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Properties

Label 15.7.44896833821...5625.1
Degree $15$
Signature $[7, 4]$
Discriminant $3^{8}\cdot 5^{14}\cdot 257^{5}$
Root discriminant $51.30$
Ramified primes $3, 5, 257$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T60

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-97, 3401, -28970, -14084, -416, 18939, -1614, -6905, 1896, 899, -71, -24, -35, -4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 4*x^13 - 35*x^12 - 24*x^11 - 71*x^10 + 899*x^9 + 1896*x^8 - 6905*x^7 - 1614*x^6 + 18939*x^5 - 416*x^4 - 14084*x^3 - 28970*x^2 + 3401*x - 97)
 
gp: K = bnfinit(x^15 - x^14 - 4*x^13 - 35*x^12 - 24*x^11 - 71*x^10 + 899*x^9 + 1896*x^8 - 6905*x^7 - 1614*x^6 + 18939*x^5 - 416*x^4 - 14084*x^3 - 28970*x^2 + 3401*x - 97, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 4 x^{13} - 35 x^{12} - 24 x^{11} - 71 x^{10} + 899 x^{9} + 1896 x^{8} - 6905 x^{7} - 1614 x^{6} + 18939 x^{5} - 416 x^{4} - 14084 x^{3} - 28970 x^{2} + 3401 x - 97 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(44896833821697857666015625=3^{8}\cdot 5^{14}\cdot 257^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.30$
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:  $3, 5, 257$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{27218067468613304559217425912965} a^{14} - \frac{11270715585168278748890875547688}{27218067468613304559217425912965} a^{13} - \frac{13196050044492112032112675520898}{27218067468613304559217425912965} a^{12} + \frac{8854474492710318354675418928256}{27218067468613304559217425912965} a^{11} + \frac{12939910210250256755545832288199}{27218067468613304559217425912965} a^{10} + \frac{7190360376672158430726380526046}{27218067468613304559217425912965} a^{9} + \frac{9155791862553261112926740635082}{27218067468613304559217425912965} a^{8} + \frac{7670371265551947160741267869582}{27218067468613304559217425912965} a^{7} - \frac{5313976626232525642583451048304}{27218067468613304559217425912965} a^{6} - \frac{1089761994906085647262960238121}{27218067468613304559217425912965} a^{5} + \frac{8363950670905993194031930808476}{27218067468613304559217425912965} a^{4} + \frac{12074209086281148152216739872722}{27218067468613304559217425912965} a^{3} + \frac{10733924086901297729576879699557}{27218067468613304559217425912965} a^{2} - \frac{8137864878522184011896983155539}{27218067468613304559217425912965} a - \frac{5385518685404675254802174961851}{27218067468613304559217425912965}$

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:  $10$
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:  \( 36339727.7705 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T60:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6000
The 40 conjugacy class representatives for [D(5)^3]S(3)=D(5)wrS(3)
Character table for [D(5)^3]S(3)=D(5)wrS(3) is not computed

Intermediate fields

3.3.257.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ R R ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ $15$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ $15$ $15$ ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ $15$

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
$3$3.5.0.1$x^{5} - x + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.10.12.11$x^{10} + 5 x^{7} + 10 x^{5} + 25 x^{4} + 25 x^{2} + 25$$5$$2$$12$$C_5^2 : C_4$$[3/2, 3/2]_{2}^{2}$
257Data not computed