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Properties

Label 14.14.6438392352...3713.1
Degree $14$
Signature $[14, 0]$
Discriminant $13^{7}\cdot 29^{13}$
Root discriminant $82.21$
Ramified primes $13, 29$
Class number $2$ (GRH)
Class group $[2]$ (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![40193, -98752, -820295, 1955033, -905100, -556143, 392145, 60585, -56106, -3448, 3546, 99, -100, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 - 100*x^12 + 99*x^11 + 3546*x^10 - 3448*x^9 - 56106*x^8 + 60585*x^7 + 392145*x^6 - 556143*x^5 - 905100*x^4 + 1955033*x^3 - 820295*x^2 - 98752*x + 40193)
 
gp: K = bnfinit(x^14 - x^13 - 100*x^12 + 99*x^11 + 3546*x^10 - 3448*x^9 - 56106*x^8 + 60585*x^7 + 392145*x^6 - 556143*x^5 - 905100*x^4 + 1955033*x^3 - 820295*x^2 - 98752*x + 40193, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} - 100 x^{12} + 99 x^{11} + 3546 x^{10} - 3448 x^{9} - 56106 x^{8} + 60585 x^{7} + 392145 x^{6} - 556143 x^{5} - 905100 x^{4} + 1955033 x^{3} - 820295 x^{2} - 98752 x + 40193 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(643839235225770969752703713=13^{7}\cdot 29^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.21$
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:  $13, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(377=13\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{377}(64,·)$, $\chi_{377}(1,·)$, $\chi_{377}(324,·)$, $\chi_{377}(326,·)$, $\chi_{377}(38,·)$, $\chi_{377}(129,·)$, $\chi_{377}(170,·)$, $\chi_{377}(207,·)$, $\chi_{377}(248,·)$, $\chi_{377}(339,·)$, $\chi_{377}(53,·)$, $\chi_{377}(376,·)$, $\chi_{377}(313,·)$, $\chi_{377}(51,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{41} a^{12} - \frac{13}{41} a^{11} + \frac{10}{41} a^{10} + \frac{3}{41} a^{9} + \frac{16}{41} a^{8} - \frac{6}{41} a^{7} + \frac{15}{41} a^{6} + \frac{1}{41} a^{5} + \frac{16}{41} a^{4} - \frac{11}{41} a^{3} - \frac{14}{41} a^{2} + \frac{7}{41} a + \frac{19}{41}$, $\frac{1}{6136939860308177773805548110896431} a^{13} - \frac{104973750024496558998164412977}{149681460007516531068428002704791} a^{12} + \frac{2274511929793934389557865534351582}{6136939860308177773805548110896431} a^{11} - \frac{791738555072517462074223707717740}{6136939860308177773805548110896431} a^{10} - \frac{1980092736982982844095425775036744}{6136939860308177773805548110896431} a^{9} - \frac{1306479248442413547924016298248953}{6136939860308177773805548110896431} a^{8} - \frac{2034066367213930873893846371151437}{6136939860308177773805548110896431} a^{7} + \frac{1550230605912532949429398518431524}{6136939860308177773805548110896431} a^{6} + \frac{2688500841712291442564445123203831}{6136939860308177773805548110896431} a^{5} + \frac{217321005548970568489522878245595}{6136939860308177773805548110896431} a^{4} - \frac{1595297953073112082368488446531494}{6136939860308177773805548110896431} a^{3} - \frac{1290274560838652298919696144430884}{6136939860308177773805548110896431} a^{2} + \frac{2148244362241661552530864129698248}{6136939860308177773805548110896431} a - \frac{1745121860485322635931634462688029}{6136939860308177773805548110896431}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $13$
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:  \( 278026941.252397 \) (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

\(\Q(\sqrt{377}) \), 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$13$13.14.7.1$x^{14} - 43940 x^{8} + 482680900 x^{2} - 250994068$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$