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Properties

Label 14.0.53588606915...5607.3
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{25}\cdot 43^{12}$
Root discriminant $811.37$
Ramified primes $7, 43$
Class number $108437$ (GRH)
Class group $[7, 15491]$ (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![22085122836303, 4291822604559, 886043817568, 98387288769, 69559163667, 6124265196, 2010254988, 117159090, 16012899, -413574, -20769, -7224, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7224*x^11 - 20769*x^10 - 413574*x^9 + 16012899*x^8 + 117159090*x^7 + 2010254988*x^6 + 6124265196*x^5 + 69559163667*x^4 + 98387288769*x^3 + 886043817568*x^2 + 4291822604559*x + 22085122836303)
 
gp: K = bnfinit(x^14 - 7224*x^11 - 20769*x^10 - 413574*x^9 + 16012899*x^8 + 117159090*x^7 + 2010254988*x^6 + 6124265196*x^5 + 69559163667*x^4 + 98387288769*x^3 + 886043817568*x^2 + 4291822604559*x + 22085122836303, 1)
 

Normalized defining polynomial

\( x^{14} - 7224 x^{11} - 20769 x^{10} - 413574 x^{9} + 16012899 x^{8} + 117159090 x^{7} + 2010254988 x^{6} + 6124265196 x^{5} + 69559163667 x^{4} + 98387288769 x^{3} + 886043817568 x^{2} + 4291822604559 x + 22085122836303 \)

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:  \(-53588606915566584613059849086223224055607=-\,7^{25}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $811.37$
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:  $7, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2107=7^{2}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{2107}(1,·)$, $\chi_{2107}(1896,·)$, $\chi_{2107}(41,·)$, $\chi_{2107}(1420,·)$, $\chi_{2107}(1392,·)$, $\chi_{2107}(1681,·)$, $\chi_{2107}(274,·)$, $\chi_{2107}(1331,·)$, $\chi_{2107}(1268,·)$, $\chi_{2107}(183,·)$, $\chi_{2107}(1497,·)$, $\chi_{2107}(699,·)$, $\chi_{2107}(1884,·)$, $\chi_{2107}(1182,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{129} a^{7} + \frac{1}{3} a^{3}$, $\frac{1}{129} a^{8} + \frac{1}{3} a^{4}$, $\frac{1}{129} a^{9} - \frac{1}{3} a$, $\frac{1}{16641} a^{10} - \frac{10}{387} a^{6} + \frac{19}{129} a^{5} + \frac{11}{43} a^{4} - \frac{12}{43} a^{3} - \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{16641} a^{11} - \frac{1}{387} a^{7} + \frac{19}{129} a^{6} - \frac{10}{129} a^{5} - \frac{12}{43} a^{4} - \frac{2}{9} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{715563} a^{12} - \frac{13}{5547} a^{9} - \frac{10}{16641} a^{8} + \frac{19}{5547} a^{7} + \frac{248}{5547} a^{6} + \frac{117}{1849} a^{5} + \frac{1}{9} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{330996212258482438141938543794913415002563931685107455145543} a^{13} + \frac{46300904034821194572346226195989306376046578640258125}{330996212258482438141938543794913415002563931685107455145543} a^{12} + \frac{208937566680568923029496783051632233610255667595677653}{7697586331592614840510198692904963139594510039188545468501} a^{11} + \frac{38352101918777248331202672660374117135214316184603648}{7697586331592614840510198692904963139594510039188545468501} a^{10} - \frac{5333446543730042238308732155141155599687654755074059929}{7697586331592614840510198692904963139594510039188545468501} a^{9} - \frac{3083927383212619782633766828140062936168570403759316553}{7697586331592614840510198692904963139594510039188545468501} a^{8} - \frac{16658628928087549128412771099086333793977952818228149222}{7697586331592614840510198692904963139594510039188545468501} a^{7} + \frac{1261380959973812296215968036832630172929156585216273822616}{7697586331592614840510198692904963139594510039188545468501} a^{6} - \frac{77952378233694170465790633449663682954048285466616715794}{7697586331592614840510198692904963139594510039188545468501} a^{5} + \frac{54799827096443486060410560156038909165945562385270190099}{179013635618432903267679039369882863711500233469501057407} a^{4} + \frac{85905262865773354538845168714169483132695219220464437056}{179013635618432903267679039369882863711500233469501057407} a^{3} - \frac{906975328593478848542156113503780270570812357705868082}{4163107805079834959713466031857741016546517057430257149} a^{2} - \frac{207204547459178350068384023707631724746597238324796093}{462567533897759439968162892428637890727390784158917461} a - \frac{184308807336419397864991424804828109233780788005295975}{462567533897759439968162892428637890727390784158917461}$

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

Class group and class number

$C_{7}\times C_{15491}$, which has order $108437$ (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:  \( 5135341547.611097 \) (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

\(\Q(\sqrt{-7}) \), 7.7.87495801462998035849.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ R ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$7$7.14.25.59$x^{14} - 168 x^{13} + 70 x^{12} - 147 x^{11} + 147 x^{10} - 98 x^{9} + 49 x^{8} + 168 x^{7} - 49 x^{4} - 147 x^{3} - 49 x^{2} + 98 x + 126$$14$$1$$25$$C_{14}$$[2]_{2}$
$43$43.7.6.5$x^{7} - 282123$$7$$1$$6$$C_7$$[\ ]_{7}$
43.7.6.5$x^{7} - 282123$$7$$1$$6$$C_7$$[\ ]_{7}$