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Properties

Label 15.15.1286032269...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{8}\cdot 3^{20}\cdot 5^{8}\cdot 11^{6}\cdot 113^{6}$
Root discriminant $255.44$
Ramified primes $2, 3, 5, 11, 113$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T88

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-12500000, 48750000, 156375000, 87752500, -45475500, -40117500, 4605500, 6395700, -231300, -457975, 7605, 14715, -75, -204, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 204*x^13 - 75*x^12 + 14715*x^11 + 7605*x^10 - 457975*x^9 - 231300*x^8 + 6395700*x^7 + 4605500*x^6 - 40117500*x^5 - 45475500*x^4 + 87752500*x^3 + 156375000*x^2 + 48750000*x - 12500000)
 
gp: K = bnfinit(x^15 - 204*x^13 - 75*x^12 + 14715*x^11 + 7605*x^10 - 457975*x^9 - 231300*x^8 + 6395700*x^7 + 4605500*x^6 - 40117500*x^5 - 45475500*x^4 + 87752500*x^3 + 156375000*x^2 + 48750000*x - 12500000, 1)
 

Normalized defining polynomial

\( x^{15} - 204 x^{13} - 75 x^{12} + 14715 x^{11} + 7605 x^{10} - 457975 x^{9} - 231300 x^{8} + 6395700 x^{7} + 4605500 x^{6} - 40117500 x^{5} - 45475500 x^{4} + 87752500 x^{3} + 156375000 x^{2} + 48750000 x - 12500000 \)

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:  \(1286032269717057992679562824900000000=2^{8}\cdot 3^{20}\cdot 5^{8}\cdot 11^{6}\cdot 113^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $255.44$
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:  $2, 3, 5, 11, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{6} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{25} a^{8} + \frac{1}{25} a^{6} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3}$, $\frac{1}{50} a^{9} - \frac{2}{25} a^{7} - \frac{1}{10} a^{6} - \frac{1}{10} a^{5} - \frac{1}{2} a^{4} + \frac{1}{10} a^{3}$, $\frac{1}{50} a^{10} - \frac{1}{10} a^{7} - \frac{1}{50} a^{6} - \frac{1}{10} a^{5} - \frac{3}{10} a^{4} - \frac{1}{5} a^{3}$, $\frac{1}{500} a^{11} - \frac{1}{125} a^{9} - \frac{1}{100} a^{8} + \frac{3}{100} a^{7} + \frac{1}{20} a^{6} - \frac{1}{20} a^{5} - \frac{1}{10} a^{4} + \frac{3}{10} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2500} a^{12} - \frac{1}{625} a^{10} - \frac{1}{100} a^{9} + \frac{3}{500} a^{8} - \frac{19}{500} a^{7} - \frac{1}{20} a^{6} - \frac{1}{50} a^{5} + \frac{9}{50} a^{4} - \frac{3}{10} a^{3} - \frac{1}{5} a$, $\frac{1}{50000} a^{13} - \frac{1}{12500} a^{11} - \frac{3}{2000} a^{10} - \frac{17}{10000} a^{9} - \frac{79}{10000} a^{8} + \frac{41}{2000} a^{7} + \frac{17}{500} a^{6} - \frac{43}{500} a^{5} + \frac{41}{100} a^{4} + \frac{7}{20} a^{3} - \frac{1}{100} a^{2} + \frac{1}{20} a - \frac{1}{2}$, $\frac{1}{228191244423788940329896093000000} a^{14} - \frac{136615384929581869398218557}{22819124442378894032989609300000} a^{13} + \frac{278535333010006789824281637}{28523905552973617541237011625000} a^{12} + \frac{26542084701039676073062327841}{45638248884757788065979218600000} a^{11} + \frac{209786256665153562327797641853}{45638248884757788065979218600000} a^{10} - \frac{63050408427318540242883128889}{45638248884757788065979218600000} a^{9} + \frac{178774363007799216696325969067}{9127649776951557613195843720000} a^{8} + \frac{370100635701845590163749503419}{4563824888475778806597921860000} a^{7} - \frac{108764072018548718907987210529}{1140956222118944701649480465000} a^{6} - \frac{35385019919195858853409048617}{456382488847577880659792186000} a^{5} + \frac{44309883619562470919265211363}{91276497769515576131958437200} a^{4} - \frac{24472829766644979861395876901}{456382488847577880659792186000} a^{3} - \frac{2392141729663018732167567277}{18255299553903115226391687440} a^{2} + \frac{414563676848467808592031553}{2281912444237889403298960930} a - \frac{176012700452036004306350237}{912764977695155761319584372}$

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:  \( 207449918613000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T88:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 233280
The 48 conjugacy class representatives for [1/2.S(3)^5]A(5)
Character table for [1/2.S(3)^5]A(5) is not computed

Intermediate fields

5.5.6180196.1

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

Sibling fields

Degree 30 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 R R R ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ R $15$ $15$ $15$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ $15$ $15$ $15$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.6.6.8$x^{6} + 2 x + 2$$6$$1$$6$$S_4$$[4/3, 4/3]_{3}^{2}$
3Data not computed
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.9.6.1$x^{9} - 25 x^{3} + 250$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.3.2.1$x^{3} - 11$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
113Data not computed