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Properties

Label 14.0.12046775773...6327.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,23^{7}\cdot 29^{12}$
Root discriminant $85.97$
Ramified primes $23, 29$
Class number $7224$ (GRH)
Class group $[2, 2, 1806]$ (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![3780979, -1446221, 2179894, -714115, 561187, -158128, 80950, -18671, 7314, -1362, 468, -79, 27, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 + 27*x^12 - 79*x^11 + 468*x^10 - 1362*x^9 + 7314*x^8 - 18671*x^7 + 80950*x^6 - 158128*x^5 + 561187*x^4 - 714115*x^3 + 2179894*x^2 - 1446221*x + 3780979)
 
gp: K = bnfinit(x^14 - 5*x^13 + 27*x^12 - 79*x^11 + 468*x^10 - 1362*x^9 + 7314*x^8 - 18671*x^7 + 80950*x^6 - 158128*x^5 + 561187*x^4 - 714115*x^3 + 2179894*x^2 - 1446221*x + 3780979, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} + 27 x^{12} - 79 x^{11} + 468 x^{10} - 1362 x^{9} + 7314 x^{8} - 18671 x^{7} + 80950 x^{6} - 158128 x^{5} + 561187 x^{4} - 714115 x^{3} + 2179894 x^{2} - 1446221 x + 3780979 \)

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:  \(-1204677577382769220367486327=-\,23^{7}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $85.97$
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:  $23, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(667=23\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{667}(576,·)$, $\chi_{667}(1,·)$, $\chi_{667}(139,·)$, $\chi_{667}(484,·)$, $\chi_{667}(645,·)$, $\chi_{667}(459,·)$, $\chi_{667}(45,·)$, $\chi_{667}(24,·)$, $\chi_{667}(436,·)$, $\chi_{667}(277,·)$, $\chi_{667}(344,·)$, $\chi_{667}(252,·)$, $\chi_{667}(413,·)$, $\chi_{667}(574,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{17} a^{12} + \frac{7}{17} a^{11} - \frac{3}{17} a^{10} + \frac{5}{17} a^{9} + \frac{3}{17} a^{8} + \frac{8}{17} a^{7} - \frac{4}{17} a^{6} + \frac{4}{17} a^{5} + \frac{7}{17} a^{4} + \frac{8}{17} a^{3} - \frac{5}{17} a^{2} + \frac{1}{17} a + \frac{5}{17}$, $\frac{1}{208071437219101694314384748513107} a^{13} - \frac{4392387248346367229524110885991}{208071437219101694314384748513107} a^{12} - \frac{54369695941961029858202352549935}{208071437219101694314384748513107} a^{11} + \frac{48813049183577516627679752828877}{208071437219101694314384748513107} a^{10} + \frac{101809804286664164927424979359562}{208071437219101694314384748513107} a^{9} - \frac{1064050440026151165423858626336}{12239496307005982018493220500771} a^{8} - \frac{939141213962009720103929838529}{208071437219101694314384748513107} a^{7} - \frac{80243570404166203456512577999602}{208071437219101694314384748513107} a^{6} + \frac{58177909531752849807760162097257}{208071437219101694314384748513107} a^{5} + \frac{31656923750780872553610125843879}{208071437219101694314384748513107} a^{4} + \frac{9633221974091668062277752334203}{208071437219101694314384748513107} a^{3} - \frac{40000164456397197333680885700029}{208071437219101694314384748513107} a^{2} - \frac{92582016003277223513510854616593}{208071437219101694314384748513107} a - \frac{2344005673491104096641018072831}{5074913102904919373521579232027}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{1806}$, which has order $7224$ (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:  \( 6020.985100147561 \) (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

\(\Q(\sqrt{-23}) \), 7.7.594823321.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ R R ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{14}$

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
$23$23.14.7.2$x^{14} - 148035889 x^{2} + 27238603576$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$