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Properties

Label 14.0.84500689521...5627.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{7}\cdot 29^{13}$
Root discriminant $60.32$
Ramified primes $7, 29$
Class number $928$ (GRH)
Class group $[2, 2, 2, 116]$ (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![77023, -135147, 128440, -15372, 34790, -27038, 7170, -5970, 3489, -693, 646, -46, 45, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 45*x^12 - 46*x^11 + 646*x^10 - 693*x^9 + 3489*x^8 - 5970*x^7 + 7170*x^6 - 27038*x^5 + 34790*x^4 - 15372*x^3 + 128440*x^2 - 135147*x + 77023)
 
gp: K = bnfinit(x^14 - x^13 + 45*x^12 - 46*x^11 + 646*x^10 - 693*x^9 + 3489*x^8 - 5970*x^7 + 7170*x^6 - 27038*x^5 + 34790*x^4 - 15372*x^3 + 128440*x^2 - 135147*x + 77023, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 45 x^{12} - 46 x^{11} + 646 x^{10} - 693 x^{9} + 3489 x^{8} - 5970 x^{7} + 7170 x^{6} - 27038 x^{5} + 34790 x^{4} - 15372 x^{3} + 128440 x^{2} - 135147 x + 77023 \)

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:  \(-8450068952156066122535627=-\,7^{7}\cdot 29^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.32$
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:  $7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(203=7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{203}(1,·)$, $\chi_{203}(34,·)$, $\chi_{203}(36,·)$, $\chi_{203}(197,·)$, $\chi_{203}(6,·)$, $\chi_{203}(167,·)$, $\chi_{203}(169,·)$, $\chi_{203}(202,·)$, $\chi_{203}(13,·)$, $\chi_{203}(78,·)$, $\chi_{203}(141,·)$, $\chi_{203}(62,·)$, $\chi_{203}(125,·)$, $\chi_{203}(190,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{17} a^{10} - \frac{8}{17} a^{9} + \frac{4}{17} a^{8} - \frac{8}{17} a^{7} - \frac{8}{17} a^{6} - \frac{7}{17} a^{5} - \frac{6}{17} a^{4} - \frac{3}{17} a^{3} + \frac{1}{17} a^{2} + \frac{4}{17} a - \frac{4}{17}$, $\frac{1}{17} a^{11} + \frac{8}{17} a^{9} + \frac{7}{17} a^{8} - \frac{4}{17} a^{7} - \frac{3}{17} a^{6} + \frac{6}{17} a^{5} - \frac{6}{17} a^{3} - \frac{5}{17} a^{2} - \frac{6}{17} a + \frac{2}{17}$, $\frac{1}{697} a^{12} - \frac{16}{697} a^{11} + \frac{11}{697} a^{10} - \frac{9}{697} a^{9} - \frac{19}{697} a^{8} - \frac{320}{697} a^{7} + \frac{47}{697} a^{6} - \frac{134}{697} a^{5} + \frac{197}{697} a^{4} + \frac{286}{697} a^{3} + \frac{179}{697} a^{2} + \frac{93}{697} a + \frac{58}{697}$, $\frac{1}{17515968588892736811704960087} a^{13} - \frac{208250387514610650867465}{427218746070554556383047807} a^{12} - \frac{20496319689700993845068939}{1030351093464278635982644711} a^{11} - \frac{393914450568108526074253451}{17515968588892736811704960087} a^{10} - \frac{596544138450682088376147193}{17515968588892736811704960087} a^{9} + \frac{5158945767986441666569700023}{17515968588892736811704960087} a^{8} + \frac{5698607791699085372241141597}{17515968588892736811704960087} a^{7} + \frac{94561380322425330051431709}{1030351093464278635982644711} a^{6} + \frac{7827492783542765442451885994}{17515968588892736811704960087} a^{5} - \frac{371216771792449433810781793}{1030351093464278635982644711} a^{4} + \frac{5322101942640727457868157442}{17515968588892736811704960087} a^{3} - \frac{8118806959814327111475563078}{17515968588892736811704960087} a^{2} + \frac{8633469501660822950170805498}{17515968588892736811704960087} a + \frac{1618390807100722236900706452}{17515968588892736811704960087}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{116}$, which has order $928$ (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:  \( 6020.98510015 \) (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

\(\Q(\sqrt{-203}) \), 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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
7Data not computed
$29$29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$