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Properties

Label 15.7.30697627022...1241.1
Degree $15$
Signature $[7, 4]$
Discriminant $11^{12}\cdot 23^{2}\cdot 43^{2}$
Root discriminant $17.08$
Ramified primes $11, 23, 43$
Class number $1$
Class group Trivial
Galois group 15T71

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, 5, 34, -38, 47, -48, -19, 45, -38, 30, -9, -10, 13, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 + 13*x^13 - 10*x^12 - 9*x^11 + 30*x^10 - 38*x^9 + 45*x^8 - 19*x^7 - 48*x^6 + 47*x^5 - 38*x^4 + 34*x^3 + 5*x^2 - 7*x + 1)
 
gp: K = bnfinit(x^15 - 6*x^14 + 13*x^13 - 10*x^12 - 9*x^11 + 30*x^10 - 38*x^9 + 45*x^8 - 19*x^7 - 48*x^6 + 47*x^5 - 38*x^4 + 34*x^3 + 5*x^2 - 7*x + 1, 1)
 

Normalized defining polynomial

\( x^{15} - 6 x^{14} + 13 x^{13} - 10 x^{12} - 9 x^{11} + 30 x^{10} - 38 x^{9} + 45 x^{8} - 19 x^{7} - 48 x^{6} + 47 x^{5} - 38 x^{4} + 34 x^{3} + 5 x^{2} - 7 x + 1 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3069762702266721241=11^{12}\cdot 23^{2}\cdot 43^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.08$
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:  $11, 23, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{47485327} a^{14} - \frac{1534654}{47485327} a^{13} - \frac{23556741}{47485327} a^{12} + \frac{13737153}{47485327} a^{11} - \frac{13631579}{47485327} a^{10} + \frac{14639372}{47485327} a^{9} + \frac{22434473}{47485327} a^{8} - \frac{20205744}{47485327} a^{7} - \frac{21163466}{47485327} a^{6} - \frac{20852943}{47485327} a^{5} - \frac{1611980}{47485327} a^{4} - \frac{21197717}{47485327} a^{3} + \frac{23604125}{47485327} a^{2} + \frac{16022974}{47485327} a - \frac{13210787}{47485327}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $10$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3397.28173407 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T71:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 19440
The 39 conjugacy class representatives for [1/2.S(3)^5]5
Character table for [1/2.S(3)^5]5 is not computed

Intermediate fields

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

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 30 siblings: data not computed
Degree 45 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $15$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ R $15$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ R $15$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ R $15$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ $15$

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
11Data not computed
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
43Data not computed