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Properties

Label 15.15.4973877463...0000.2
Degree $15$
Signature $[15, 0]$
Discriminant $2^{12}\cdot 3^{20}\cdot 5^{15}\cdot 7^{6}\cdot 97$
Root discriminant $111.29$
Ramified primes $2, 3, 5, 7, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T87

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21168, 105840, 136080, -83720, -281160, -111204, 104340, 68520, -14760, -14180, 900, 1350, -20, -60, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 60*x^13 - 20*x^12 + 1350*x^11 + 900*x^10 - 14180*x^9 - 14760*x^8 + 68520*x^7 + 104340*x^6 - 111204*x^5 - 281160*x^4 - 83720*x^3 + 136080*x^2 + 105840*x + 21168)
 
gp: K = bnfinit(x^15 - 60*x^13 - 20*x^12 + 1350*x^11 + 900*x^10 - 14180*x^9 - 14760*x^8 + 68520*x^7 + 104340*x^6 - 111204*x^5 - 281160*x^4 - 83720*x^3 + 136080*x^2 + 105840*x + 21168, 1)
 

Normalized defining polynomial

\( x^{15} - 60 x^{13} - 20 x^{12} + 1350 x^{11} + 900 x^{10} - 14180 x^{9} - 14760 x^{8} + 68520 x^{7} + 104340 x^{6} - 111204 x^{5} - 281160 x^{4} - 83720 x^{3} + 136080 x^{2} + 105840 x + 21168 \)

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:  \(4973877463168144125000000000000=2^{12}\cdot 3^{20}\cdot 5^{15}\cdot 7^{6}\cdot 97\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $111.29$
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:  $2, 3, 5, 7, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{4} a^{12}$, $\frac{1}{168} a^{13} - \frac{3}{28} a^{11} - \frac{5}{42} a^{10} + \frac{1}{28} a^{9} - \frac{1}{7} a^{8} + \frac{2}{21} a^{7} + \frac{1}{7} a^{6} - \frac{1}{7} a^{5} + \frac{1}{14} a^{4} + \frac{1}{14} a^{3} + \frac{3}{7} a^{2} - \frac{1}{3} a$, $\frac{1}{202275133876981675944} a^{14} - \frac{6222492590273069}{5618753718805046554} a^{13} - \frac{2074180910409991535}{16856261156415139662} a^{12} - \frac{1142606923201285141}{14448223848355833996} a^{11} - \frac{89249600866387496}{2809376859402523277} a^{10} + \frac{1063666208623557823}{11237507437610093108} a^{9} - \frac{3636200869271321597}{50568783469245418986} a^{8} + \frac{21780483607303399}{802679102686435222} a^{7} + \frac{1635913490992081442}{8428130578207569831} a^{6} - \frac{1666750654578776044}{8428130578207569831} a^{5} + \frac{1340390719640074554}{2809376859402523277} a^{4} - \frac{22479812663360095}{2809376859402523277} a^{3} + \frac{7170886031138508179}{25284391734622709493} a^{2} + \frac{41043613884111418}{401339551343217611} a + \frac{175931319548707983}{401339551343217611}$

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:  \( 420248050137 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T87:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 155520
The 63 conjugacy class representatives for [S(3)^5]F(5)=S(3)wrF(5) are not computed
Character table for [S(3)^5]F(5)=S(3)wrF(5) is not computed

Intermediate fields

5.5.2450000.1

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

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 R ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$3$3.3.4.4$x^{3} + 3 x^{2} + 3$$3$$1$$4$$S_3$$[2]^{2}$
3.12.16.6$x^{12} + 120 x^{11} - 117 x^{10} - 57 x^{9} + 36 x^{8} + 54 x^{7} - 18 x^{6} + 81 x^{5} + 81$$3$$4$$16$12T46$[2, 2]^{8}$
$5$5.5.5.1$x^{5} + 20 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.10.10.10$x^{10} + 10 x^{8} + 5 x^{6} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 2$$5$$2$$10$$F_{5}\times C_2$$[5/4]_{4}^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
97Data not computed