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Properties

Label 15.15.1037696513...5169.1
Degree $15$
Signature $[15, 0]$
Discriminant $7^{12}\cdot 83^{2}\cdot 349^{2}\cdot 94524289^{2}$
Root discriminant $215.98$
Ramified primes $7, 83, 349, 94524289$
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![-1512, 322560, -17470362, 272554703, -2899414, -68843164, 798140, 7059483, -54292, -375360, 1470, 10901, -14, -164, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 164*x^13 - 14*x^12 + 10901*x^11 + 1470*x^10 - 375360*x^9 - 54292*x^8 + 7059483*x^7 + 798140*x^6 - 68843164*x^5 - 2899414*x^4 + 272554703*x^3 - 17470362*x^2 + 322560*x - 1512)
 
gp: K = bnfinit(x^15 - 164*x^13 - 14*x^12 + 10901*x^11 + 1470*x^10 - 375360*x^9 - 54292*x^8 + 7059483*x^7 + 798140*x^6 - 68843164*x^5 - 2899414*x^4 + 272554703*x^3 - 17470362*x^2 + 322560*x - 1512, 1)
 

Normalized defining polynomial

\( x^{15} - 164 x^{13} - 14 x^{12} + 10901 x^{11} + 1470 x^{10} - 375360 x^{9} - 54292 x^{8} + 7059483 x^{7} + 798140 x^{6} - 68843164 x^{5} - 2899414 x^{4} + 272554703 x^{3} - 17470362 x^{2} + 322560 x - 1512 \)

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:  \(103769651336273628596409096948945169=7^{12}\cdot 83^{2}\cdot 349^{2}\cdot 94524289^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $215.98$
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:  $7, 83, 349, 94524289$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{7} - \frac{1}{8} a^{4} - \frac{1}{16} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{96} a^{8} - \frac{1}{96} a^{7} + \frac{1}{24} a^{6} - \frac{5}{48} a^{5} - \frac{7}{96} a^{4} + \frac{1}{32} a^{3} - \frac{1}{16} a^{2} + \frac{5}{12} a - \frac{1}{4}$, $\frac{1}{192} a^{9} + \frac{1}{64} a^{7} - \frac{1}{32} a^{6} + \frac{7}{192} a^{5} + \frac{5}{48} a^{4} + \frac{7}{64} a^{3} + \frac{5}{96} a^{2} + \frac{1}{3} a + \frac{3}{8}$, $\frac{1}{4608} a^{10} + \frac{1}{4608} a^{9} - \frac{13}{4608} a^{8} - \frac{35}{4608} a^{7} - \frac{5}{1536} a^{6} + \frac{571}{4608} a^{5} + \frac{17}{512} a^{4} + \frac{991}{4608} a^{3} - \frac{35}{2304} a^{2} - \frac{31}{64} a - \frac{23}{64}$, $\frac{1}{36864} a^{11} - \frac{1}{9216} a^{10} - \frac{11}{6144} a^{9} + \frac{5}{6144} a^{8} - \frac{71}{2304} a^{7} + \frac{1043}{18432} a^{6} - \frac{2095}{18432} a^{5} + \frac{1361}{18432} a^{4} - \frac{2011}{12288} a^{3} - \frac{1181}{18432} a^{2} - \frac{149}{384} a + \frac{195}{512}$, $\frac{1}{884736} a^{12} - \frac{1}{98304} a^{11} + \frac{41}{442368} a^{10} - \frac{83}{110592} a^{9} - \frac{1091}{442368} a^{8} + \frac{1835}{442368} a^{7} - \frac{1639}{221184} a^{6} - \frac{2545}{110592} a^{5} - \frac{10043}{884736} a^{4} + \frac{118939}{884736} a^{3} + \frac{29195}{147456} a^{2} - \frac{4913}{12288} a - \frac{69}{4096}$, $\frac{1}{21233664} a^{13} + \frac{1}{10616832} a^{12} - \frac{17}{21233664} a^{11} + \frac{119}{10616832} a^{10} + \frac{497}{1179648} a^{9} + \frac{13349}{5308416} a^{8} - \frac{158197}{10616832} a^{7} - \frac{253519}{5308416} a^{6} + \frac{309071}{7077888} a^{5} - \frac{281213}{3538944} a^{4} + \frac{3308267}{21233664} a^{3} + \frac{661549}{3538944} a^{2} + \frac{44555}{147456} a + \frac{38153}{98304}$, $\frac{1}{509607936} a^{14} - \frac{11}{509607936} a^{13} - \frac{43}{509607936} a^{12} + \frac{17}{18874368} a^{11} + \frac{1463}{127401984} a^{10} - \frac{31451}{254803968} a^{9} + \frac{158281}{254803968} a^{8} - \frac{1768237}{254803968} a^{7} - \frac{17740295}{509607936} a^{6} - \frac{6540287}{56623104} a^{5} - \frac{58364671}{509607936} a^{4} - \frac{125299937}{509607936} a^{3} + \frac{9721535}{84934656} a^{2} - \frac{2076163}{7077888} a - \frac{643445}{2359296}$

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:  \( 11611996840600 \) (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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{5}$ R $15$ ${\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $15$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ $15$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$83$$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.5.0.1$x^{5} - x + 3$$1$$5$$0$$C_5$$[\ ]^{5}$
83.5.0.1$x^{5} - x + 3$$1$$5$$0$$C_5$$[\ ]^{5}$
349Data not computed
94524289Data not computed