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Properties

Label 15.15.2847534053...6752.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{21}\cdot 3^{20}\cdot 7^{10}\cdot 13^{10}$
Root discriminant $231.01$
Ramified primes $2, 3, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T52

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11603494, 73948203, -161153748, 125382621, 13338780, -50782473, 4842362, 7528962, -602154, -514584, 20328, 16317, -172, -219, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 219*x^13 - 172*x^12 + 16317*x^11 + 20328*x^10 - 514584*x^9 - 602154*x^8 + 7528962*x^7 + 4842362*x^6 - 50782473*x^5 + 13338780*x^4 + 125382621*x^3 - 161153748*x^2 + 73948203*x - 11603494)
 
gp: K = bnfinit(x^15 - 219*x^13 - 172*x^12 + 16317*x^11 + 20328*x^10 - 514584*x^9 - 602154*x^8 + 7528962*x^7 + 4842362*x^6 - 50782473*x^5 + 13338780*x^4 + 125382621*x^3 - 161153748*x^2 + 73948203*x - 11603494, 1)
 

Normalized defining polynomial

\( x^{15} - 219 x^{13} - 172 x^{12} + 16317 x^{11} + 20328 x^{10} - 514584 x^{9} - 602154 x^{8} + 7528962 x^{7} + 4842362 x^{6} - 50782473 x^{5} + 13338780 x^{4} + 125382621 x^{3} - 161153748 x^{2} + 73948203 x - 11603494 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(284753405390815870852035855884746752=2^{21}\cdot 3^{20}\cdot 7^{10}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $231.01$
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:  $2, 3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7} a^{9} - \frac{2}{7} a^{7} + \frac{3}{7} a^{6}$, $\frac{1}{7} a^{10} - \frac{2}{7} a^{8} + \frac{3}{7} a^{7}$, $\frac{1}{7} a^{11} + \frac{3}{7} a^{8} + \frac{3}{7} a^{7} - \frac{1}{7} a^{6}$, $\frac{1}{7} a^{12} + \frac{3}{7} a^{8} - \frac{2}{7} a^{7} - \frac{2}{7} a^{6}$, $\frac{1}{14} a^{13} - \frac{1}{14} a^{12} - \frac{1}{14} a^{11} + \frac{3}{7} a^{8} - \frac{2}{7} a^{7} - \frac{3}{7} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4674929772695879075069169608939723391581628770} a^{14} + \frac{9618643869696867927548418150480956187665251}{2337464886347939537534584804469861695790814385} a^{13} - \frac{31601820240369365956516240474845271286963178}{467492977269587907506916960893972339158162877} a^{12} + \frac{288352222284877421826054346648308012656257583}{4674929772695879075069169608939723391581628770} a^{11} - \frac{18485024203003959558924300727446110313167768}{333923555192562791076369257781408813684402055} a^{10} + \frac{101206054275778115735654459064732867405502122}{2337464886347939537534584804469861695790814385} a^{9} - \frac{721305098569466452732689920079978280324154978}{2337464886347939537534584804469861695790814385} a^{8} + \frac{1037293472833502669296997636638297014573505182}{2337464886347939537534584804469861695790814385} a^{7} - \frac{101701310203427870621718210419669606037630411}{467492977269587907506916960893972339158162877} a^{6} - \frac{84847060938531709038831336006192980167878852}{333923555192562791076369257781408813684402055} a^{5} + \frac{267993293826431095520046218011863054396189833}{667847110385125582152738515562817627368804110} a^{4} + \frac{31227175378875488148804440311953606883566033}{333923555192562791076369257781408813684402055} a^{3} - \frac{15801745123303500083643605399065194254624768}{66784711038512558215273851556281762736880411} a^{2} - \frac{259585041725480834356889869024595038384383889}{667847110385125582152738515562817627368804110} a - \frac{147648971584968819325714186490798608091524022}{333923555192562791076369257781408813684402055}$

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:  $14$
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:  \( 85887003561800 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T52:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 3240
The 24 conjugacy class representatives for [3^4:2]F(5)
Character table for [3^4:2]F(5) is not computed

Intermediate fields

5.5.6889792.1

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 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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
3Data not computed
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.12.10.5$x^{12} + 56 x^{6} + 1323$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$