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Properties

Label 15.15.1162175951...3009.1
Degree $15$
Signature $[15, 0]$
Discriminant $7^{10}\cdot 283^{8}$
Root discriminant $74.31$
Ramified primes $7, 283$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $A_6$ (as 15T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-245, 2058, -2037, -16184, 33201, 6328, -37522, 2757, 15116, -2219, -2646, 476, 196, -38, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 38*x^13 + 196*x^12 + 476*x^11 - 2646*x^10 - 2219*x^9 + 15116*x^8 + 2757*x^7 - 37522*x^6 + 6328*x^5 + 33201*x^4 - 16184*x^3 - 2037*x^2 + 2058*x - 245)
 
gp: K = bnfinit(x^15 - 5*x^14 - 38*x^13 + 196*x^12 + 476*x^11 - 2646*x^10 - 2219*x^9 + 15116*x^8 + 2757*x^7 - 37522*x^6 + 6328*x^5 + 33201*x^4 - 16184*x^3 - 2037*x^2 + 2058*x - 245, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 38 x^{13} + 196 x^{12} + 476 x^{11} - 2646 x^{10} - 2219 x^{9} + 15116 x^{8} + 2757 x^{7} - 37522 x^{6} + 6328 x^{5} + 33201 x^{4} - 16184 x^{3} - 2037 x^{2} + 2058 x - 245 \)

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:  \(11621759510846642583679213009=7^{10}\cdot 283^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.31$
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:  $7, 283$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7} a^{9} - \frac{3}{7} a^{8} - \frac{2}{7} a^{6} + \frac{1}{7} a^{5} + \frac{3}{7} a^{4}$, $\frac{1}{7} a^{10} - \frac{2}{7} a^{8} - \frac{2}{7} a^{7} + \frac{2}{7} a^{6} - \frac{1}{7} a^{5} + \frac{2}{7} a^{4}$, $\frac{1}{119} a^{11} + \frac{6}{119} a^{10} - \frac{6}{119} a^{9} + \frac{54}{119} a^{8} - \frac{59}{119} a^{7} - \frac{9}{119} a^{6} + \frac{13}{119} a^{5} - \frac{1}{17} a^{4} - \frac{1}{17} a^{3} + \frac{6}{17} a^{2} + \frac{5}{17} a + \frac{5}{17}$, $\frac{1}{119} a^{12} - \frac{8}{119} a^{10} + \frac{5}{119} a^{9} + \frac{6}{17} a^{8} + \frac{39}{119} a^{7} - \frac{52}{119} a^{6} + \frac{2}{7} a^{5} - \frac{33}{119} a^{4} - \frac{5}{17} a^{3} + \frac{3}{17} a^{2} - \frac{8}{17} a + \frac{4}{17}$, $\frac{1}{119} a^{13} + \frac{2}{119} a^{10} - \frac{6}{119} a^{9} - \frac{22}{119} a^{8} + \frac{54}{119} a^{7} - \frac{3}{17} a^{6} + \frac{3}{119} a^{5} + \frac{45}{119} a^{4} - \frac{5}{17} a^{3} + \frac{6}{17} a^{2} - \frac{7}{17} a + \frac{6}{17}$, $\frac{1}{1292035650152345} a^{14} - \frac{2218816459322}{1292035650152345} a^{13} - \frac{177356103447}{184576521450335} a^{12} + \frac{315151008177}{184576521450335} a^{11} + \frac{52816190011}{1551063205465} a^{10} + \frac{1852946366367}{26368074492905} a^{9} + \frac{2189814917013}{36915304290067} a^{8} + \frac{309902078496286}{1292035650152345} a^{7} + \frac{62482364217810}{258407130030469} a^{6} - \frac{23885772550576}{184576521450335} a^{5} + \frac{77018898749431}{184576521450335} a^{4} + \frac{80804241405446}{184576521450335} a^{3} + \frac{1945164765426}{184576521450335} a^{2} + \frac{10095914468986}{26368074492905} a + \frac{71866310433}{5273614898581}$

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:  \( 4903571851.42 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_6$ (as 15T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 360
The 7 conjugacy class representatives for $A_6$
Character table for $A_6$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 6 siblings: 6.6.15400609258321.1, 6.6.192293689.1
Degree 10 sibling: 10.10.60437550349593857881.1
Degree 15 sibling: Deg 15
Degree 20 sibling: Deg 20
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 sibling: 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ R ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.3.1$x^{4} + 14$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
283Data not computed