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Properties

Label 15.15.2639300305...2721.2
Degree $15$
Signature $[15, 0]$
Discriminant $3^{20}\cdot 31^{14}$
Root discriminant $106.68$
Ramified primes $3, 31$
Class number $3$ (GRH)
Class group $[3]$ (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![96875, 1023000, 2046465, 541198, -1558308, -837000, 447981, 290067, -59985, -43617, 3534, 3069, -62, -93, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 93*x^13 - 62*x^12 + 3069*x^11 + 3534*x^10 - 43617*x^9 - 59985*x^8 + 290067*x^7 + 447981*x^6 - 837000*x^5 - 1558308*x^4 + 541198*x^3 + 2046465*x^2 + 1023000*x + 96875)
 
gp: K = bnfinit(x^15 - 93*x^13 - 62*x^12 + 3069*x^11 + 3534*x^10 - 43617*x^9 - 59985*x^8 + 290067*x^7 + 447981*x^6 - 837000*x^5 - 1558308*x^4 + 541198*x^3 + 2046465*x^2 + 1023000*x + 96875, 1)
 

Normalized defining polynomial

\( x^{15} - 93 x^{13} - 62 x^{12} + 3069 x^{11} + 3534 x^{10} - 43617 x^{9} - 59985 x^{8} + 290067 x^{7} + 447981 x^{6} - 837000 x^{5} - 1558308 x^{4} + 541198 x^{3} + 2046465 x^{2} + 1023000 x + 96875 \)

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:  \(2639300305759427100493462112721=3^{20}\cdot 31^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $106.68$
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:  $3, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(279=3^{2}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{279}(64,·)$, $\chi_{279}(1,·)$, $\chi_{279}(67,·)$, $\chi_{279}(7,·)$, $\chi_{279}(40,·)$, $\chi_{279}(169,·)$, $\chi_{279}(103,·)$, $\chi_{279}(109,·)$, $\chi_{279}(205,·)$, $\chi_{279}(49,·)$, $\chi_{279}(163,·)$, $\chi_{279}(214,·)$, $\chi_{279}(25,·)$, $\chi_{279}(175,·)$, $\chi_{279}(190,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{25} a^{8} - \frac{1}{25} a^{6} + \frac{4}{25} a^{4} - \frac{1}{5} a^{3} - \frac{4}{25} a^{2} + \frac{1}{5} a$, $\frac{1}{25} a^{9} - \frac{1}{25} a^{7} - \frac{1}{25} a^{5} - \frac{1}{5} a^{4} - \frac{4}{25} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{125} a^{10} + \frac{2}{125} a^{9} + \frac{1}{125} a^{8} - \frac{12}{125} a^{7} + \frac{12}{125} a^{6} + \frac{3}{125} a^{5} + \frac{44}{125} a^{4} - \frac{3}{125} a^{3} - \frac{33}{125} a^{2} - \frac{3}{25} a$, $\frac{1}{125} a^{11} + \frac{2}{125} a^{9} + \frac{1}{125} a^{8} + \frac{6}{125} a^{7} - \frac{11}{125} a^{6} + \frac{8}{125} a^{5} - \frac{56}{125} a^{4} + \frac{28}{125} a^{3} - \frac{9}{125} a^{2} + \frac{6}{25} a$, $\frac{1}{625} a^{12} - \frac{2}{625} a^{11} - \frac{2}{625} a^{10} - \frac{6}{625} a^{9} - \frac{2}{125} a^{8} - \frac{11}{125} a^{7} - \frac{33}{625} a^{6} - \frac{39}{625} a^{5} + \frac{149}{625} a^{4} + \frac{302}{625} a^{3} + \frac{29}{125} a^{2} + \frac{7}{25} a$, $\frac{1}{3125} a^{13} + \frac{2}{3125} a^{12} + \frac{2}{625} a^{11} + \frac{6}{3125} a^{10} - \frac{4}{3125} a^{9} - \frac{6}{625} a^{8} - \frac{198}{3125} a^{7} - \frac{51}{3125} a^{6} + \frac{138}{3125} a^{5} - \frac{867}{3125} a^{4} - \frac{1322}{3125} a^{3} + \frac{13}{625} a^{2} - \frac{7}{25} a$, $\frac{1}{503060578191041796875} a^{14} - \frac{30981815651586449}{503060578191041796875} a^{13} + \frac{248471233170700133}{503060578191041796875} a^{12} + \frac{1138052893053555971}{503060578191041796875} a^{11} + \frac{82768055284390093}{100612115638208359375} a^{10} + \frac{9720992023190112874}{503060578191041796875} a^{9} + \frac{893426435092082582}{503060578191041796875} a^{8} - \frac{21333325098449197553}{503060578191041796875} a^{7} - \frac{2744164099335656261}{503060578191041796875} a^{6} + \frac{7118901912424934409}{100612115638208359375} a^{5} - \frac{42558277973848935541}{100612115638208359375} a^{4} - \frac{198237324748006101888}{503060578191041796875} a^{3} + \frac{22682386902271089382}{100612115638208359375} a^{2} - \frac{1398657105982163384}{4024484625528334375} a + \frac{69178693381109394}{160979385021133375}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 123454772619 \) (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

3.3.77841.1, 5.5.923521.1

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 $15$ R ${\href{/LocalNumberField/5.1.0.1}{1} }^{15}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ $15$ $15$ $15$ $15$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ $15$ $15$ $15$

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
3Data not computed
31Data not computed