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Properties

Label 15.15.4326590027...7129.1
Degree $15$
Signature $[15, 0]$
Discriminant $11^{12}\cdot 13^{10}$
Root discriminant $37.65$
Ramified primes $11, 13$
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![131, -373, -1103, 2758, 2926, -6581, -2740, 5700, 1335, -2214, -362, 402, 46, -33, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 33*x^13 + 46*x^12 + 402*x^11 - 362*x^10 - 2214*x^9 + 1335*x^8 + 5700*x^7 - 2740*x^6 - 6581*x^5 + 2926*x^4 + 2758*x^3 - 1103*x^2 - 373*x + 131)
 
gp: K = bnfinit(x^15 - 2*x^14 - 33*x^13 + 46*x^12 + 402*x^11 - 362*x^10 - 2214*x^9 + 1335*x^8 + 5700*x^7 - 2740*x^6 - 6581*x^5 + 2926*x^4 + 2758*x^3 - 1103*x^2 - 373*x + 131, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 33 x^{13} + 46 x^{12} + 402 x^{11} - 362 x^{10} - 2214 x^{9} + 1335 x^{8} + 5700 x^{7} - 2740 x^{6} - 6581 x^{5} + 2926 x^{4} + 2758 x^{3} - 1103 x^{2} - 373 x + 131 \)

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:  \(432659002790862279847129=11^{12}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.65$
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:  $11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(143=11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{143}(1,·)$, $\chi_{143}(3,·)$, $\chi_{143}(100,·)$, $\chi_{143}(133,·)$, $\chi_{143}(16,·)$, $\chi_{143}(113,·)$, $\chi_{143}(9,·)$, $\chi_{143}(42,·)$, $\chi_{143}(14,·)$, $\chi_{143}(48,·)$, $\chi_{143}(81,·)$, $\chi_{143}(53,·)$, $\chi_{143}(27,·)$, $\chi_{143}(92,·)$, $\chi_{143}(126,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{210473000101397461} a^{14} + \frac{86646957983386562}{210473000101397461} a^{13} - \frac{44444239855040036}{210473000101397461} a^{12} + \frac{45138934947666029}{210473000101397461} a^{11} - \frac{60372376613571248}{210473000101397461} a^{10} + \frac{12886799768084115}{210473000101397461} a^{9} + \frac{84129775593377451}{210473000101397461} a^{8} - \frac{80096312558798962}{210473000101397461} a^{7} - \frac{38966385761037156}{210473000101397461} a^{6} - \frac{60634527264102097}{210473000101397461} a^{5} - \frac{97356638096463563}{210473000101397461} a^{4} + \frac{25518505785938775}{210473000101397461} a^{3} - \frac{69726274149756796}{210473000101397461} a^{2} - \frac{39904378281570716}{210473000101397461} a - \frac{577463602703962}{1606664122911431}$

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:  \( 5243886.21648 \) (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

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

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 $15$ $15$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ $15$ R R $15$ $15$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ $15$

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
11Data not computed
13Data not computed