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Properties

Label 15.3.12246301270...1129.1
Degree $15$
Signature $[3, 6]$
Discriminant $3^{5}\cdot 11^{12}\cdot 107^{7}$
Root discriminant $86.94$
Ramified primes $3, 11, 107$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $((C_5^2 : C_3):C_2):C_2$ (as 15T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-295769, 651327, -167023, -32759, 36562, -173362, 53930, -1890, 15525, 1338, 708, 365, -15, 29, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 29*x^13 - 15*x^12 + 365*x^11 + 708*x^10 + 1338*x^9 + 15525*x^8 - 1890*x^7 + 53930*x^6 - 173362*x^5 + 36562*x^4 - 32759*x^3 - 167023*x^2 + 651327*x - 295769)
 
gp: K = bnfinit(x^15 + 29*x^13 - 15*x^12 + 365*x^11 + 708*x^10 + 1338*x^9 + 15525*x^8 - 1890*x^7 + 53930*x^6 - 173362*x^5 + 36562*x^4 - 32759*x^3 - 167023*x^2 + 651327*x - 295769, 1)
 

Normalized defining polynomial

\( x^{15} + 29 x^{13} - 15 x^{12} + 365 x^{11} + 708 x^{10} + 1338 x^{9} + 15525 x^{8} - 1890 x^{7} + 53930 x^{6} - 173362 x^{5} + 36562 x^{4} - 32759 x^{3} - 167023 x^{2} + 651327 x - 295769 \)

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:  \(122463012708006247915836261129=3^{5}\cdot 11^{12}\cdot 107^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.94$
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:  $3, 11, 107$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{62} a^{13} + \frac{9}{62} a^{12} - \frac{10}{31} a^{11} + \frac{15}{31} a^{10} + \frac{11}{62} a^{9} + \frac{7}{62} a^{8} - \frac{29}{62} a^{7} + \frac{1}{62} a^{6} + \frac{29}{62} a^{5} + \frac{14}{31} a^{4} + \frac{3}{31} a^{3} + \frac{23}{62} a^{2} - \frac{7}{62} a + \frac{21}{62}$, $\frac{1}{238930512228261527687946414375287656611238} a^{14} - \frac{1467713306221338666177721880612510955881}{238930512228261527687946414375287656611238} a^{13} - \frac{19832188730238277833212155366623605255231}{119465256114130763843973207187643828305619} a^{12} + \frac{43290950428674017562201958707186626236001}{119465256114130763843973207187643828305619} a^{11} - \frac{15504270559451192801414848936251812671861}{238930512228261527687946414375287656611238} a^{10} - \frac{62107104270686249494925605730404383405305}{238930512228261527687946414375287656611238} a^{9} + \frac{65816129968824044554182188405442888618009}{238930512228261527687946414375287656611238} a^{8} + \frac{114654846358245643203051886097360340832499}{238930512228261527687946414375287656611238} a^{7} + \frac{107664798246220679735687560340893119580477}{238930512228261527687946414375287656611238} a^{6} - \frac{56000323828394451318531669026268693515114}{119465256114130763843973207187643828305619} a^{5} - \frac{56216293910714537501945053783178830756730}{119465256114130763843973207187643828305619} a^{4} + \frac{72727052671187117188269700661759640986185}{238930512228261527687946414375287656611238} a^{3} + \frac{95457240129781724916059302168504467910119}{238930512228261527687946414375287656611238} a^{2} + \frac{82248333636111785865219446652544641331407}{238930512228261527687946414375287656611238} a - \frac{58323468085016394450977416109360856473130}{119465256114130763843973207187643828305619}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( 66275507.7691 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2:D_6$ (as 15T18):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 300
The 14 conjugacy class representatives for $((C_5^2 : C_3):C_2):C_2$
Character table for $((C_5^2 : C_3):C_2):C_2$

Intermediate fields

3.3.321.1

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 25 sibling: data not computed
Degree 30 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$3$3.5.0.1$x^{5} - x + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.5.4.5$x^{5} - 99$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.8.3$x^{10} - 11 x^{5} + 847$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
107Data not computed