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Properties

Label 15.15.2971550633...7921.1
Degree $15$
Signature $[15, 0]$
Discriminant $11^{12}\cdot 79^{10}$
Root discriminant $125.37$
Ramified primes $11, 79$
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![-31658639, -160431365, -288431111, -235831716, -75224192, 12430527, 14598886, 2075724, -786687, -219872, 14904, 8246, 0, -143, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 143*x^13 + 8246*x^11 + 14904*x^10 - 219872*x^9 - 786687*x^8 + 2075724*x^7 + 14598886*x^6 + 12430527*x^5 - 75224192*x^4 - 235831716*x^3 - 288431111*x^2 - 160431365*x - 31658639)
 
gp: K = bnfinit(x^15 - 2*x^14 - 143*x^13 + 8246*x^11 + 14904*x^10 - 219872*x^9 - 786687*x^8 + 2075724*x^7 + 14598886*x^6 + 12430527*x^5 - 75224192*x^4 - 235831716*x^3 - 288431111*x^2 - 160431365*x - 31658639, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 143 x^{13} + 8246 x^{11} + 14904 x^{10} - 219872 x^{9} - 786687 x^{8} + 2075724 x^{7} + 14598886 x^{6} + 12430527 x^{5} - 75224192 x^{4} - 235831716 x^{3} - 288431111 x^{2} - 160431365 x - 31658639 \)

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:  \(29715506336344844930608532407921=11^{12}\cdot 79^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $125.37$
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:  $11, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(869=11\cdot 79\)
Dirichlet character group:    $\lbrace$$\chi_{869}(576,·)$, $\chi_{869}(1,·)$, $\chi_{869}(102,·)$, $\chi_{869}(554,·)$, $\chi_{869}(845,·)$, $\chi_{869}(687,·)$, $\chi_{869}(80,·)$, $\chi_{869}(529,·)$, $\chi_{869}(339,·)$, $\chi_{869}(213,·)$, $\chi_{869}(23,·)$, $\chi_{869}(608,·)$, $\chi_{869}(159,·)$, $\chi_{869}(317,·)$, $\chi_{869}(181,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{223067537909890884217300218957468426947} a^{14} + \frac{56886692286278735510646654840065791561}{223067537909890884217300218957468426947} a^{13} + \frac{101318305369077898643052249354844834796}{223067537909890884217300218957468426947} a^{12} + \frac{21678463956372739290103200590042158906}{223067537909890884217300218957468426947} a^{11} - \frac{71526624288994460266849611564575326260}{223067537909890884217300218957468426947} a^{10} - \frac{30913609047569453833449913177190218609}{223067537909890884217300218957468426947} a^{9} - \frac{109094159413731616192685638615680394529}{223067537909890884217300218957468426947} a^{8} + \frac{15713207295443673690540121940292371915}{223067537909890884217300218957468426947} a^{7} - \frac{51120570810107325675971981339845789970}{223067537909890884217300218957468426947} a^{6} + \frac{48925773291469038285858169985298924512}{223067537909890884217300218957468426947} a^{5} + \frac{66439627061255413177608191999094326213}{223067537909890884217300218957468426947} a^{4} - \frac{80625202064355176122685683602469540122}{223067537909890884217300218957468426947} a^{3} + \frac{82864884068779128876572465759464789076}{223067537909890884217300218957468426947} a^{2} - \frac{21594301065229192854312470787730726967}{223067537909890884217300218957468426947} a + \frac{406648275031361465029135647150858}{923029175897160513832143216372263}$

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:  \( 61238384087.82688 \) (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

3.3.6241.1, \(\Q(\zeta_{11})^+\)

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 $15$ $15$ $15$ $15$ R $15$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}$ $15$ $15$ $15$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{5}$ $15$ $15$ $15$

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$11.15.12.1$x^{15} + 165 x^{10} + 5324 x^{5} + 323433$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$
$79$79.15.10.1$x^{15} + 23665872 x^{6} - 38950081 x^{3} + 12603623010304$$3$$5$$10$$C_{15}$$[\ ]_{3}^{5}$