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Properties

Label 15.15.3643029633...0576.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{14}\cdot 11^{8}\cdot 467^{6}$
Root discriminant $80.19$
Ramified primes $2, 11, 467$
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![-736, -7408, -8352, 40656, 21376, -71584, -4512, 39452, -696, -9480, 224, 1084, -12, -56, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 56*x^13 - 12*x^12 + 1084*x^11 + 224*x^10 - 9480*x^9 - 696*x^8 + 39452*x^7 - 4512*x^6 - 71584*x^5 + 21376*x^4 + 40656*x^3 - 8352*x^2 - 7408*x - 736)
 
gp: K = bnfinit(x^15 - 56*x^13 - 12*x^12 + 1084*x^11 + 224*x^10 - 9480*x^9 - 696*x^8 + 39452*x^7 - 4512*x^6 - 71584*x^5 + 21376*x^4 + 40656*x^3 - 8352*x^2 - 7408*x - 736, 1)
 

Normalized defining polynomial

\( x^{15} - 56 x^{13} - 12 x^{12} + 1084 x^{11} + 224 x^{10} - 9480 x^{9} - 696 x^{8} + 39452 x^{7} - 4512 x^{6} - 71584 x^{5} + 21376 x^{4} + 40656 x^{3} - 8352 x^{2} - 7408 x - 736 \)

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:  \(36430296336664162471668760576=2^{14}\cdot 11^{8}\cdot 467^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $80.19$
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:  $2, 11, 467$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{7472} a^{13} + \frac{89}{3736} a^{12} - \frac{55}{3736} a^{11} + \frac{53}{934} a^{10} - \frac{9}{467} a^{9} + \frac{113}{934} a^{8} - \frac{75}{934} a^{7} + \frac{77}{1868} a^{6} + \frac{321}{1868} a^{5} + \frac{211}{934} a^{4} - \frac{257}{934} a^{3} + \frac{419}{934} a^{2} + \frac{159}{467} a - \frac{205}{467}$, $\frac{1}{1285932242859568} a^{14} + \frac{4962684681}{321483060714892} a^{13} + \frac{4694831336623}{1285932242859568} a^{12} - \frac{12329697067979}{321483060714892} a^{11} - \frac{23121996487305}{642966121429784} a^{10} - \frac{926500769450}{80370765178723} a^{9} - \frac{46525949458405}{642966121429784} a^{8} - \frac{2283657915167}{160741530357446} a^{7} + \frac{4252875586266}{80370765178723} a^{6} - \frac{44073811989291}{321483060714892} a^{5} + \frac{12263586250001}{160741530357446} a^{4} + \frac{6139307559343}{80370765178723} a^{3} - \frac{21755939101722}{80370765178723} a^{2} - \frac{23677728717253}{80370765178723} a + \frac{19799198459673}{80370765178723}$

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:  \( 15873657608.2 \) (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: data not computed
Degree 10 sibling: data not computed
Degree 15 sibling: data not computed
Degree 20 sibling: data not computed
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 R ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/13.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.6.6.7$x^{6} + 2 x^{2} + 2 x + 2$$6$$1$$6$$S_4$$[4/3, 4/3]_{3}^{2}$
2.8.8.11$x^{8} + 20 x^{2} + 4$$4$$2$$8$$S_4$$[4/3, 4/3]_{3}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.3.2.1$x^{3} - 11$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
11.3.2.1$x^{3} - 11$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
467Data not computed