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Properties

Label 15.15.1924191232...0521.1
Degree $15$
Signature $[15, 0]$
Discriminant $11^{12}\cdot 19^{10}$
Root discriminant $48.49$
Ramified primes $11, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{15}$ (as 15T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![197, 1723, -6663, -6520, 23300, 3839, -27366, 4836, 11013, -3380, -1668, 622, 100, -43, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 43*x^13 + 100*x^12 + 622*x^11 - 1668*x^10 - 3380*x^9 + 11013*x^8 + 4836*x^7 - 27366*x^6 + 3839*x^5 + 23300*x^4 - 6520*x^3 - 6663*x^2 + 1723*x + 197)
 
gp: K = bnfinit(x^15 - 2*x^14 - 43*x^13 + 100*x^12 + 622*x^11 - 1668*x^10 - 3380*x^9 + 11013*x^8 + 4836*x^7 - 27366*x^6 + 3839*x^5 + 23300*x^4 - 6520*x^3 - 6663*x^2 + 1723*x + 197, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 43 x^{13} + 100 x^{12} + 622 x^{11} - 1668 x^{10} - 3380 x^{9} + 11013 x^{8} + 4836 x^{7} - 27366 x^{6} + 3839 x^{5} + 23300 x^{4} - 6520 x^{3} - 6663 x^{2} + 1723 x + 197 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(19241912323039288533050521=11^{12}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.49$
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:  $11, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(209=11\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{209}(64,·)$, $\chi_{209}(1,·)$, $\chi_{209}(163,·)$, $\chi_{209}(102,·)$, $\chi_{209}(201,·)$, $\chi_{209}(45,·)$, $\chi_{209}(144,·)$, $\chi_{209}(49,·)$, $\chi_{209}(115,·)$, $\chi_{209}(20,·)$, $\chi_{209}(26,·)$, $\chi_{209}(159,·)$, $\chi_{209}(58,·)$, $\chi_{209}(125,·)$, $\chi_{209}(191,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{79558723009865103705733} a^{14} - \frac{27781103092348323877477}{79558723009865103705733} a^{13} - \frac{4679131509343446783192}{79558723009865103705733} a^{12} + \frac{18375265261250659683472}{79558723009865103705733} a^{11} + \frac{5944532062656426382484}{79558723009865103705733} a^{10} - \frac{11460419137972451570578}{79558723009865103705733} a^{9} - \frac{34794616323348689036803}{79558723009865103705733} a^{8} - \frac{33807488999702967339222}{79558723009865103705733} a^{7} - \frac{36614679320120092748231}{79558723009865103705733} a^{6} - \frac{17062657115156750728896}{79558723009865103705733} a^{5} + \frac{22547000991091646944056}{79558723009865103705733} a^{4} + \frac{24482270397004665786747}{79558723009865103705733} a^{3} + \frac{4860636621020262981509}{79558723009865103705733} a^{2} + \frac{26563942987350214407321}{79558723009865103705733} a + \frac{39488382583456003237824}{79558723009865103705733}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $14$
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:  \( 39634245.0922 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{15}$ (as 15T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 15
The 15 conjugacy class representatives for $C_{15}$
Character table for $C_{15}$

Intermediate fields

3.3.361.1, \(\Q(\zeta_{11})^+\)

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 $15$ $15$ $15$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ R $15$ $15$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{5}$ $15$ $15$ $15$

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
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
19Data not computed