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Properties

Label 14.0.94796020201...8747.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 113^{12}$
Root discriminant $99.62$
Ramified primes $3, 113$
Class number $1016$ (GRH)
Class group $[2, 2, 254]$ (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![1, -49, 2389, -1212, 15469, 70, 95434, -10342, 16259, 2412, 1955, 122, 49, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 49*x^12 + 122*x^11 + 1955*x^10 + 2412*x^9 + 16259*x^8 - 10342*x^7 + 95434*x^6 + 70*x^5 + 15469*x^4 - 1212*x^3 + 2389*x^2 - 49*x + 1)
 
gp: K = bnfinit(x^14 - x^13 + 49*x^12 + 122*x^11 + 1955*x^10 + 2412*x^9 + 16259*x^8 - 10342*x^7 + 95434*x^6 + 70*x^5 + 15469*x^4 - 1212*x^3 + 2389*x^2 - 49*x + 1, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 49 x^{12} + 122 x^{11} + 1955 x^{10} + 2412 x^{9} + 16259 x^{8} - 10342 x^{7} + 95434 x^{6} + 70 x^{5} + 15469 x^{4} - 1212 x^{3} + 2389 x^{2} - 49 x + 1 \)

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:  \(-9479602020119218896677148747=-\,3^{7}\cdot 113^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $99.62$
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:  $3, 113$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(339=3\cdot 113\)
Dirichlet character group:    $\lbrace$$\chi_{339}(256,·)$, $\chi_{339}(1,·)$, $\chi_{339}(227,·)$, $\chi_{339}(106,·)$, $\chi_{339}(332,·)$, $\chi_{339}(109,·)$, $\chi_{339}(143,·)$, $\chi_{339}(16,·)$, $\chi_{339}(49,·)$, $\chi_{339}(242,·)$, $\chi_{339}(275,·)$, $\chi_{339}(335,·)$, $\chi_{339}(28,·)$, $\chi_{339}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11461} a^{12} + \frac{2330}{11461} a^{11} - \frac{2282}{11461} a^{10} + \frac{367}{11461} a^{9} + \frac{3223}{11461} a^{8} + \frac{2234}{11461} a^{7} + \frac{4011}{11461} a^{6} - \frac{2370}{11461} a^{5} - \frac{1313}{11461} a^{4} + \frac{5368}{11461} a^{3} + \frac{795}{11461} a^{2} + \frac{2306}{11461} a - \frac{4707}{11461}$, $\frac{1}{56007829185520976378452093} a^{13} - \frac{1255143675287956948076}{56007829185520976378452093} a^{12} - \frac{17655808311079184724819918}{56007829185520976378452093} a^{11} + \frac{19110133762521936969135968}{56007829185520976378452093} a^{10} - \frac{9256869010040444197716421}{56007829185520976378452093} a^{9} + \frac{14391252331755303139164915}{56007829185520976378452093} a^{8} + \frac{25076608041260211990523337}{56007829185520976378452093} a^{7} + \frac{12797253305694874144427499}{56007829185520976378452093} a^{6} + \frac{19670449581544548023364542}{56007829185520976378452093} a^{5} - \frac{4061919949123979950125620}{56007829185520976378452093} a^{4} - \frac{541261677733520127158773}{56007829185520976378452093} a^{3} - \frac{10004125770984172050289433}{56007829185520976378452093} a^{2} - \frac{21244104489870396800466903}{56007829185520976378452093} a + \frac{8592204200471366721334101}{56007829185520976378452093}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{254}$, which has order $1016$ (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:  \( \frac{99946859735858651928}{4886818705655787137113} a^{13} - \frac{99985379946876851604}{4886818705655787137113} a^{12} + \frac{4897519259676980553108}{4886818705655787137113} a^{11} + \frac{12191869583297379753383}{4886818705655787137113} a^{10} + \frac{195395211113601788375089}{4886818705655787137113} a^{9} + \frac{241022726756247429283319}{4886818705655787137113} a^{8} + \frac{1625147176216015904600302}{4886818705655787137113} a^{7} - \frac{1033442481937703953024631}{4886818705655787137113} a^{6} + \frac{9540839855256628826161968}{4886818705655787137113} a^{5} + \frac{7595338756841366691336}{4886818705655787137113} a^{4} + \frac{1550448176363054728534571}{4886818705655787137113} a^{3} - \frac{90570472761689533781447}{4886818705655787137113} a^{2} + \frac{239469235798813858304113}{4886818705655787137113} a - \frac{24857798687317192620}{4886818705655787137113} \) (order $6$)
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:  \( 222748.97284811488 \) (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

\(\Q(\sqrt{-3}) \), 7.7.2081951752609.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ R ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$3$3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$113$113.14.12.1$x^{14} + 640597 x^{7} + 127690000000$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$