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Properties

Label 15.7.33455467633...3489.1
Degree $15$
Signature $[7, 4]$
Discriminant $11^{12}\cdot 23^{2}\cdot 67^{4}$
Root discriminant $31.74$
Ramified primes $11, 23, 67$
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![859, -2460, 2020, 445, -2082, 2450, -1896, 603, 345, -478, 267, -81, -2, 14, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 + 14*x^13 - 2*x^12 - 81*x^11 + 267*x^10 - 478*x^9 + 345*x^8 + 603*x^7 - 1896*x^6 + 2450*x^5 - 2082*x^4 + 445*x^3 + 2020*x^2 - 2460*x + 859)
 
gp: K = bnfinit(x^15 - 6*x^14 + 14*x^13 - 2*x^12 - 81*x^11 + 267*x^10 - 478*x^9 + 345*x^8 + 603*x^7 - 1896*x^6 + 2450*x^5 - 2082*x^4 + 445*x^3 + 2020*x^2 - 2460*x + 859, 1)
 

Normalized defining polynomial

\( x^{15} - 6 x^{14} + 14 x^{13} - 2 x^{12} - 81 x^{11} + 267 x^{10} - 478 x^{9} + 345 x^{8} + 603 x^{7} - 1896 x^{6} + 2450 x^{5} - 2082 x^{4} + 445 x^{3} + 2020 x^{2} - 2460 x + 859 \)

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:  \(33455467633674242293489=11^{12}\cdot 23^{2}\cdot 67^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.74$
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:  $11, 23, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{879107737270024217} a^{14} - \frac{260825519780186402}{879107737270024217} a^{13} - \frac{308027841552522970}{879107737270024217} a^{12} + \frac{340569343986615610}{879107737270024217} a^{11} - \frac{100201998029196743}{879107737270024217} a^{10} - \frac{135623134720473803}{879107737270024217} a^{9} + \frac{339581517866733450}{879107737270024217} a^{8} + \frac{6874727396203384}{879107737270024217} a^{7} + \frac{191610931063860238}{879107737270024217} a^{6} + \frac{7149641149422220}{20444365983023819} a^{5} + \frac{302731134573268382}{879107737270024217} a^{4} + \frac{160496230888982030}{879107737270024217} a^{3} + \frac{91897857262594212}{879107737270024217} a^{2} - \frac{215857576283500374}{879107737270024217} a + \frac{291991300061784854}{879107737270024217}$

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:  \( 316012.234645 \)
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 $15$ $15$ $15$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ R $15$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ $15$ R $15$ $15$ $15$ $15$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ $15$ $15$

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
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$$0$Trivial$[\ ]$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
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}$
23.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
23.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
$67$$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.3.0.1$x^{3} - x + 16$$1$$3$$0$$C_3$$[\ ]^{3}$
67.3.2.2$x^{3} + 268$$3$$1$$2$$C_3$$[\ ]_{3}$
67.3.2.2$x^{3} + 268$$3$$1$$2$$C_3$$[\ ]_{3}$