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Properties

Label 15.3.17440648431...8061.1
Degree $15$
Signature $[3, 6]$
Discriminant $229^{5}\cdot 120721^{2}\cdot 435923^{2}$
Root discriminant $164.48$
Ramified primes $229, 120721, 435923$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T96

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-16384, -512000, 2560000, 16000768, 70400, 618496, 164960, 254304, -648, 2624, 780, 1117, 24, -6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 - 6*x^13 + 24*x^12 + 1117*x^11 + 780*x^10 + 2624*x^9 - 648*x^8 + 254304*x^7 + 164960*x^6 + 618496*x^5 + 70400*x^4 + 16000768*x^3 + 2560000*x^2 - 512000*x - 16384)
 
gp: K = bnfinit(x^15 - 4*x^14 - 6*x^13 + 24*x^12 + 1117*x^11 + 780*x^10 + 2624*x^9 - 648*x^8 + 254304*x^7 + 164960*x^6 + 618496*x^5 + 70400*x^4 + 16000768*x^3 + 2560000*x^2 - 512000*x - 16384, 1)
 

Normalized defining polynomial

\( x^{15} - 4 x^{14} - 6 x^{13} + 24 x^{12} + 1117 x^{11} + 780 x^{10} + 2624 x^{9} - 648 x^{8} + 254304 x^{7} + 164960 x^{6} + 618496 x^{5} + 70400 x^{4} + 16000768 x^{3} + 2560000 x^{2} - 512000 x - 16384 \)

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:  \(1744064843158258782713344097088061=229^{5}\cdot 120721^{2}\cdot 435923^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $164.48$
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:  $229, 120721, 435923$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} + \frac{1}{16} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{7} - \frac{1}{16} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{32} a^{8} - \frac{1}{32} a^{7} - \frac{1}{32} a^{6} + \frac{1}{32} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{64} a^{9} - \frac{1}{64} a^{8} - \frac{1}{64} a^{7} + \frac{1}{64} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4}$, $\frac{1}{64} a^{10} - \frac{1}{32} a^{7} - \frac{1}{64} a^{6} + \frac{1}{32} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{896} a^{11} - \frac{1}{448} a^{10} + \frac{1}{448} a^{9} + \frac{25}{896} a^{7} + \frac{9}{448} a^{6} + \frac{3}{224} a^{5} - \frac{11}{112} a^{4} - \frac{1}{8} a^{3} - \frac{1}{7} a^{2} - \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{3584} a^{12} - \frac{1}{1792} a^{11} - \frac{13}{1792} a^{10} - \frac{1}{128} a^{9} + \frac{53}{3584} a^{8} + \frac{51}{1792} a^{7} + \frac{3}{896} a^{6} - \frac{1}{112} a^{5} - \frac{1}{16} a^{4} + \frac{3}{112} a^{3} + \frac{3}{56} a^{2} - \frac{11}{28} a$, $\frac{1}{14336} a^{13} - \frac{1}{7168} a^{12} - \frac{1}{7168} a^{11} + \frac{9}{3584} a^{10} - \frac{11}{14336} a^{9} - \frac{5}{7168} a^{8} - \frac{99}{3584} a^{7} + \frac{25}{896} a^{6} - \frac{19}{448} a^{5} - \frac{1}{64} a^{4} + \frac{17}{224} a^{3} - \frac{1}{16} a^{2} + \frac{9}{28} a + \frac{1}{7}$, $\frac{1}{198516752534832735401792909312} a^{14} - \frac{1747652612161602982957757}{99258376267416367700896454656} a^{13} + \frac{9841818798254026951907843}{99258376267416367700896454656} a^{12} - \frac{11736245571935227209057439}{49629188133708183850448227328} a^{11} + \frac{1375704891696291261974291013}{198516752534832735401792909312} a^{10} + \frac{62575588479643685450237551}{99258376267416367700896454656} a^{9} - \frac{1128542328375064366432603}{90729777209704175229338624} a^{8} - \frac{1944449497092843315269335}{443117751193823070093287744} a^{7} - \frac{2429734458427531435517227}{96932008073648796582906694} a^{6} + \frac{52520641702758545495597939}{886235502387646140186575488} a^{5} - \frac{339685073698211090695868475}{3101824258356761490653014208} a^{4} - \frac{155583665231300462412098081}{1550912129178380745326507104} a^{3} + \frac{3486011369308699663947737}{48466004036824398291453347} a^{2} - \frac{22952855602415745302976909}{193864016147297593165813388} a - \frac{2070501400885834805376822}{6923714862403485470207621}$

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:  \( 60181699125.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T96:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1296000
The 65 conjugacy class representatives for [A(5)^3]S(3)=A(5)wrS(3) are not computed
Character table for [A(5)^3]S(3)=A(5)wrS(3) is not computed

Intermediate fields

3.3.229.1

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

Sibling fields

Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 siblings: data not computed
Degree 45 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ $15$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
229Data not computed
120721Data not computed
435923Data not computed