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Properties

Label 14.0.26117884524...1967.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{25}\cdot 41^{7}$
Root discriminant $206.77$
Ramified primes $7, 41$
Class number $842506$ (GRH)
Class group $[842506]$ (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![1353299719, -1278452721, 1175109670, -855223397, 519273797, -220994886, 59795582, -12306090, 2635479, -216776, 53977, -938, 420, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 420*x^12 - 938*x^11 + 53977*x^10 - 216776*x^9 + 2635479*x^8 - 12306090*x^7 + 59795582*x^6 - 220994886*x^5 + 519273797*x^4 - 855223397*x^3 + 1175109670*x^2 - 1278452721*x + 1353299719)
 
gp: K = bnfinit(x^14 + 420*x^12 - 938*x^11 + 53977*x^10 - 216776*x^9 + 2635479*x^8 - 12306090*x^7 + 59795582*x^6 - 220994886*x^5 + 519273797*x^4 - 855223397*x^3 + 1175109670*x^2 - 1278452721*x + 1353299719, 1)
 

Normalized defining polynomial

\( x^{14} + 420 x^{12} - 938 x^{11} + 53977 x^{10} - 216776 x^{9} + 2635479 x^{8} - 12306090 x^{7} + 59795582 x^{6} - 220994886 x^{5} + 519273797 x^{4} - 855223397 x^{3} + 1175109670 x^{2} - 1278452721 x + 1353299719 \)

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:  \(-261178845247250442478137475921967=-\,7^{25}\cdot 41^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $206.77$
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:  $7, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2009=7^{2}\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{2009}(288,·)$, $\chi_{2009}(1,·)$, $\chi_{2009}(1147,·)$, $\chi_{2009}(1436,·)$, $\chi_{2009}(1149,·)$, $\chi_{2009}(286,·)$, $\chi_{2009}(2008,·)$, $\chi_{2009}(1721,·)$, $\chi_{2009}(1434,·)$, $\chi_{2009}(1723,·)$, $\chi_{2009}(860,·)$, $\chi_{2009}(573,·)$, $\chi_{2009}(862,·)$, $\chi_{2009}(575,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{19} a^{10} + \frac{6}{19} a^{9} - \frac{2}{19} a^{8} + \frac{3}{19} a^{7} + \frac{2}{19} a^{6} + \frac{7}{19} a^{5} + \frac{3}{19} a^{4} - \frac{4}{19} a^{3} + \frac{9}{19} a^{2} + \frac{1}{19} a$, $\frac{1}{19} a^{11} - \frac{4}{19} a^{8} + \frac{3}{19} a^{7} - \frac{5}{19} a^{6} - \frac{1}{19} a^{5} - \frac{3}{19} a^{4} - \frac{5}{19} a^{3} + \frac{4}{19} a^{2} - \frac{6}{19} a$, $\frac{1}{19} a^{12} - \frac{4}{19} a^{9} + \frac{3}{19} a^{8} - \frac{5}{19} a^{7} - \frac{1}{19} a^{6} - \frac{3}{19} a^{5} - \frac{5}{19} a^{4} + \frac{4}{19} a^{3} - \frac{6}{19} a^{2}$, $\frac{1}{438736019189971475287379338577071474654235652460828629277919} a^{13} + \frac{5283642533699376598321981523144636068227067238158914586035}{438736019189971475287379338577071474654235652460828629277919} a^{12} + \frac{2917160233479656827276092021003440273438694482745643856559}{438736019189971475287379338577071474654235652460828629277919} a^{11} - \frac{10650563296387303251672684462196571457778458673782254530391}{438736019189971475287379338577071474654235652460828629277919} a^{10} + \frac{5343470449082921458541072452617167910570732514792199993545}{23091369431051130278283123083003761823907139603201506804101} a^{9} - \frac{128038249894492741137912741064581859487865838177402935600703}{438736019189971475287379338577071474654235652460828629277919} a^{8} + \frac{89107900244056845668570295540025018791370801996488808157646}{438736019189971475287379338577071474654235652460828629277919} a^{7} + \frac{114923256777164869644699769663090071858084633772644567621628}{438736019189971475287379338577071474654235652460828629277919} a^{6} + \frac{207606452493231248923601236128567964816908237728472142870026}{438736019189971475287379338577071474654235652460828629277919} a^{5} - \frac{140697263157451101964490619216748744367631226600848398039253}{438736019189971475287379338577071474654235652460828629277919} a^{4} + \frac{212710937949017818034495859623342318284837731872843243568126}{438736019189971475287379338577071474654235652460828629277919} a^{3} - \frac{176494513727759747654943967437683086411517931450281113219032}{438736019189971475287379338577071474654235652460828629277919} a^{2} - \frac{167906527126991077827288211889431963205233473695284811621847}{438736019189971475287379338577071474654235652460828629277919} a + \frac{11285512490088906031508774375398108703124710855494760595322}{23091369431051130278283123083003761823907139603201506804101}$

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

Class group and class number

$C_{842506}$, which has order $842506$ (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

\(\Q(\sqrt{-287}) \), 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$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}$$[2]_{2}$
$41$41.14.7.2$x^{14} - 14250312723 x^{2} + 1363279917167$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$