↑

Properties

Label 15.11.5788859223...5009.2
Degree $15$
Signature $[11, 2]$
Discriminant $3^{20}\cdot 11^{12}\cdot 23^{2}$
Root discriminant $44.75$
Ramified primes $3, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T71

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![659, 3186, -3159, -9231, -753, 8388, 5490, -1641, -3123, -664, 564, 252, -31, -27, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 27*x^13 - 31*x^12 + 252*x^11 + 564*x^10 - 664*x^9 - 3123*x^8 - 1641*x^7 + 5490*x^6 + 8388*x^5 - 753*x^4 - 9231*x^3 - 3159*x^2 + 3186*x + 659)
 
gp: K = bnfinit(x^15 - 27*x^13 - 31*x^12 + 252*x^11 + 564*x^10 - 664*x^9 - 3123*x^8 - 1641*x^7 + 5490*x^6 + 8388*x^5 - 753*x^4 - 9231*x^3 - 3159*x^2 + 3186*x + 659, 1)
 

Normalized defining polynomial

\( x^{15} - 27 x^{13} - 31 x^{12} + 252 x^{11} + 564 x^{10} - 664 x^{9} - 3123 x^{8} - 1641 x^{7} + 5490 x^{6} + 8388 x^{5} - 753 x^{4} - 9231 x^{3} - 3159 x^{2} + 3186 x + 659 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5788859223923856660105009=3^{20}\cdot 11^{12}\cdot 23^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.75$
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:  $3, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{591480291533192480431} a^{14} - \frac{39212148885143262580}{591480291533192480431} a^{13} + \frac{106811622707094152678}{591480291533192480431} a^{12} + \frac{160788350478527233018}{591480291533192480431} a^{11} - \frac{76643015507835899133}{591480291533192480431} a^{10} - \frac{17450530767281614246}{591480291533192480431} a^{9} + \frac{21086887607645602030}{591480291533192480431} a^{8} + \frac{204446597141800529088}{591480291533192480431} a^{7} + \frac{120121784412637189877}{591480291533192480431} a^{6} - \frac{236351757433995363373}{591480291533192480431} a^{5} + \frac{231156615422844625226}{591480291533192480431} a^{4} + \frac{69126362358288461780}{591480291533192480431} a^{3} - \frac{104845929854314233035}{591480291533192480431} a^{2} + \frac{277117986275604429246}{591480291533192480431} a - \frac{151591425364157227416}{591480291533192480431}$

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:  $12$
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:  \( 11715038.8016 \) (assuming GRH)
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 $15$ R $15$ $15$ R $15$ $15$ $15$ R $15$ $15$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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
3Data not computed
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$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$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}$