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Properties

Label 15.15.2375614325...7441.1
Degree $15$
Signature $[15, 0]$
Discriminant $11^{12}\cdot 31^{14}$
Root discriminant $167.90$
Ramified primes $11, 31$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_{15}$ (as 15T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-141824, 154496, 972256, -627480, -1783924, 477210, 1300271, 54921, -360148, -98056, 13450, 6278, -80, -138, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 138*x^13 - 80*x^12 + 6278*x^11 + 13450*x^10 - 98056*x^9 - 360148*x^8 + 54921*x^7 + 1300271*x^6 + 477210*x^5 - 1783924*x^4 - 627480*x^3 + 972256*x^2 + 154496*x - 141824)
 
gp: K = bnfinit(x^15 - x^14 - 138*x^13 - 80*x^12 + 6278*x^11 + 13450*x^10 - 98056*x^9 - 360148*x^8 + 54921*x^7 + 1300271*x^6 + 477210*x^5 - 1783924*x^4 - 627480*x^3 + 972256*x^2 + 154496*x - 141824, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 138 x^{13} - 80 x^{12} + 6278 x^{11} + 13450 x^{10} - 98056 x^{9} - 360148 x^{8} + 54921 x^{7} + 1300271 x^{6} + 477210 x^{5} - 1783924 x^{4} - 627480 x^{3} + 972256 x^{2} + 154496 x - 141824 \)

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:  \(2375614325883809574306005975647441=11^{12}\cdot 31^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $167.90$
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:  $11, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(341=11\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{341}(1,·)$, $\chi_{341}(67,·)$, $\chi_{341}(324,·)$, $\chi_{341}(71,·)$, $\chi_{341}(202,·)$, $\chi_{341}(267,·)$, $\chi_{341}(289,·)$, $\chi_{341}(157,·)$, $\chi_{341}(20,·)$, $\chi_{341}(97,·)$, $\chi_{341}(235,·)$, $\chi_{341}(56,·)$, $\chi_{341}(225,·)$, $\chi_{341}(59,·)$, $\chi_{341}(317,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{32} a^{7} - \frac{1}{16} a^{5} - \frac{3}{32} a^{3} - \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{128} a^{8} + \frac{1}{64} a^{6} + \frac{1}{128} a^{4} - \frac{1}{4} a^{3} - \frac{1}{32} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{256} a^{9} - \frac{1}{256} a^{8} + \frac{1}{128} a^{7} - \frac{1}{128} a^{6} + \frac{1}{256} a^{5} - \frac{1}{256} a^{4} + \frac{7}{64} a^{3} - \frac{15}{64} a^{2} - \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{512} a^{10} - \frac{1}{512} a^{9} - \frac{1}{256} a^{7} + \frac{13}{512} a^{6} - \frac{17}{512} a^{5} + \frac{5}{256} a^{4} - \frac{11}{128} a^{3} - \frac{3}{64} a^{2} - \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{1024} a^{11} - \frac{1}{1024} a^{9} - \frac{1}{512} a^{8} + \frac{11}{1024} a^{7} - \frac{1}{256} a^{6} + \frac{57}{1024} a^{5} + \frac{15}{512} a^{4} + \frac{63}{256} a^{3} - \frac{19}{128} a^{2} - \frac{1}{16} a + \frac{3}{8}$, $\frac{1}{4096} a^{12} - \frac{1}{2048} a^{11} + \frac{1}{4096} a^{10} - \frac{1}{2048} a^{9} - \frac{9}{4096} a^{8} - \frac{15}{2048} a^{7} + \frac{107}{4096} a^{6} + \frac{37}{2048} a^{5} + \frac{63}{1024} a^{4} + \frac{107}{512} a^{3} - \frac{27}{128} a^{2} - \frac{15}{32} a - \frac{3}{8}$, $\frac{1}{16384} a^{13} - \frac{1}{16384} a^{12} + \frac{7}{16384} a^{11} + \frac{15}{16384} a^{10} - \frac{19}{16384} a^{9} - \frac{39}{16384} a^{8} + \frac{37}{16384} a^{7} + \frac{133}{16384} a^{6} + \frac{263}{8192} a^{5} - \frac{67}{4096} a^{4} + \frac{43}{2048} a^{3} + \frac{85}{512} a^{2} - \frac{47}{128} a - \frac{15}{32}$, $\frac{1}{5637818623197184} a^{14} - \frac{19765035243}{2818909311598592} a^{13} - \frac{91428392447}{1409454655799296} a^{12} + \frac{459242087223}{1409454655799296} a^{11} - \frac{2621394266115}{2818909311598592} a^{10} - \frac{1019217603839}{704727327899648} a^{9} + \frac{521378543325}{352363663949824} a^{8} - \frac{6924064513625}{1409454655799296} a^{7} + \frac{125852046007869}{5637818623197184} a^{6} + \frac{80292896350071}{2818909311598592} a^{5} - \frac{75586316127675}{1409454655799296} a^{4} - \frac{19830503956483}{704727327899648} a^{3} - \frac{20169071449817}{176181831974912} a^{2} + \frac{6886292425167}{44045457993728} a - \frac{3380923733569}{11011364498432}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{5}$, which has order $5$ (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:  \( 2170143402898142.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{15}$ (as 15T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 15
The 15 conjugacy class representatives for $C_{15}$
Character table for $C_{15}$

Intermediate fields

3.3.961.1, 5.5.13521270961.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.1.0.1}{1} }^{15}$ $15$ $15$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{5}$ R $15$ $15$ $15$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ R $15$ $15$ $15$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ $15$ $15$

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
$11$11.15.12.4$x^{15} + 99 x^{10} + 3146 x^{5} + 35937$$5$$3$$12$$C_{15}$$[\ ]_{5}^{3}$
$31$31.15.14.13$x^{15} - 429079903231$$15$$1$$14$$C_{15}$$[\ ]_{15}$