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Properties

Label 15.15.2238775636...1321.1
Degree $15$
Signature $[15, 0]$
Discriminant $11^{12}\cdot 61^{10}$
Root discriminant $105.52$
Ramified primes $11, 61$
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![1201663, -6747207, -6538649, 10129434, 5161658, -4903173, -1336418, 1004274, 150883, -98676, -8166, 4834, 208, -113, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 113*x^13 + 208*x^12 + 4834*x^11 - 8166*x^10 - 98676*x^9 + 150883*x^8 + 1004274*x^7 - 1336418*x^6 - 4903173*x^5 + 5161658*x^4 + 10129434*x^3 - 6538649*x^2 - 6747207*x + 1201663)
 
gp: K = bnfinit(x^15 - 2*x^14 - 113*x^13 + 208*x^12 + 4834*x^11 - 8166*x^10 - 98676*x^9 + 150883*x^8 + 1004274*x^7 - 1336418*x^6 - 4903173*x^5 + 5161658*x^4 + 10129434*x^3 - 6538649*x^2 - 6747207*x + 1201663, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 113 x^{13} + 208 x^{12} + 4834 x^{11} - 8166 x^{10} - 98676 x^{9} + 150883 x^{8} + 1004274 x^{7} - 1336418 x^{6} - 4903173 x^{5} + 5161658 x^{4} + 10129434 x^{3} - 6538649 x^{2} - 6747207 x + 1201663 \)

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:  \(2238775636295572340244024331321=11^{12}\cdot 61^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $105.52$
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:  $11, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(671=11\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{671}(1,·)$, $\chi_{671}(291,·)$, $\chi_{671}(196,·)$, $\chi_{671}(257,·)$, $\chi_{671}(169,·)$, $\chi_{671}(135,·)$, $\chi_{671}(108,·)$, $\chi_{671}(306,·)$, $\chi_{671}(47,·)$, $\chi_{671}(562,·)$, $\chi_{671}(367,·)$, $\chi_{671}(245,·)$, $\chi_{671}(489,·)$, $\chi_{671}(474,·)$, $\chi_{671}(379,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{69} a^{12} - \frac{2}{69} a^{10} - \frac{14}{69} a^{9} + \frac{7}{69} a^{8} - \frac{8}{23} a^{7} + \frac{1}{23} a^{6} + \frac{28}{69} a^{5} - \frac{4}{23} a^{4} + \frac{4}{23} a^{3} + \frac{7}{69} a^{2} + \frac{4}{69} a + \frac{13}{69}$, $\frac{1}{16629} a^{13} - \frac{110}{16629} a^{12} - \frac{890}{5543} a^{11} - \frac{813}{5543} a^{10} + \frac{2950}{16629} a^{9} + \frac{2978}{16629} a^{8} - \frac{1313}{16629} a^{7} + \frac{2573}{16629} a^{6} + \frac{1507}{5543} a^{5} + \frac{7289}{16629} a^{4} + \frac{6967}{16629} a^{3} - \frac{5987}{16629} a^{2} + \frac{379}{5543} a - \frac{7525}{16629}$, $\frac{1}{75922359385401101032090624362953435673} a^{14} + \frac{743850357750581821424903734150455}{25307453128467033677363541454317811891} a^{13} - \frac{121795379357446499690102489835832517}{25307453128467033677363541454317811891} a^{12} + \frac{1406423361540643322961042484712992856}{25307453128467033677363541454317811891} a^{11} + \frac{3557113998329059461538526016415984972}{25307453128467033677363541454317811891} a^{10} - \frac{3571047896060997521718765802937933873}{25307453128467033677363541454317811891} a^{9} + \frac{8625230672575051315349227652643554477}{25307453128467033677363541454317811891} a^{8} - \frac{242477654869727747397734979329178684}{1100324049063784072928849628448600517} a^{7} - \frac{21560357853200825162303785123756517717}{75922359385401101032090624362953435673} a^{6} + \frac{35160862653988296955893253948339950884}{75922359385401101032090624362953435673} a^{5} + \frac{25486169004637888695379388081235933658}{75922359385401101032090624362953435673} a^{4} - \frac{21965960312205191732603471417705484001}{75922359385401101032090624362953435673} a^{3} - \frac{9824414387955724869203069913638830181}{75922359385401101032090624362953435673} a^{2} + \frac{117768998841020025448197739360822134}{284353405937831839071500465778851819} a - \frac{32766217821436901207004713524356834707}{75922359385401101032090624362953435673}$

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:  \( 31742014185.46094 \) (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

3.3.3721.1, \(\Q(\zeta_{11})^+\)

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 $15$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ $15$ $15$ R $15$ $15$ $15$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{15}$ $15$ $15$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ $15$

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$$[\ ]_{5}$
$61$61.15.10.1$x^{15} + 4085658 x^{6} - 13845841 x^{3} + 182432801016$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$