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Properties

Label 15.3.11956105637...0625.1
Degree $15$
Signature $[3, 6]$
Discriminant $5^{18}\cdot 7^{10}\cdot 3331^{2}$
Root discriminant $74.45$
Ramified primes $5, 7, 3331$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T50

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7279, -39745, -58420, -28890, 5450, 16836, 6980, -1950, -1670, 50, 139, 0, -10, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^13 - 10*x^12 + 139*x^10 + 50*x^9 - 1670*x^8 - 1950*x^7 + 6980*x^6 + 16836*x^5 + 5450*x^4 - 28890*x^3 - 58420*x^2 - 39745*x - 7279)
 
gp: K = bnfinit(x^15 - 5*x^13 - 10*x^12 + 139*x^10 + 50*x^9 - 1670*x^8 - 1950*x^7 + 6980*x^6 + 16836*x^5 + 5450*x^4 - 28890*x^3 - 58420*x^2 - 39745*x - 7279, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{13} - 10 x^{12} + 139 x^{10} + 50 x^{9} - 1670 x^{8} - 1950 x^{7} + 6980 x^{6} + 16836 x^{5} + 5450 x^{4} - 28890 x^{3} - 58420 x^{2} - 39745 x - 7279 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(11956105637625461578369140625=5^{18}\cdot 7^{10}\cdot 3331^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.45$
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:  $5, 7, 3331$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{4}{9} a^{9} - \frac{2}{9} a^{7} + \frac{4}{9} a^{6} + \frac{4}{9} a^{5} + \frac{1}{3} a^{4} + \frac{4}{9} a^{3} - \frac{1}{9} a^{2} - \frac{2}{9} a + \frac{4}{9}$, $\frac{1}{6144937860798243589029831} a^{14} - \frac{34824207719850082205870}{6144937860798243589029831} a^{13} + \frac{9475751182554040130653}{211894408993042882380339} a^{12} - \frac{16165345256531606796293}{6144937860798243589029831} a^{11} + \frac{2738263225803473445258364}{6144937860798243589029831} a^{10} + \frac{975741638023252559987378}{6144937860798243589029831} a^{9} - \frac{2481498844620022624235885}{6144937860798243589029831} a^{8} + \frac{1340878305100461645441458}{6144937860798243589029831} a^{7} - \frac{1160701159827375222889384}{6144937860798243589029831} a^{6} + \frac{1068599863862933032160365}{6144937860798243589029831} a^{5} - \frac{1990972900920426481972613}{6144937860798243589029831} a^{4} - \frac{465873280574681692951837}{2048312620266081196343277} a^{3} - \frac{253128519715832202422017}{682770873422027065447759} a^{2} - \frac{120041593024896895238680}{472687527753711045309987} a + \frac{92290048208835422672054}{211894408993042882380339}$

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:  $8$
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:  \( 191983513.22 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T50:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 3000
The 32 conjugacy class representatives for [D(5)^3]3=D(5)wr3
Character table for [D(5)^3]3=D(5)wr3 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\)

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

Sibling fields

Degree 20 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: 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 $15$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{5}$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
5Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
3331Data not computed