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Properties

Label 14.0.84150067079...3219.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{25}\cdot 13^{7}$
Root discriminant $116.43$
Ramified primes $7, 13$
Class number $26098$ (GRH)
Class group $[26098]$ (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![2817187, -2274769, 2580193, -3819046, 4145911, -2371978, 975541, -354843, 88998, -20776, 4879, -301, 126, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 126*x^12 - 301*x^11 + 4879*x^10 - 20776*x^9 + 88998*x^8 - 354843*x^7 + 975541*x^6 - 2371978*x^5 + 4145911*x^4 - 3819046*x^3 + 2580193*x^2 - 2274769*x + 2817187)
 
gp: K = bnfinit(x^14 + 126*x^12 - 301*x^11 + 4879*x^10 - 20776*x^9 + 88998*x^8 - 354843*x^7 + 975541*x^6 - 2371978*x^5 + 4145911*x^4 - 3819046*x^3 + 2580193*x^2 - 2274769*x + 2817187, 1)
 

Normalized defining polynomial

\( x^{14} + 126 x^{12} - 301 x^{11} + 4879 x^{10} - 20776 x^{9} + 88998 x^{8} - 354843 x^{7} + 975541 x^{6} - 2371978 x^{5} + 4145911 x^{4} - 3819046 x^{3} + 2580193 x^{2} - 2274769 x + 2817187 \)

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:  \(-84150067079150835865691353219=-\,7^{25}\cdot 13^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $116.43$
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:  $7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(637=7^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{637}(1,·)$, $\chi_{637}(547,·)$, $\chi_{637}(454,·)$, $\chi_{637}(545,·)$, $\chi_{637}(456,·)$, $\chi_{637}(92,·)$, $\chi_{637}(363,·)$, $\chi_{637}(365,·)$, $\chi_{637}(272,·)$, $\chi_{637}(274,·)$, $\chi_{637}(181,·)$, $\chi_{637}(183,·)$, $\chi_{637}(90,·)$, $\chi_{637}(636,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{19} a^{10} - \frac{2}{19} a^{9} + \frac{1}{19} a^{8} - \frac{3}{19} a^{7} + \frac{6}{19} a^{6} - \frac{7}{19} a^{5} - \frac{2}{19} a^{4} - \frac{4}{19} a^{3} + \frac{6}{19} a^{2} + \frac{4}{19} a$, $\frac{1}{1501} a^{11} - \frac{34}{1501} a^{10} - \frac{49}{1501} a^{9} - \frac{415}{1501} a^{8} - \frac{126}{1501} a^{7} - \frac{617}{1501} a^{6} + \frac{735}{1501} a^{5} + \frac{41}{1501} a^{4} - \frac{702}{1501} a^{3} + \frac{40}{1501} a^{2} + \frac{157}{1501} a + \frac{16}{79}$, $\frac{1}{4513507} a^{12} - \frac{949}{4513507} a^{11} - \frac{71481}{4513507} a^{10} + \frac{2089730}{4513507} a^{9} + \frac{1662480}{4513507} a^{8} + \frac{2107922}{4513507} a^{7} - \frac{2065805}{4513507} a^{6} - \frac{748719}{4513507} a^{5} - \frac{7249}{4513507} a^{4} + \frac{410110}{4513507} a^{3} + \frac{325456}{4513507} a^{2} - \frac{401918}{4513507} a + \frac{1964}{7663}$, $\frac{1}{125363252779435555280614086513064493} a^{13} + \frac{61193608247322996684599467}{1292404667829232528666124603227469} a^{12} + \frac{13713915101627716993258519020923}{125363252779435555280614086513064493} a^{11} + \frac{278512419886098991711856076393124}{125363252779435555280614086513064493} a^{10} + \frac{1067521596688405442243752656309848}{6598065935759766067400741395424447} a^{9} + \frac{705220992441384869720678553167118}{125363252779435555280614086513064493} a^{8} + \frac{26156341276360779544927100609868013}{125363252779435555280614086513064493} a^{7} - \frac{27447013859193532636649308484313071}{125363252779435555280614086513064493} a^{6} + \frac{31497760686178906666141042896187762}{125363252779435555280614086513064493} a^{5} - \frac{51039269569728829309316364243776417}{125363252779435555280614086513064493} a^{4} + \frac{58325879218279442448664757176195802}{125363252779435555280614086513064493} a^{3} - \frac{33421629385235509557540709792302776}{125363252779435555280614086513064493} a^{2} - \frac{29943969133378598759406784858936758}{125363252779435555280614086513064493} a + \frac{100395513588643332429146377839124}{212840836637411808625830367594337}$

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

Class group and class number

$C_{26098}$, which has order $26098$ (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:  \( 35256.68973693789 \) (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

\(\Q(\sqrt{-91}) \), 7.7.13841287201.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ R ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$7$7.14.25.75$x^{14} + 112 x^{13} + 28 x^{12} - 98 x^{11} + 147 x^{10} + 98 x^{9} + 49 x^{8} - 161 x^{7} + 49 x^{6} + 147 x^{5} + 147 x^{2} - 147 x - 126$$14$$1$$25$$C_{14}$$[2]_{2}$
$13$13.14.7.2$x^{14} - 48268090 x^{2} + 125497034$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$