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Properties

Label 14.0.11791701300...7344.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{14}\cdot 3^{7}\cdot 7^{7}\cdot 43^{12}$
Root discriminant $230.28$
Ramified primes $2, 3, 7, 43$
Class number $2813696$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 21982]$ (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![7521613141, -1379805112, 1594403790, -207970434, 147472662, -13385716, 7907601, -461308, 276271, -9524, 6693, -146, 112, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 112*x^12 - 146*x^11 + 6693*x^10 - 9524*x^9 + 276271*x^8 - 461308*x^7 + 7907601*x^6 - 13385716*x^5 + 147472662*x^4 - 207970434*x^3 + 1594403790*x^2 - 1379805112*x + 7521613141)
 
gp: K = bnfinit(x^14 - 2*x^13 + 112*x^12 - 146*x^11 + 6693*x^10 - 9524*x^9 + 276271*x^8 - 461308*x^7 + 7907601*x^6 - 13385716*x^5 + 147472662*x^4 - 207970434*x^3 + 1594403790*x^2 - 1379805112*x + 7521613141, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} + 112 x^{12} - 146 x^{11} + 6693 x^{10} - 9524 x^{9} + 276271 x^{8} - 461308 x^{7} + 7907601 x^{6} - 13385716 x^{5} + 147472662 x^{4} - 207970434 x^{3} + 1594403790 x^{2} - 1379805112 x + 7521613141 \)

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:  \(-1179170130027156611464857533497344=-\,2^{14}\cdot 3^{7}\cdot 7^{7}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $230.28$
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:  $2, 3, 7, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3612=2^{2}\cdot 3\cdot 7\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{3612}(1,·)$, $\chi_{3612}(1091,·)$, $\chi_{3612}(3527,·)$, $\chi_{3612}(2185,·)$, $\chi_{3612}(1933,·)$, $\chi_{3612}(3107,·)$, $\chi_{3612}(1681,·)$, $\chi_{3612}(2099,·)$, $\chi_{3612}(2773,·)$, $\chi_{3612}(1847,·)$, $\chi_{3612}(1177,·)$, $\chi_{3612}(3193,·)$, $\chi_{3612}(1595,·)$, $\chi_{3612}(2687,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} + \frac{3}{7} a^{9} - \frac{2}{7} a^{7} - \frac{2}{7} a^{6} + \frac{2}{7} a^{5} - \frac{3}{7} a^{3} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{11} - \frac{2}{7} a^{9} - \frac{2}{7} a^{8} - \frac{3}{7} a^{7} + \frac{1}{7} a^{6} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} + \frac{3}{7} a^{3} - \frac{3}{7} a^{2}$, $\frac{1}{1813} a^{12} + \frac{123}{1813} a^{11} + \frac{108}{1813} a^{10} + \frac{5}{1813} a^{9} + \frac{360}{1813} a^{8} + \frac{3}{259} a^{7} + \frac{156}{1813} a^{6} - \frac{717}{1813} a^{5} + \frac{572}{1813} a^{4} - \frac{244}{1813} a^{3} - \frac{25}{259} a^{2} + \frac{45}{259} a - \frac{3}{37}$, $\frac{1}{9980134320712919925863499495883889090977} a^{13} - \frac{1508798227560681857723122409929427948}{9980134320712919925863499495883889090977} a^{12} - \frac{668723318414241590837985250079869176172}{9980134320712919925863499495883889090977} a^{11} - \frac{165211177227060414071860306707345754547}{9980134320712919925863499495883889090977} a^{10} + \frac{1071542674899693095774191783703393547016}{9980134320712919925863499495883889090977} a^{9} + \frac{4137041546173303808189052621034588552098}{9980134320712919925863499495883889090977} a^{8} + \frac{1287482639342254722855676782643855610422}{9980134320712919925863499495883889090977} a^{7} - \frac{1476163339869170356758738504669294529223}{9980134320712919925863499495883889090977} a^{6} - \frac{2594704282916083561758996299646815093150}{9980134320712919925863499495883889090977} a^{5} - \frac{2787618771388663684080614082137199841299}{9980134320712919925863499495883889090977} a^{4} - \frac{110575033358467720471234864232877290040}{9980134320712919925863499495883889090977} a^{3} - \frac{677638577490151117864506980491693608370}{1425733474387559989409071356554841298711} a^{2} + \frac{157355125991780414379043567378927014022}{1425733474387559989409071356554841298711} a + \frac{14319120475701606436102479471194437269}{203676210626794284201295908079263042673}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{21982}$, which has order $2813696$ (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

\(\Q(\sqrt{-21}) \), 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 R R ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$2$2.14.14.15$x^{14} + 2 x^{13} + x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
$3$3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
$43$43.14.12.1$x^{14} + 3569 x^{7} + 4043763$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$