Properties

Label 15.5.154972454814106259.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,11^{13}\cdot 67^{2}$
Root discriminant $14.00$
Ramified primes $11, 67$
Class number $1$
Class group Trivial
Galois group 15T44

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, -7, 27, -7, -34, 28, 8, -15, 2, -1, 1, 1, -2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 2*x^13 + x^12 + x^11 - x^10 + 2*x^9 - 15*x^8 + 8*x^7 + 28*x^6 - 34*x^5 - 7*x^4 + 27*x^3 - 7*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^15 - x^14 - 2*x^13 + x^12 + x^11 - x^10 + 2*x^9 - 15*x^8 + 8*x^7 + 28*x^6 - 34*x^5 - 7*x^4 + 27*x^3 - 7*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 2 x^{13} + x^{12} + x^{11} - x^{10} + 2 x^{9} - 15 x^{8} + 8 x^{7} + 28 x^{6} - 34 x^{5} - 7 x^{4} + 27 x^{3} - 7 x^{2} - 3 x + 1 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-154972454814106259=-\,11^{13}\cdot 67^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.00$
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:  $11, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1375967} a^{14} + \frac{175735}{1375967} a^{13} - \frac{613357}{1375967} a^{12} + \frac{221128}{1375967} a^{11} + \frac{90195}{1375967} a^{10} - \frac{631321}{1375967} a^{9} - \frac{232077}{1375967} a^{8} - \frac{621807}{1375967} a^{7} - \frac{79672}{1375967} a^{6} + \frac{601628}{1375967} a^{5} - \frac{230139}{1375967} a^{4} + \frac{90720}{1375967} a^{3} - \frac{559682}{1375967} a^{2} + \frac{597135}{1375967} a - \frac{6898}{1375967}$

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:  $9$
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:  \( 492.79424719 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T44:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2430
The 39 conjugacy class representatives for [3^5:2]5
Character table for [3^5:2]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 15 siblings: data not computed
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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ $15$ $15$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ $15$ $15$ $15$

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$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
67Data not computed