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Properties

Label 15.15.9142350653...2256.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{10}\cdot 3^{17}\cdot 101^{2}\cdot 163\cdot 401^{6}$
Root discriminant $157.54$
Ramified primes $2, 3, 101, 163, 401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T86

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-29088, -218160, -654480, -961448, -614658, 52785, 268296, 78435, -36288, -17460, 2052, 1539, -42, -63, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 63*x^13 - 42*x^12 + 1539*x^11 + 2052*x^10 - 17460*x^9 - 36288*x^8 + 78435*x^7 + 268296*x^6 + 52785*x^5 - 614658*x^4 - 961448*x^3 - 654480*x^2 - 218160*x - 29088)
 
gp: K = bnfinit(x^15 - 63*x^13 - 42*x^12 + 1539*x^11 + 2052*x^10 - 17460*x^9 - 36288*x^8 + 78435*x^7 + 268296*x^6 + 52785*x^5 - 614658*x^4 - 961448*x^3 - 654480*x^2 - 218160*x - 29088, 1)
 

Normalized defining polynomial

\( x^{15} - 63 x^{13} - 42 x^{12} + 1539 x^{11} + 2052 x^{10} - 17460 x^{9} - 36288 x^{8} + 78435 x^{7} + 268296 x^{6} + 52785 x^{5} - 614658 x^{4} - 961448 x^{3} - 654480 x^{2} - 218160 x - 29088 \)

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:  \(914235065331822156454227288032256=2^{10}\cdot 3^{17}\cdot 101^{2}\cdot 163\cdot 401^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $157.54$
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:  $2, 3, 101, 163, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{8} a^{11} + \frac{1}{8} a^{9} - \frac{1}{4} a^{8} + \frac{3}{8} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{3}{8} a^{3} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{64} a^{12} - \frac{1}{32} a^{11} + \frac{25}{64} a^{10} - \frac{7}{16} a^{9} + \frac{31}{64} a^{8} - \frac{1}{32} a^{7} + \frac{7}{16} a^{6} - \frac{1}{8} a^{5} + \frac{19}{64} a^{4} - \frac{15}{32} a^{3} - \frac{23}{64} a^{2} + \frac{3}{16} a + \frac{1}{16}$, $\frac{1}{512} a^{13} - \frac{1}{128} a^{12} + \frac{29}{512} a^{11} + \frac{25}{256} a^{10} - \frac{41}{512} a^{9} - \frac{1}{4} a^{8} - \frac{3}{16} a^{7} - \frac{1}{8} a^{6} + \frac{227}{512} a^{5} - \frac{33}{128} a^{4} - \frac{91}{512} a^{3} - \frac{99}{256} a^{2} + \frac{27}{128} a + \frac{31}{64}$, $\frac{1}{12288} a^{14} - \frac{1}{2048} a^{13} - \frac{9}{4096} a^{12} + \frac{5}{512} a^{11} + \frac{273}{4096} a^{10} - \frac{477}{2048} a^{9} - \frac{3}{128} a^{8} + \frac{3}{16} a^{7} + \frac{1057}{4096} a^{6} + \frac{585}{2048} a^{5} - \frac{1713}{4096} a^{4} + \frac{125}{256} a^{3} - \frac{265}{1536} a^{2} - \frac{29}{128} a - \frac{101}{256}$

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:  \( 690382095935 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T86:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 77760
The 72 conjugacy class representatives for [S(3)^5]D(5)=S(3)wrD(5) are not computed
Character table for [S(3)^5]D(5)=S(3)wrD(5) is not computed

Intermediate fields

5.5.160801.1

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 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 $15$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ $15$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.1$x^{10} - 9 x^{8} + 54 x^{6} - 38 x^{4} + 41 x^{2} - 17$$2$$5$$10$$C_2^4 : C_5$$[2, 2, 2, 2]^{5}$
$3$3.3.5.3$x^{3} + 12$$3$$1$$5$$S_3$$[5/2]_{2}$
3.6.6.4$x^{6} + 3 x^{4} + 6 x^{3} + 9 x^{2} + 63 x + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
3.6.6.1$x^{6} + 3 x^{5} - 2$$3$$2$$6$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
101Data not computed
163Data not computed
401Data not computed