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Properties

Label 15.15.3091133177...6129.2
Degree $15$
Signature $[15, 0]$
Discriminant $3^{20}\cdot 7^{10}\cdot 11^{12}$
Root discriminant $107.81$
Ramified primes $3, 7, 11$
Class number $3$ (GRH)
Class group $[3]$ (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![831403, 4535715, -3294435, -19922550, -17750946, -2654712, 3201011, 1201692, -163851, -119015, 309, 5277, 165, -114, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - 114*x^13 + 165*x^12 + 5277*x^11 + 309*x^10 - 119015*x^9 - 163851*x^8 + 1201692*x^7 + 3201011*x^6 - 2654712*x^5 - 17750946*x^4 - 19922550*x^3 - 3294435*x^2 + 4535715*x + 831403)
 
gp: K = bnfinit(x^15 - 3*x^14 - 114*x^13 + 165*x^12 + 5277*x^11 + 309*x^10 - 119015*x^9 - 163851*x^8 + 1201692*x^7 + 3201011*x^6 - 2654712*x^5 - 17750946*x^4 - 19922550*x^3 - 3294435*x^2 + 4535715*x + 831403, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{14} - 114 x^{13} + 165 x^{12} + 5277 x^{11} + 309 x^{10} - 119015 x^{9} - 163851 x^{8} + 1201692 x^{7} + 3201011 x^{6} - 2654712 x^{5} - 17750946 x^{4} - 19922550 x^{3} - 3294435 x^{2} + 4535715 x + 831403 \)

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:  \(3091133177133909578645502426129=3^{20}\cdot 7^{10}\cdot 11^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $107.81$
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:  $3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(693=3^{2}\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{693}(64,·)$, $\chi_{693}(1,·)$, $\chi_{693}(529,·)$, $\chi_{693}(631,·)$, $\chi_{693}(466,·)$, $\chi_{693}(592,·)$, $\chi_{693}(625,·)$, $\chi_{693}(562,·)$, $\chi_{693}(499,·)$, $\chi_{693}(214,·)$, $\chi_{693}(247,·)$, $\chi_{693}(25,·)$, $\chi_{693}(58,·)$, $\chi_{693}(379,·)$, $\chi_{693}(190,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{215425782410011849433604372193490013574} a^{14} + \frac{44087925091157478556735995716545418357}{215425782410011849433604372193490013574} a^{13} - \frac{24492180681615608234255414944310603061}{215425782410011849433604372193490013574} a^{12} + \frac{203185972427616345904936937841769485}{1607655092612028727116450538757388161} a^{11} + \frac{1276772543324158003675221037352420694}{107712891205005924716802186096745006787} a^{10} - \frac{73566715492660463632853925194138981471}{215425782410011849433604372193490013574} a^{9} - \frac{37989194393314503047335337951111689713}{107712891205005924716802186096745006787} a^{8} + \frac{62958159753364753905078861273594348653}{215425782410011849433604372193490013574} a^{7} + \frac{23249882737170747775707796806974831719}{107712891205005924716802186096745006787} a^{6} + \frac{26355745021474411685349676979328047276}{107712891205005924716802186096745006787} a^{5} + \frac{22930629291761277321122099836877746839}{107712891205005924716802186096745006787} a^{4} + \frac{34707375451218952534878240828859771552}{107712891205005924716802186096745006787} a^{3} - \frac{51549587519271903953201958596666342051}{107712891205005924716802186096745006787} a^{2} + \frac{12861101179154450086376025664985170055}{107712891205005924716802186096745006787} a + \frac{1187108358755416700450704133809464525}{3215310185224057454232901077514776322}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 5252645612.166227 \) (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

3.3.3969.1, \(\Q(\zeta_{11})^+\)

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{3}$ R $15$ R R $15$ $15$ $15$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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$3.15.20.100$x^{15} + 39 x^{14} + 42 x^{13} + 55 x^{12} + 15 x^{11} + 57 x^{10} + 50 x^{9} + 15 x^{8} + 45 x^{7} + 35 x^{6} + 51 x^{5} + 45 x^{4} + 24 x^{3} + 69 x^{2} + 69 x + 73$$3$$5$$20$$C_{15}$$[2]^{5}$
$7$7.15.10.3$x^{15} - 98 x^{9} + 2401 x^{3} - 268912$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$
$11$11.15.12.1$x^{15} + 165 x^{10} + 5324 x^{5} + 323433$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$