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Properties

Label 15.3.60500833661...2241.1
Degree $15$
Signature $[3, 6]$
Discriminant $3^{20}\cdot 1609^{6}$
Root discriminant $82.94$
Ramified primes $3, 1609$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $\GL(2,4):C_2$ (as 15T21)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-152252029, -315011013, -281605005, -144815383, -45591102, -6688395, 1019238, 675537, 107811, -5103, -4551, -477, 0, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 18*x^13 - 477*x^11 - 4551*x^10 - 5103*x^9 + 107811*x^8 + 675537*x^7 + 1019238*x^6 - 6688395*x^5 - 45591102*x^4 - 144815383*x^3 - 281605005*x^2 - 315011013*x - 152252029)
 
gp: K = bnfinit(x^15 - 18*x^13 - 477*x^11 - 4551*x^10 - 5103*x^9 + 107811*x^8 + 675537*x^7 + 1019238*x^6 - 6688395*x^5 - 45591102*x^4 - 144815383*x^3 - 281605005*x^2 - 315011013*x - 152252029, 1)
 

Normalized defining polynomial

\( x^{15} - 18 x^{13} - 477 x^{11} - 4551 x^{10} - 5103 x^{9} + 107811 x^{8} + 675537 x^{7} + 1019238 x^{6} - 6688395 x^{5} - 45591102 x^{4} - 144815383 x^{3} - 281605005 x^{2} - 315011013 x - 152252029 \)

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:  \(60500833661891746134479882241=3^{20}\cdot 1609^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.94$
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:  $3, 1609$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{19353167455538682107017342735089268427451049653046037} a^{14} - \frac{970642853440073441394743808518257606508568328404534}{19353167455538682107017342735089268427451049653046037} a^{13} - \frac{1012439943018849684837841811406555058363566959540576}{19353167455538682107017342735089268427451049653046037} a^{12} - \frac{2142948129477390388286866266810574451637265767196321}{19353167455538682107017342735089268427451049653046037} a^{11} + \frac{7406249388911122754394759852699555301409898612501124}{19353167455538682107017342735089268427451049653046037} a^{10} + \frac{5224049982094068402904242549740219791335901571475831}{19353167455538682107017342735089268427451049653046037} a^{9} - \frac{8732121511208478588307909086603917306700171235978337}{19353167455538682107017342735089268427451049653046037} a^{8} - \frac{4382162326107451071060643327379039302245511800638715}{19353167455538682107017342735089268427451049653046037} a^{7} + \frac{1218485047641512822739904946320163965002348498866454}{19353167455538682107017342735089268427451049653046037} a^{6} + \frac{2687734901445717284955703519861798190708910798148710}{19353167455538682107017342735089268427451049653046037} a^{5} - \frac{3921361255202553893711448684870460780451485239070379}{19353167455538682107017342735089268427451049653046037} a^{4} + \frac{6645851321340572095571787612291407010184566616067229}{19353167455538682107017342735089268427451049653046037} a^{3} - \frac{9001573984276716513615680457246990210521056648567050}{19353167455538682107017342735089268427451049653046037} a^{2} + \frac{4447592314004868959572294083245899862278995975274680}{19353167455538682107017342735089268427451049653046037} a - \frac{6161099884889560479002109747504533305405117527001989}{19353167455538682107017342735089268427451049653046037}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 84559284.0776 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_5$ (as 15T21):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 360
The 12 conjugacy class representatives for $\GL(2,4):C_2$
Character table for $\GL(2,4):C_2$

Intermediate fields

5.1.1609.1

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

Sibling fields

Degree 15 sibling: data not computed
Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 45 sibling: data not computed
Arithmetically equvalently 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{3}$ R $15$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
3Data not computed
1609Data not computed