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Properties

Label 15.15.2130745982...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{14}\cdot 5^{15}\cdot 37^{7}\cdot 67^{2}$
Root discriminant $90.21$
Ramified primes $2, 5, 37, 67$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T68

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 160, -1200, 1620, 3200, -5836, -2480, 6000, 720, -2600, -72, 500, 0, -40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 40*x^13 + 500*x^11 - 72*x^10 - 2600*x^9 + 720*x^8 + 6000*x^7 - 2480*x^6 - 5836*x^5 + 3200*x^4 + 1620*x^3 - 1200*x^2 + 160*x + 4)
 
gp: K = bnfinit(x^15 - 40*x^13 + 500*x^11 - 72*x^10 - 2600*x^9 + 720*x^8 + 6000*x^7 - 2480*x^6 - 5836*x^5 + 3200*x^4 + 1620*x^3 - 1200*x^2 + 160*x + 4, 1)
 

Normalized defining polynomial

\( x^{15} - 40 x^{13} + 500 x^{11} - 72 x^{10} - 2600 x^{9} + 720 x^{8} + 6000 x^{7} - 2480 x^{6} - 5836 x^{5} + 3200 x^{4} + 1620 x^{3} - 1200 x^{2} + 160 x + 4 \)

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:  \(213074598225018500000000000000=2^{14}\cdot 5^{15}\cdot 37^{7}\cdot 67^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.21$
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:  $2, 5, 37, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{7178951688638914} a^{14} + \frac{1086412505372797}{7178951688638914} a^{13} + \frac{892065614848773}{3589475844319457} a^{12} + \frac{279786988753556}{3589475844319457} a^{11} - \frac{419721369745618}{3589475844319457} a^{10} + \frac{369856533292211}{7178951688638914} a^{9} + \frac{659276106690487}{3589475844319457} a^{8} + \frac{477043166037909}{3589475844319457} a^{7} + \frac{1108381611527344}{3589475844319457} a^{6} + \frac{427401841715827}{3589475844319457} a^{5} - \frac{1268724132991850}{3589475844319457} a^{4} - \frac{1750210015924707}{3589475844319457} a^{3} + \frac{360535164889232}{3589475844319457} a^{2} + \frac{1230547073202163}{3589475844319457} a - \frac{1346549081095160}{3589475844319457}$

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:  \( 13946109292.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T68:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 12000
The 38 conjugacy class representatives for [D(5)^3:2]S(3)
Character table for [D(5)^3:2]S(3) is not computed

Intermediate fields

3.3.148.1

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 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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
2Data not computed
$5$5.5.5.1$x^{5} + 20 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.10.10.13$x^{10} + 15 x^{6} + 10 x^{5} + 100 x^{2} + 75 x + 25$$5$$2$$10$$(C_5^2 : C_4) : C_2$$[5/4, 5/4]_{4}^{2}$
37Data not computed
$67$$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
67.4.2.2$x^{4} - 67 x^{2} + 53868$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
67.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
67.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$