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Properties

Label 15.3.38771729885...4048.2
Degree $15$
Signature $[3, 6]$
Discriminant $2^{20}\cdot 3^{20}\cdot 13^{9}$
Root discriminant $50.80$
Ramified primes $2, 3, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T85

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65248, -75456, -100368, -67080, 2208, 19062, 13736, 2202, -324, 518, 72, -108, -38, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 38*x^12 - 108*x^11 + 72*x^10 + 518*x^9 - 324*x^8 + 2202*x^7 + 13736*x^6 + 19062*x^5 + 2208*x^4 - 67080*x^3 - 100368*x^2 - 75456*x + 65248)
 
gp: K = bnfinit(x^15 - 38*x^12 - 108*x^11 + 72*x^10 + 518*x^9 - 324*x^8 + 2202*x^7 + 13736*x^6 + 19062*x^5 + 2208*x^4 - 67080*x^3 - 100368*x^2 - 75456*x + 65248, 1)
 

Normalized defining polynomial

\( x^{15} - 38 x^{12} - 108 x^{11} + 72 x^{10} + 518 x^{9} - 324 x^{8} + 2202 x^{7} + 13736 x^{6} + 19062 x^{5} + 2208 x^{4} - 67080 x^{3} - 100368 x^{2} - 75456 x + 65248 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(38771729885236487072514048=2^{20}\cdot 3^{20}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.80$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{26} a^{11} + \frac{3}{26} a^{10} + \frac{2}{13} a^{9} - \frac{3}{13} a^{8} + \frac{1}{13} a^{7} + \frac{1}{13} a^{5} - \frac{6}{13} a^{4} + \frac{3}{13} a^{3} + \frac{2}{13} a - \frac{5}{13}$, $\frac{1}{52} a^{12} + \frac{2}{13} a^{10} - \frac{9}{26} a^{9} + \frac{5}{13} a^{8} + \frac{5}{13} a^{7} + \frac{1}{26} a^{6} + \frac{2}{13} a^{5} - \frac{5}{26} a^{4} + \frac{2}{13} a^{3} - \frac{11}{26} a^{2} + \frac{1}{13} a + \frac{1}{13}$, $\frac{1}{104} a^{13} + \frac{5}{52} a^{10} - \frac{3}{26} a^{9} + \frac{2}{13} a^{8} + \frac{19}{52} a^{7} - \frac{11}{26} a^{6} + \frac{1}{4} a^{5} - \frac{9}{52} a^{3} - \frac{6}{13} a^{2} + \frac{3}{13} a - \frac{3}{13}$, $\frac{1}{71971866548279218467625864015376} a^{14} + \frac{2538989448764865742052696417}{35985933274139609233812932007688} a^{13} + \frac{11218434947401752322610427289}{1384074356697677278223574307988} a^{12} - \frac{45094295603486212746797445723}{2768148713395354556447148615976} a^{11} + \frac{345646273833121947877567248577}{4498241659267451154226616500961} a^{10} - \frac{564860507891014599109779793763}{1285211902647843186921890428846} a^{9} - \frac{1496356286477377259598544600387}{5140847610591372747687561715384} a^{8} - \frac{3539815534114774308137496683737}{8996483318534902308453233001922} a^{7} - \frac{9197491672256306686498105870011}{35985933274139609233812932007688} a^{6} + \frac{78333353198075030427618238229}{2570423805295686373843780857692} a^{5} - \frac{12139833523110054456148756155465}{35985933274139609233812932007688} a^{4} + \frac{256689734108387231037323257861}{2570423805295686373843780857692} a^{3} + \frac{1407366982745639880129366118782}{4498241659267451154226616500961} a^{2} - \frac{1402734080558254642821347227929}{4498241659267451154226616500961} a + \frac{291130044558704575735733895539}{4498241659267451154226616500961}$

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:  $8$
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:  \( 27522585.6641 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T85:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 77760
The 45 conjugacy class representatives for [1/2.S(3)^5]F(5)
Character table for [1/2.S(3)^5]F(5) is not computed

Intermediate fields

5.1.35152.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 30 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 R ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.10.16.4$x^{10} - 2 x^{9} - 2 x^{8} + 2 x^{7} - 2 x^{6} - 2 x^{2} + 4 x - 2$$10$$1$$16$$(C_2^4 : C_5):C_4$$[12/5, 12/5, 12/5, 12/5]_{5}^{4}$
3Data not computed
$13$13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$