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Properties

Label 15.7.15630956656...1536.2
Degree $15$
Signature $[7, 4]$
Discriminant $2^{26}\cdot 137^{10}$
Root discriminant $88.36$
Ramified primes $2, 137$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $A_8$ (as 15T72)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1648, -8528, -4272, 20212, 7348, -22856, -816, 11245, -112, -3322, 132, 421, 32, -30, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 - 30*x^13 + 32*x^12 + 421*x^11 + 132*x^10 - 3322*x^9 - 112*x^8 + 11245*x^7 - 816*x^6 - 22856*x^5 + 7348*x^4 + 20212*x^3 - 4272*x^2 - 8528*x - 1648)
 
gp: K = bnfinit(x^15 - 4*x^14 - 30*x^13 + 32*x^12 + 421*x^11 + 132*x^10 - 3322*x^9 - 112*x^8 + 11245*x^7 - 816*x^6 - 22856*x^5 + 7348*x^4 + 20212*x^3 - 4272*x^2 - 8528*x - 1648, 1)
 

Normalized defining polynomial

\( x^{15} - 4 x^{14} - 30 x^{13} + 32 x^{12} + 421 x^{11} + 132 x^{10} - 3322 x^{9} - 112 x^{8} + 11245 x^{7} - 816 x^{6} - 22856 x^{5} + 7348 x^{4} + 20212 x^{3} - 4272 x^{2} - 8528 x - 1648 \)

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:  \(156309566567541201405567041536=2^{26}\cdot 137^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.36$
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:  $2, 137$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} + \frac{1}{4} a^{8} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{1976} a^{13} - \frac{105}{988} a^{12} + \frac{33}{247} a^{11} + \frac{54}{247} a^{10} - \frac{435}{1976} a^{9} + \frac{3}{76} a^{8} - \frac{101}{494} a^{7} - \frac{115}{247} a^{6} + \frac{293}{1976} a^{5} + \frac{449}{988} a^{4} - \frac{153}{988} a^{3} + \frac{18}{247} a^{2} - \frac{35}{247} a - \frac{24}{247}$, $\frac{1}{854451284267867952406000} a^{14} - \frac{13668295693554953273}{213612821066966988101500} a^{13} - \frac{4914249941484051487523}{106806410533483494050750} a^{12} - \frac{7163743407106244209611}{53403205266741747025375} a^{11} + \frac{3334468719732010418511}{44971120224624629074000} a^{10} - \frac{16790432907759563780963}{42722564213393397620300} a^{9} - \frac{88882282772784110353923}{213612821066966988101500} a^{8} + \frac{2430309328912814363674}{53403205266741747025375} a^{7} + \frac{346014916301792464144253}{854451284267867952406000} a^{6} - \frac{4895725808802944096101}{10680641053348349405075} a^{5} - \frac{109775609062577700453}{1183450532226963923000} a^{4} + \frac{73824960204388436528539}{213612821066966988101500} a^{3} + \frac{18110754327421061843374}{53403205266741747025375} a^{2} - \frac{1107766288053459958907}{5621390028078078634250} a - \frac{14104906367306746970031}{53403205266741747025375}$

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:  \( 12284484047.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_8$ (as 15T72):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 20160
The 14 conjugacy class representatives for $A_8$
Character table for $A_8$

Intermediate fields

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

Sibling fields

Degree 8 sibling: data not computed
Degree 28 sibling: data not computed
Degree 35 sibling: data not computed
Arithmetically equvalently 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 $15$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.4.8.7$x^{4} + 4 x^{2} + 4 x + 2$$4$$1$$8$$S_4$$[8/3, 8/3]_{3}^{2}$
2.8.16.20$x^{8} + 2 x^{4} + 8 x^{2} + 8 x + 4$$4$$2$$16$$V_4^2:S_3$$[4/3, 4/3, 8/3, 8/3]_{3}^{2}$
$137$137.6.4.2$x^{6} - 137 x^{3} + 112614$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
137.9.6.1$x^{9} - 18769 x^{3} + 12856765$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$