↑

Properties

Label 15.15.2639300305...2721.1
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)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5921, -209901, 645327, 723943, -871131, -681318, 375162, 258261, -62496, -43338, 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 - 43338*x^9 - 62496*x^8 + 258261*x^7 + 375162*x^6 - 681318*x^5 - 871131*x^4 + 723943*x^3 + 645327*x^2 - 209901*x + 5921)
 
gp: K = bnfinit(x^15 - 93*x^13 - 62*x^12 + 3069*x^11 + 3534*x^10 - 43338*x^9 - 62496*x^8 + 258261*x^7 + 375162*x^6 - 681318*x^5 - 871131*x^4 + 723943*x^3 + 645327*x^2 - 209901*x + 5921, 1)
 

Normalized defining polynomial

\( x^{15} - 93 x^{13} - 62 x^{12} + 3069 x^{11} + 3534 x^{10} - 43338 x^{9} - 62496 x^{8} + 258261 x^{7} + 375162 x^{6} - 681318 x^{5} - 871131 x^{4} + 723943 x^{3} + 645327 x^{2} - 209901 x + 5921 \)

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$$\chi_{279}(64,·)$, $\chi_{279}(1,·)$, $\chi_{279}(163,·)$, $\chi_{279}(196,·)$, $\chi_{279}(133,·)$, $\chi_{279}(193,·)$, $\chi_{279}(268,·)$, $\chi_{279}(160,·)$, $\chi_{279}(76,·)$, $\chi_{279}(109,·)$, $\chi_{279}(142,·)$, $\chi_{279}(112,·)$, $\chi_{279}(211,·)$, $\chi_{279}(121,·)$, $\chi_{279}(190,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{75669950457494895861561729218830405} a^{14} + \frac{2567169241314249616043243923273222}{75669950457494895861561729218830405} a^{13} - \frac{893578782406639662722633201529403}{15133990091498979172312345843766081} a^{12} + \frac{19875917299103109906420125885099042}{75669950457494895861561729218830405} a^{11} + \frac{4865281334367610216597521191813192}{15133990091498979172312345843766081} a^{10} + \frac{540209574380249951285333680715012}{15133990091498979172312345843766081} a^{9} - \frac{13040525402175349481658789966201496}{75669950457494895861561729218830405} a^{8} - \frac{33225837578086208182582658516169899}{75669950457494895861561729218830405} a^{7} + \frac{29017963814194924068058354952117002}{75669950457494895861561729218830405} a^{6} - \frac{3137492910729002141259702607955713}{15133990091498979172312345843766081} a^{5} + \frac{27196691529578968318719540025078678}{75669950457494895861561729218830405} a^{4} + \frac{18528776913441595472743156575315933}{75669950457494895861561729218830405} a^{3} - \frac{37364643268814642166241185175122537}{75669950457494895861561729218830405} a^{2} - \frac{28725732408418291761263917213984532}{75669950457494895861561729218830405} a + \frac{187102737260684948620484078762991}{396177751086360711317077116328955}$

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:  \( 5067617902.44 \) (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

3.3.77841.2, 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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{5}$ $15$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ $15$ $15$ $15$ $15$ $15$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{5}$ $15$ $15$ $15$ $15$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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
3Data not computed
31Data not computed