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Properties

Label 14.0.35887976153...5067.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 71^{12}$
Root discriminant $66.89$
Ramified primes $3, 71$
Class number $43$ (GRH)
Class group $[43]$ (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![18769, 33565, 93727, 9326, 122335, 65124, 58178, 14106, 7385, 844, 643, 36, 31, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 31*x^12 + 36*x^11 + 643*x^10 + 844*x^9 + 7385*x^8 + 14106*x^7 + 58178*x^6 + 65124*x^5 + 122335*x^4 + 9326*x^3 + 93727*x^2 + 33565*x + 18769)
 
gp: K = bnfinit(x^14 - x^13 + 31*x^12 + 36*x^11 + 643*x^10 + 844*x^9 + 7385*x^8 + 14106*x^7 + 58178*x^6 + 65124*x^5 + 122335*x^4 + 9326*x^3 + 93727*x^2 + 33565*x + 18769, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 31 x^{12} + 36 x^{11} + 643 x^{10} + 844 x^{9} + 7385 x^{8} + 14106 x^{7} + 58178 x^{6} + 65124 x^{5} + 122335 x^{4} + 9326 x^{3} + 93727 x^{2} + 33565 x + 18769 \)

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:  \(-35887976153781453950585067=-\,3^{7}\cdot 71^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.89$
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:  $3, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(213=3\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{213}(32,·)$, $\chi_{213}(1,·)$, $\chi_{213}(91,·)$, $\chi_{213}(101,·)$, $\chi_{213}(103,·)$, $\chi_{213}(172,·)$, $\chi_{213}(143,·)$, $\chi_{213}(179,·)$, $\chi_{213}(20,·)$, $\chi_{213}(119,·)$, $\chi_{213}(116,·)$, $\chi_{213}(187,·)$, $\chi_{213}(190,·)$, $\chi_{213}(37,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{4} + \frac{1}{5}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{5} + \frac{1}{5} a$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{425} a^{12} + \frac{18}{425} a^{11} + \frac{12}{425} a^{10} - \frac{29}{425} a^{9} + \frac{2}{85} a^{8} + \frac{28}{425} a^{7} - \frac{198}{425} a^{6} + \frac{66}{425} a^{5} + \frac{2}{85} a^{4} - \frac{162}{425} a^{3} - \frac{208}{425} a^{2} - \frac{64}{425} a + \frac{149}{425}$, $\frac{1}{2636457236403252015821127475} a^{13} - \frac{2820697635013844684972958}{2636457236403252015821127475} a^{12} - \frac{41881246293457237148127406}{2636457236403252015821127475} a^{11} - \frac{19076686543189072778062736}{2636457236403252015821127475} a^{10} - \frac{251347519537116501005845921}{2636457236403252015821127475} a^{9} + \frac{241738514107073198992334318}{2636457236403252015821127475} a^{8} - \frac{123031560039040298642307226}{2636457236403252015821127475} a^{7} + \frac{207518102005677621879542044}{2636457236403252015821127475} a^{6} - \frac{942562831769625306425867416}{2636457236403252015821127475} a^{5} + \frac{575234369803909535867581653}{2636457236403252015821127475} a^{4} + \frac{139392538248150688453740979}{2636457236403252015821127475} a^{3} + \frac{1218407435663719822228753974}{2636457236403252015821127475} a^{2} + \frac{604342940433162814333049728}{2636457236403252015821127475} a - \frac{151922069782918766019656}{1132012553200194081503275}$

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

Class group and class number

$C_{43}$, which has order $43$ (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:  \( \frac{152810655005034307380}{6203428791537063566637947} a^{13} - \frac{192854454713800815780}{6203428791537063566637947} a^{12} + \frac{4727589257104845174486}{6203428791537063566637947} a^{11} + \frac{4261278814402922331539}{6203428791537063566637947} a^{10} + \frac{95397845363786510354887}{6203428791537063566637947} a^{9} + \frac{99991389165952853472467}{6203428791537063566637947} a^{8} + \frac{1063202695506832851780056}{6203428791537063566637947} a^{7} + \frac{1791858473237167563216969}{6203428791537063566637947} a^{6} + \frac{7969138954769793291376702}{6203428791537063566637947} a^{5} + \frac{6672723099021205946246114}{6203428791537063566637947} a^{4} + \frac{13092953778048760535747401}{6203428791537063566637947} a^{3} - \frac{8498269567856723160895751}{6203428791537063566637947} a^{2} + \frac{9714561672974891254392137}{6203428791537063566637947} a + \frac{25085929376777113490677}{45280502128007763260131} \) (order $6$)
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:  \( 315114.696625 \) (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

\(\Q(\sqrt{-3}) \), 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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$3$3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$71$71.14.12.1$x^{14} + 546629 x^{7} + 98234829011$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$