947229383519062960551} a^{6} + \frac{59272410191325475082298479799163345844349849270709403}{214991445195987433587116827589012947229383519062960551} a^{5} + \frac{53044703487012725454387810192215734606128650653820122}{214991445195987433587116827589012947229383519062960551} a^{4} + \frac{59231330855637394415219132011736783982364139286704944}{214991445195987433587116827589012947229383519062960551} a^{3} - \frac{93505184866608967699850987769583926782075508493451833}{214991445195987433587116827589012947229383519062960551} a^{2} - \frac{12997413845546266582701046421178803016349346579796951}{214991445195987433587116827589012947229383519062960551} a - \frac{26998521062271692638517343954670263881703785124565202}{214991445195987433587116827589012947229383519062960551}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{1044}$, which has order $133632$ (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:  \( 2313806.343858355 \) (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

\(\Q(\sqrt{-491}) \), 7.7.14011639427134441.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
491Data not computed