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Properties

Label 14.0.55430298865...7411.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{7}\cdot 197^{13}$
Root discriminant $357.38$
Ramified primes $7, 197$
Class number $290432$ (GRH)
Class group $[2, 2, 2, 36304]$ (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![53109997117, -32497939521, 6537290446, -56932186, 118057159, 7832183, 17975510, 3680571, 1369102, 89886, 32127, 420, 303, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 303*x^12 + 420*x^11 + 32127*x^10 + 89886*x^9 + 1369102*x^8 + 3680571*x^7 + 17975510*x^6 + 7832183*x^5 + 118057159*x^4 - 56932186*x^3 + 6537290446*x^2 - 32497939521*x + 53109997117)
 
gp: K = bnfinit(x^14 - x^13 + 303*x^12 + 420*x^11 + 32127*x^10 + 89886*x^9 + 1369102*x^8 + 3680571*x^7 + 17975510*x^6 + 7832183*x^5 + 118057159*x^4 - 56932186*x^3 + 6537290446*x^2 - 32497939521*x + 53109997117, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 303 x^{12} + 420 x^{11} + 32127 x^{10} + 89886 x^{9} + 1369102 x^{8} + 3680571 x^{7} + 17975510 x^{6} + 7832183 x^{5} + 118057159 x^{4} - 56932186 x^{3} + 6537290446 x^{2} - 32497939521 x + 53109997117 \)

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:  \(-554302988656067112295385411481137411=-\,7^{7}\cdot 197^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $357.38$
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:  $7, 197$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1379=7\cdot 197\)
Dirichlet character group:    $\lbrace$$\chi_{1379}(1,·)$, $\chi_{1379}(1378,·)$, $\chi_{1379}(36,·)$, $\chi_{1379}(6,·)$, $\chi_{1379}(230,·)$, $\chi_{1379}(1163,·)$, $\chi_{1379}(1149,·)$, $\chi_{1379}(1296,·)$, $\chi_{1379}(881,·)$, $\chi_{1379}(498,·)$, $\chi_{1379}(83,·)$, $\chi_{1379}(216,·)$, $\chi_{1379}(1373,·)$, $\chi_{1379}(1343,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{417799694974035573084435815463356245652238816611406970333566506281} a^{13} - \frac{165575095248928599699611293000724906738835638304031335874415676379}{417799694974035573084435815463356245652238816611406970333566506281} a^{12} - \frac{55562299371396589161223490523335970512680469163522991881642395185}{417799694974035573084435815463356245652238816611406970333566506281} a^{11} + \frac{161375477972943859534430782591964968426907002460766875260047712473}{417799694974035573084435815463356245652238816611406970333566506281} a^{10} - \frac{146047432649477111062641667201318298228517626720543823616379087396}{417799694974035573084435815463356245652238816611406970333566506281} a^{9} + \frac{51831490909751514777145837561422114034563827875976049550917855093}{417799694974035573084435815463356245652238816611406970333566506281} a^{8} + \frac{90749277517208188768860052723920873749333579920875787842096125976}{417799694974035573084435815463356245652238816611406970333566506281} a^{7} + \frac{63124834531384833571841112069925197150724971593385572577902629410}{417799694974035573084435815463356245652238816611406970333566506281} a^{6} - \frac{190129733188323646642113663382235198618404808183221563375815706527}{417799694974035573084435815463356245652238816611406970333566506281} a^{5} + \frac{200868495099015044193385054769234723224099514072099330202321664820}{417799694974035573084435815463356245652238816611406970333566506281} a^{4} - \frac{192641125104194413475092241299462804592602340787003989474602316086}{417799694974035573084435815463356245652238816611406970333566506281} a^{3} - \frac{41636971970349274994298049408848746796653345114520256979492308463}{417799694974035573084435815463356245652238816611406970333566506281} a^{2} + \frac{186843215663722532083211746759569297449342362451499952110027298727}{417799694974035573084435815463356245652238816611406970333566506281} a + \frac{151115259522302281884603956001490170480228249106772307699533360749}{417799694974035573084435815463356245652238816611406970333566506281}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{36304}$, which has order $290432$ (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:  \( 1553055.199048291 \) (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

\(\Q(\sqrt{-1379}) \), 7.7.58451728309129.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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$7.14.7.1$x^{14} - 117649 x^{2} + 1647086$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$197$197.14.13.1$x^{14} - 197$$14$$1$$13$$C_{14}$$[\ ]_{14}$